Mathematics Concept Book for IIT-JEE is primarily a revision book for engineering aspirants and with detailed theory students can revise complete syllabus more frequently. The purpose of the…Full description
Mathematics Concept Book for IIT-JEE is primarily a revision book for engineering aspirants and with detailed theory students can revise complete syllabus more frequently. The purpose of the…Descrição completa
Mathematics Concept Book for IIT-JEE is primarily a revision book for engineering aspirants and with detailed theory students can revise complete syllabus more frequently. The purpose of theory boo...
Mathematics Concept Book for IIT-JEE is primarily a revision book for engineering aspirants and with detailed theory students can revise complete syllabus more frequently. The purpose of the…Full description
In view of the new pattern in IIT-JEE ,this book provides ample scope to the engineering aspirants to practice quality questions in Mathematics.This book is completely objective in its natur…Descrição completa
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BRILLIANT PUBLIC SCHOOL, SITAMARHI
(Affiliated up to +2 level to C.B.S.E., New Delhi)
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: MATHEMATICS TOPIC: 1 XI M 1. Trigonometric Ratio and Identity Index: 1. Key Concepts 2. Exercise I to V 3. Answer Key 4. Assertion and Reasons 5. 34 Yrs. Que. from IIT-JEE 6. 10 Yrs. Que. from AIEEE
(c) cosec² θ − cot² θ = 1 ; cosec θ ≥ 1 ∀ θ ∈ R – {nπ , n ∈ Ι} Solved Example # 1 Prove that (i) cos4A – sin4A + 1 = 2 cos2A (ii) Solution (i)
(ii)
tan A + sec A − 1 1+ sin A = tan A − sec A + 1 cos A
cos4A – sin4A + 1 = (cos2A – sin2A) (cos2A + sin2A) + 1 = cos2A – sin2A + 1 [∴ cos2A + sin2A = 1] 2 = 2 cos A tan A + sec A − 1 tan A − sec A + 1
=
tan A + sec A − (sec 2 A − tan 2 A ) tan A − sec A + 1
=
(tan A + sec A )(1 − sec A + tan A ) tan A − sec A + 1
= tan A + sec A =
1+ sin A cos A
Solved Example # 2 If sin x + sin2x = 1, then find the value of cos12x + 3 cos10x + 3 cos8x + cos6x – 1 Solution cos12x + 3 cos10x + 3 cos8x + cos6x – 1 = (cos4x + cos2x)3 – 1 = (sin2x + sinx)3 – 1 [∵ cos2x = sin x] =1–1=0 Solved Example # 3 If tan θ = m –
1 1 , then show that sec θ – tan θ = – 2m or 4m 2m
Solution Depending on quadrant in which θ falls, sec θ can be ±
So, if sec θ =
1 4m2 + 1 =m+ 4m 4m
2
4m 2 + 1 4m
1 1 and if sec θ = – m + 4m 2m
⇒
sec θ – tan θ =
⇒
sec θ – tan θ = – 2m
Self Practice Problem 1.
Prove the followings : (i) cos6A + sin6A + 3 sin2A cos2A = 1 (ii) sec 2A + cosec2A = (tan A + cot A)2 (iii) sec 2A cosec2A = tan2A + cot 2A + 2 (iv) (tan α + cosec β)2 – (cot β – sec α)2 = 2 tan α cot β (cosec α + sec β) 1 1 1 − sin 2 α cos 2 α + cos2α sin2α = 2 2 2 2 sec α − cos α cos ec α − sin α 2 + sin 2 α cos 2 α
(v)
m 2 + 2mn
2.
If sin θ =
2.
C irc ul ar sin θ =
m 2 + 2mn + 2n 2
PM OP
, then prove that tan θ =
Defi nit i o n cos θ =
Of
m 2 + 2mn 2mn + 2n 2
T rig o no met ri c
Func t i o ns:
OM OP
sin θ tan θ = cos θ , cos θ ≠ 0 cos θ cot θ = sin θ , sin θ ≠ 0
sec θ =
3.
1 , cos θ ≠ 0 cos θ
T ri g o no met ri c
cosec θ =
Fu nc t io ns
If θ is any angle, then − θ, 90 (a) sin (− θ) = − sin θ (b) sin (90° − θ) = cos θ (c) sin (90° + θ) = cos θ (d) sin (180° − θ) = sin θ (e) sin (180° + θ) = − sin θ (f) sin (270° − θ) = − cos θ (g) sin (270° + θ) = − cos θ (h) tan (90° − θ) = cot θ Solved Example # 4
1 , sin θ ≠ 0 sin θ
Of
A ll i ed
A ngl es:
± θ, 180 ± θ, 270 ± θ, 360 ± θ etc. are called ALLIED ANGLES. ; cos (− θ) = cos θ ; cos (90°− θ) = sin θ ; cos (90° + θ) = − sin θ ; cos (180° − θ) = − cos θ ; cos (180° + θ) = − cos θ ; cos (270° − θ) = − sin θ ; cos (270° + θ) = sin θ ; cot (90° − θ) = tan θ
Prove that cot A + tan (180º + A) + tan (90º + A) + tan (360º – A) = 0 (i) (ii) sec (270º – A) sec (90º – A) – tan (270º – A) tan (90º + A) + 1 = 0 Solution (i)
cot A + tan (180º + A) + tan (90º + A) + tan (360º – A) = cot A + tan A – cot A – tan A = 0 (ii) sec (270º – A) sec (90º – A) – tan (270º – A) tan (90º + A) + 1 = – cosec 2A + cot 2A + 1 = 0 Self Practice Problem 3. Prove that (i) sin 420º cos 390º + cos (–300º) sin (–330º) = 1 (ii) tan 225º cot 405º + tan 765º cot 675º = 0 3
4.
Graphs of Trigonometric functions: (a) y = sin x
x ∈ R; y ∈ [–1, 1]
(b) y = cos x x ∈ R; y ∈ [ – 1, 1]
(c) y = tan x x ∈ R – (2n + 1) π/2, n ∈ Ι ; y ∈ R
(d) y = cot x
x ∈ R – nπ , n ∈ Ι; y ∈ R
(e) y = cosec x
x ∈ R – nπ , n ∈ Ι ; y ∈ (− − ∞, − 1] ∪ [1, ∞ )
(f) y = sec x
x ∈ R – (2n + 1) π/2, n ∈ Ι ; y ∈ (− ∞, − 1] ∪ [1, ∞) 4
Solved Example # 5
Find number of solutions of the equation cos x = |x| Solution
Clearly graph of cos x & |x| intersect at two points. Hence no. of solutions is 2 Solved Example # 6
Find range of y = sin2x + 2 sin x + 3 ∀ x ∈ R Solution We know – 1 ≤ sin x ≤ 1 ⇒ 0 ≤ sin x +1 ≤ 2 ⇒ 2 ≤ (sin x +1)2 + 2 ≤ 6 Hence range is y ∈ [2, 6] Self Practice Problem 4 xy
4.
Show that the equation sec2θ =
5.
Find range of the followings. (i) y = 2 sin2x + 5 sin x +1∀ x ∈ R
Answer
[–2, 8]
y = cos2x – cos x + 1 ∀ x ∈ R
Answer
3 4 , 3
(ii)
( x + y )2
is only possible when x = y ≠ 0
3 − 1, 2
6.
2π 2π Find range of y = sin x, x ∈ 3
5.
Trigonometric Functions of Sum or Difference of Two Angles:
Answer
(a)
sin (A ± B) = sinA cosB ± cosA sinB
(b) (c) (d)
cos (A ± B) = cosA cosB ∓ sinA sinB sin²A − sin²B = cos²B − cos²A = sin (A+B). sin (A− B) cos²A − sin²B = cos²B − sin²A = cos (A+B). cos (A − B)
(e)
tan A ± tan B tan (A ± B) = 1 ∓ tan A tan B
(f)
cot A cot B ∓ 1 cot (A ± B) = cot B ± cot A
5
(g)
tan A + tan B + tanC−tan A tan B tan C tan (A + B + C) = 1 − tan A tan B − tan B tan C− tan C tan A .
Solved Example # 7
Prove that (i) sin (45º + A) cos (45º – B) + cos (45º + A) sin (45º – B) = cos (A – B) (ii)
Clearly sin (45º + A) cos (45º – B) + cos (45º + A) sin (45º – B) = sin (45º + A + 45º – B) = sin (90º + A – B) = cos (A – B) π 3π + θ tan + θ × tan 4 4
=
1 + tan θ −1 + tan θ × =–1 1 − tan θ 1 + tan θ
Self Practice Problem 3 5 , cos β = , then find sin (α + β) 5 13
33 63 , 65 65
7.
If sin α =
8.
Find the value of sin 105º
9.
Prove that 1 + tan A tan
6.
Fa c t o ris at i o n o f t he Su m o r D i fferenc e o f T w o Si nes o r Cosines:
Answer
–
Answer
2 2
A A = tan A cot – 1 = sec A 2 2
C+D C−D cos 2 2
(a)
sinC + sinD = 2 sin
(c)
cosC + cosD = 2 cos
C+D C−D cos 2 2
sinC − sinD = 2 cos
(d)
cosC − cosD = − 2 sin
Prove that sin 5A + sin 3A = 2sin 4A cos A Solution L.H.S. sin 5A + sin 3A = 2sin 4A cos A
C+D C−D sin 2 2
(b)
Solved Example # 8
[ ∵ sin C + sin D = 2 sin
3 +1
= R.H.S.
C+D C −D cos ] 2 2
Solved Example # 9
Find the value of 2 sin 3θ cos θ – sin 4θ – sin 2θ Solution 2 sin 3θ cos θ – sin 4θ – sin 2θ = 2 sin 3θ cos θ – [2 sin 3θ cos θ ] = 0 Self Practice Problem 6
C+D C−D sin 2 2
10.
7.
Proved that 13 x 3x sin 2 2
(i)
cos 8x – cos 5x = – 2 sin
(iii)
sin A + sin 3 A + sin 5 A + sin 7 A = tan 4A cos A + cos 3 A + cos 5 A + cos 7 A
(iv)
sin A + 2 sin 3 A + sin 5 A sin 3 A = sin 3 A + 2 sin 5 A + sin 7 A sin 5 A
(v)
sin A − sin 5 A + sin 9 A − sin 13 A = cot 4A cos A − cos 5 A − cos 9 A + cos 13 A
(ii)
sin A + sin 2A A = cot cos A − cos 2 A 2
Transformat io n of Prod uc ts into Sum or Dif ference of S ines & Cosines: (a)
2 sinA cosB = sin(A+B) + sin(A−B)
(b)
2 cosA sinB = sin(A+B) − sin(A−B)
(c)
2 cosA cosB = cos(A+B) + cos(A−B)
(d)
2 sinA sinB = cos(A−B) − cos(A+B)
Solved Example # 10
Prove that (i)
sin 8θ cos θ − sin 6θ cos 3θ = tan 2θ cos 2θ cos θ − sin 3θ sin 4θ
(ii)
tan 5θ + tan 3θ = 4 cos 2θ cos 4θ tan 5θ − tan 3θ
Solution
(i)
2 sin 8θ cos θ − 2 sin 6θ cos 3θ 2 cos 2θ cos θ − 2 sin 3θ sin 4θ
=
(ii)
sin 9θ + sin 7θ − sin 9θ − sin 3θ 2 sin 2θ cos 5θ = = tan 2θ cos 3θ + cos θ − cos θ + cos 7θ 2 cos 5θ cos 2θ
tan 5θ + tan 3θ sin 5θ cos 3θ + sin 3θ cos 5θ sin 8θ = = = 4 cos2θ cos 4θ tan 5θ − tan 3θ sin 5θ cos 3θ − sin 3θ cos 5θ sin 2θ
Self Practice Problem θ 7θ 3θ 11θ sin + sin sin = sin 2θ sin 5θ 2 2 2 2
11.
Prove that sin
12.
Prove that cos A sin (B – C) + cos B sin (C – A) + cos C sin (A – B) = 0
1 − cos A + cos B − cos( A + B) A B = tan cot 1 + cos A − cos B − cos( A + B) 2 2
Solution
2 sin A cos A sin 2A = = tan A 1 + cos 2 A 2 cos 2 A
(i)
L.H.S.
(ii)
1 + tan2 A 2 1 + tan 2 A = 2 2 tan A = L.H.S. tan A + cot A = = 2 cosec 2 A sin 2 A tan A
(iii)
L.H.S.
1 − cos A + cos B − cos( A + B) 1 + cos A − cos B − cos( A + B)
A A A + 2 sin sin + B 2 2 2 = A A 2 A 2 cos − 2 cos cos + B 2 2 2 2 sin 2
A +B B A A sin + sin + B 2 sin 2 cos 2 A A 2 2 = tan = tan 2 2 + A B B A A 2 sin 2 sin 2 cos 2 − cos 2 + B
= tan
A B cot 2 2
Self Practice Problem sin θ + sin 2θ = tan θ 1 + cos θ + cos 2θ
14.
Prove that
15.
Prove that sin 20º sin 40º sin 60º sin 80º =
16.
Prove that tan 3A tan 2A tan A = tan 3A – tan 2A – tan A
17.
A Prove that tan 45 º + = sec A + tan A 2
9.
Important Trigonometric Ratios: (a)
sin n π = 0
;
3 16
cos n π = (−1)n ; tan n π = 0, 8
where n ∈ Ι
(b)
sin 15° or sin
3 −1 5π π = = cos 75° or cos 12 12 2 2
cos 15° or cos
tan 15° =
(c)
sin
3 −1 3 +1
3+1 5π π = sin 75° or sin = 12 12 2 2 = 2− 3 = cot 75° ; tan 75° =
3 +1 3 −1
π π 5−1 or sin 18° = & cos 36° or cos = 10 5 4
1 0 . C o nd it i o na l
;
;
= 2+ 3 = cot 15° 5 +1 4
Ident it ies :
If A + B + C = π then : (i)
sin2A + sin2B + sin2C = 4 sinA sinB sinC
(ii)
sinA + sinB + sinC = 4 cos
(iii)
cos 2 A + cos 2 B + cos 2 C = − 1 − 4 cos A cos B cos C
(iv)
cos A + cos B + cos C = 1 + 4 sin
(v)
tanA + tanB + tanC = tanA tanB tanC
(vi)
tan
A B C cos cos 2 2 2
A B C sin sin 2 2 2
A B B C C A tan + tan tan + tan tan =1 2 2 2 2 2 2
(viii)
A B C A B C + cot + cot = cot . cot . cot 2 2 2 2 2 2 cot A cot B + cot B cot C + cot C cot A = 1
(ix)
A+B+C=
(vii)
cot
π 2
then tan A tan B + tan B tan C + tan C tan A = 1
Solved Example # 12
If A + B + C = 180°, Prove that, sin2A + sin2B + sin2C = 2 + 2cosA cosB cosC. Solution. Let S = sin2A + sin2B + sin2C so that 2S = 2sin2A + 1 – cos2B +1 – cos2C = 2 sin2A + 2 – 2cos(B + C) cos(B – C) = 2 – 2 cos2A + 2 – 2cos(B + C) cos(B – C) ∴ S = 2 + cosA [cos(B – C) + cos(B+ C)] since cosA = – cos(B+C) ∴ S = 2 + 2 cos A cos B cos C Solved Example # 13
If x + y + z = xyz, Prove that
2x 1− x2
+
2y 1− y2
+
2z 1− z2
Solution. Put x = tanA, y = tanB and z = tanC, so that we have tanA + tanB + tanC = tanA tanB tanC ⇒ Hence L.H.S.
=
2x 1− x
2
.
2y 1− y
2
.
2z 1 − z2
.
A + B + C = nπ, where n ∈ Ι
9
∴
2y
2x 1− x
+
2
+
1− y 2
2z 1− z2
=
2 tan A 1 − tan 2 A
= tan2A + tan2B + tan2C = tan2A tan2B tan2C =
2y
2x 1− x
.
2
1− y2
.
[
+
2 tan B 1 − tan2 B
+
2 tan C 1 − tan2 C
.
∵ A + B + C = nπ ]
2z 1 − z2
Self Practice Problem 18.
19.
If A + B + C = 180°, prove that (i)
sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin
(ii)
sin 2 A + sin 2B + sin 2C A B C = 8 sin sin sin . sin A + sin B + sin C 2 2 2
B−C C−A A −B sin sin 2 2 2
If A + B + C = 2S, prove that (i) sin(S – A) sin(S – B) + sinS sin (S – C) = sinA sinB. (ii)
sin(S – A) + sin (S – B) + sin(S – C) – sin S = 4sin
1 1 . Range
of
Trigonometric
Expression: Expression :
E = a sin θ + b cos θ E = a 2 + b 2 sin (θ + α), where tan α =
b a
= a 2 + b 2 cos (θ − β), where tan β =
a b
Hence for any real value of θ, − a 2 + b 2 ≤ E ≤
a2 + b2
Solved Example # 14
Find maximum and minimum values of following : (i) 3sinx + 4cosx (ii) 1 + 2sinx + 3cos2x Solution. (i) We know – (ii)
3 2 + 4 2 ≤ 3sinx + 4cosx ≤
32 + 42
– 5 ≤ 3sinx + 4cosx ≤ 5 1+ 2sinx + 3cos2x = – 3sin2x + 2sinx + 4 2 sin x 2 +4 = – 3 sin x − 3 2
1 13 = – 3 sin x − + 3 3 2
Now
1 16 0 ≤ sin x − ≤ 3 9
⇒
–
A B C sin sin . 2 2 2
2
1 16 ≤ – 3 sin x − ≤ 0 3 3 10
2
1 13 13 – 1 ≤ – 3 sin x − + ≤ 3 3 3
Self Practice Problem 20. Find maximum and minimum values of following
(i) (ii)
3 + (sinx – 2) 2 10cos2x – 6sinx cosx + 2sin2x
Answer Answer
max = 12, min = 4. max = 11, min = 1.
(iii)
π cosθ + 3 2 sin θ + + 6 4
Answer
max = 11, min = 1
1 2 . Sine a nd Cosine
Series: Series :
(
)
sin α + sin (α + β) + sin (α + 2β ) +...... + sin α + n− 1β =
nβ
sin 2 n −1 β α + β sin 2 sin 2 nβ
sin 2 n −1 β α + cos α + cos (α + β) + cos (α + 2β ) +...... + cos α + n − 1β = β cos 2 sin 2
(
)
Solved Example # 15
Find the summation of the following (i)
cos
2π 4π 6π + cos + cos 7 7 7
(ii)
cos
π 2π 3π 4π 5π 6π + cos + cos + cos + cos + cos 7 7 7 7 7 7
(iii)
cos
π 3π 5π 7π 9π + cos + cos + cos + cos 11 11 11 11 11
Solution.
(i)
2π 6 π + 3π 7 7 cos sin 2π 4π 6π 2 7 + cos + cos = cos π 7 7 7 sin 7
cos
=
4π 3π sin 7 7 π sin 7
3π 3π sin 7 7 π sin 7
− cos
=
6π 7 =– 1 =– π 2 2 sin 7 sin
(ii)
cos
π 2π 3π 4π 5π 6π + cos + cos + cos + cos + cos 7 7 7 7 7 7
11
π 6π + 6π cos 7 7 sin 14 2 π 6π cos sin 2 14 = = =0 π π sin sin 14 14
(iii)
cos
π 3π 5π 7π 9π + cos + cos + cos + cos 11 11 11 11 11
cos
=
10 π 5π sin 22 11 π sin 11
10π 11 = 1 π 2 2 sin 11
sin
=
Self Practice Problem
Find sum of the following series :
21.
cos
π 3π 5π + cos + cos + ...... + to n terms. 2n + 1 2n + 1 2n + 1
Answer
1 2
22.
sin2α + sin3α + sin4α + ..... + sin nα, where (n + 2)α = 2π
Answer
0.
12
SHORT REVISION Trigonometric Ratios & Identities 1.
2.
3.
4.
BASIC TRIGONOMETRIC IDENTITIES : (a)sin2θ + cos2θ = 1 ; −1 ≤ sin θ ≤ 1 ; −1 ≤ cos θ ≤ 1 ∀ θ ∈ R 2 (b)sec θ − tan2θ = 1 ; sec θ ≥ 1 ∀ θ ∈ R 2 2 (c)cosec θ − cot θ = 1 ; cosec θ ≥ 1 ∀ θ ∈ R IMPORTANT T′ RATIOS: cos n π = (-1)n ; tan n π = 0 where n ∈ I (a)sin n π = 0 ; ( 2 n + 1 ) π ( 2n + 1)π = (−1)n &cos = 0 where n ∈ I (b)sin 2 2 5π π 3−1 (c)sin 15° or sin = = cos 75° or cos ; 12 12 2 2 3+1 π 5π cos 15° or cos = = sin 75° or sin ; 12 12 2 2 3 +1 3 −1 tan 15° = = 2 − 3 = cot 75° ; tan 75° = = 2 + 3 = cot 15° 3 +1 3 −1 3π π π π 2+ 2 2− 2 ; tan = 2−1 ; tan = 2+1 (d)sin = ; cos = 8 8 8 8 2 2 π π 5+1 5−1 (e) sin or sin 18° = & cos 36° or cos = 10 5 4 4 TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES : If θ is any angle, then − θ, 90 ± θ, 180 ± θ, 270 ± θ, 360 ± θ etc. are called ALLIED ANGLES. (a) sin (− θ) = − sin θ ; cos (− θ) = cos θ (b) sin (90°- θ) = cos θ ; cos (90° − θ) = sin θ (c) sin (90°+ θ) = cos θ ; cos (90°+ θ) = − sin θ (d)sin (180°− θ) = sin θ; cos (180°− θ) = − cos θ (e) sin (180°+ θ) = − sin θ ; cos (180°+ θ) = − cos θ (f) sin (270°− θ) = − cos θ ; cos (270°− θ) = − sin θ (g) sin (270°+ θ) = − cos θ ; cos (270°+ θ) = sin θ
TRIGONOMETRIC FUNCTIONS OF SUM OR DIFFERENCE OF TWO ANGLES : (a) sin (A ± B) = sinA cosB ± cosA sinB (b) cos (A ± B) = cosA cosB ∓ sinA sinB (c) sin²A − sin²B = cos²B − cos²A = sin (A+B) . sin (A− B) (d) cos²A − sin²B = cos²B − sin²A = cos (A+B) . cos (A − B) (e) tan (A ± B) = tan A ± tan B (f) cot (A ± B) = cot B ± cot A 1 ∓ tan A tan B FACTORISATION OF THE SUM OR DIFFERENCE OF TWO SINES OR COSINES : C−D C+ D C− D C+ D cos (b) sinC − sinD = 2 cos sin (a) sinC + sinD = 2 sin 2 2 2 2 C+ D C− D C+ D C−D (c) cosC + cosD = 2 cos cos (d) cosC − cosD = − 2 sin sin 2 2 2 2 TRANSFORMATION OF PRODUCTS INTO SUM OR DIFFERENCE OF SINES & COSINES : (a) 2 sinA cosB = sin(A+B) + sin(A−B) (b) 2 cosA sinB = sin(A+B) − sin(A−B) (c) 2 cosA cosB = cos(A+B) + cos(A−B) (d) 2 sinA sinB = cos(A−B) − cos(A+B) MULTIPLE ANGLES AND HALF ANGLES : cot A cot B ∓ 1
1 − cos 2A 2 cos2A = 1 + cos 2A , 2sin2A = 1 − cos 2A ; tan2A = 1 + cos 2A θ θ 2 2 2 cos = 1 + cos θ , 2 sin = 1 − cos θ. 2 2 2tan(θ 2) 2tanA (c) tan 2A = ; tan θ = 2 1−tan 2 (θ 2) 1−tan A 2tanA 1−tan 2 A (d) sin 2A = , cos 2A = (e) sin 3A = 3 sinA − 4 sin3A 2 1+ tan 2 A 1+ tan A 3tanA− tan 3 A (f) cos 3A = 4 cos3A − 3 cosA (g) tan 3A = 1−3tan 2 A THREE ANGLES : tan A + tan B+ tanC− tan A tan BtanC (a) tan (A+B+C) = 1− tan A tan B− tan BtanC− tanCtan A NOTE IF : (i) A+B+C = π then tanA + tanB + tanC = tanA tanB tanC π (ii) A+B+C = then tanA tanB + tanB tanC + tanC tanA = 1 2 (b) If A + B + C = π then : (i) sin2A + sin2B + sin2C = 4 sinA sinB sinC C A B (ii) sinA + sinB + sinC = 4 cos cos cos 2 2 2 MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS: (a) Min. value of a2tan2θ + b2cot2θ = 2ab where θ ∈ R (b) Max. and Min. value of acosθ + bsinθ are a 2 + b 2 and – a 2 + b 2 (c) If f(θ) = acos(α + θ) + bcos(β + θ) where a, b, α and β are known quantities then – a 2 + b 2 + 2ab cos(α − β) < f(θ) < a 2 + b 2 + 2ab cos(α − β) π (d) If α,β ∈ 0, and α + β = σ (constant) then the maximum values of the expression 2 cosα cosβ, cosα + cosβ, sinα + sinβ and sinα sinβ occurs when α = β = σ/2. (e) If α,β ∈ 0, π and α + β = σ(constant) then the minimum values of the expression 2 secα + secβ, tanα + tanβ, cosecα + cosecβ occurs when α = β = σ/2. If A, B, C are the angles of a triangle then maximum value of (f) sinA + sinB + sinC and sinA sinB sinC occurs when A = B = C = 600 (g) In case a quadratic in sinθ or cosθ is given then the maximum or minimum values can be interpreted by making a perfect square. Sum of sines or cosines of n angles, nβ sin 2 n−1 β sin α + sin (α + β) + sin (α + 2β ) + ...... + sin α + n − 1 β = β sin α + 2 sin 2 nβ sin 2 n−1 α + n − 1 β cos α + cos (α + β) + cos (α + 2β ) + ...... + cos = cos α + β β 2 sin
(
)
(
)
EXERCISE–I
2
that cos²α + cos² (α + β) − 2cos α cos β cos (α + β) = sin²β that cos 2α = 2 sin²β + 4cos (α + β) sin α sin β + cos 2(α + β) that , tan α + 2 tan 2α + 4 tan 4α + 8 cot 8 α = cot α . that : (a) tan 20° . tan 40° . tan 60° . tan 80° = 3 3π 5π 7π 3 4 π + sin 4 + sin 4 + sin 4 = (b) tan 9° − tan 27° − tan 63° + tan 81° = 4 . (c) sin 16 16 16 16 2 Q.5 Calculate without using trigonometric tables : 2 cos 40° − cos20° (b) 4 cos 20° − 3 cot 20° (c) (a) cosec 10° − 3 sec 10° sin 20° 3π 5π 7π π sec5° cos40° + −2sin35° (d) 2 2 sin10° (e) cos6 + cos6 + cos6 + cos6 16 16 16 16 sin5° 2 (f) tan 10° − tan 50° + tan 70° 7π 7π π 3π π 3π Q.6(a) If X = sin θ + + sin θ − + sin θ + , Y = cos θ + + cos θ − + cos θ + 14 12 12 12 12 12 12
Q.1 Q.2 Q.3 Q.4
Prove Prove Prove Prove
X Y = 2 tan2θ. then prove that − Y X (b) Prove that sin²12° + sin² 21° + sin² 39° + sin² 48° = 1+ sin² 9° + sin² 18° . Q.7
Q.8 Q.9 Q.10 Q.11 Q.12
Show that :
1° 2
(a)
cot 7
(b)
tan 142
or tan 82 1° 2
1° = 2
(
3+ 2
)(
)
2 +1 or
2 + 3+ 4 + 6
=2+ 2 − 3 − 6 .
m+ n . 2( m −n ) sin y 3 + sin 2 x π y π x If tan + = tan3 + , prove that = . sin x 1 + 3 sin 2 x 4 2 4 2 π 4 5 If cos (α + β) = ; sin (α - β) = & α , β lie between 0 & , then find the value of tan 2α. 5 13 4 tanβ 1+ tanα 1−tanα tanβ sinβ n = ( m > n ) then . Prove that if the angles α & β satisfy the relation = sin(2α +β ) m m+ n m −n (a) If y = 10 cos²x − 6 sin x cos x + 2 sin²x , then find the greatest & least value of y . (b) If y = 1 + 2 sin x + 3 cos2 x , find the maximum & minimum values of y ∀ x ∈ R . (c) If y = 9 sec2x + 16 cosec2x, find the minimum value of y ∀ x ∈ R.
If m tan (θ - 30°) = n tan (θ + 120°), show that cos 2 θ =
π (d) Prove that 3 cos θ + + 5 cos θ + 3 lies from - 4 & 10 .
(
3
)
(
)
(
)
(e) Prove that 2 3 + 4 sin θ + 4 cos θ lies between − 2 2+ 5 & 2 2+ 5 . tan A = ∑ (tan A) − 2 ∑ (cot A). tan B.tanC If α + β = c where α, β > 0 each lying between 0 and π/2 and c is a constant, find the maximum or minimum value of (a) sin α + sin β (b) sin α sin β (c) tan α + tan β (d) cosec α + cosec β Let A1 , A2 , ...... , An be the vertices of an n-sided regular polygon such that ; 1 1 1 . Find the value of n. = + A1 A 2 A1 A 3 A1 A 4 Prove that : cosec θ + cosec 2 θ + cosec 22 θ + ...... + cosec 2 n − 1 θ = cot (θ/2) − cot 2 n - 1θ For all values of α , β , γ prove that; β+ γ α +β γ +α cos α + cos β + cos γ + cos (α + β + γ) = 4 cos .cos . cos . 2 2 2 1 + sin A cos B 2 sin A − 2 sin B + = . Show that cos A 1 − sin B sin(A − B) + cos A − cos B
Q.13 If A + B + C = π, prove that Q.14
Q.15
Q.16 Q.17
Q.18
Q.19 If tan β =
tan α + tan γ 1 + tan α . tan γ
∑
, prove that sin 2β =
sin 2 α + sin 2 γ 1 + sin 2 α . sin 2 γ
.
Q.20 If α + β = γ , prove that cos² α + cos² β + cos² γ = 1 + 2 cos α cos β cos γ . (1 − tan α2 ) 1 − tan β2 1 − tan 2γ sin α + sin β + sin γ − 1 π Q.21 If α + β + γ = , show that = . 2 (1 + tan α2 ) 1 + tan β2 1 + tan 2γ cos α + cos β + cos γ Q.22 If A + B + C = π and cot θ = cot A + cot B + cot C, show that , sin (A − θ) . sin (B − θ) . sin (C − θ) = sin3 θ . 3π 5π 17 π π + cos + ......... + cos Q.23 If P = cos + cos and 19 19 19 19 2π 4π 6π 20π + cos + cos + ......... + cos Q = cos , then find P – Q. 21 21 21 21 Q.24 If A, B, C denote the angles of a triangle ABC then prove that the triangle is right angled if and only if sin4A + sin4B + sin4C = 0. Q.25 Given that (1 + tan 1°)(1 + tan 2°)......(1 + tan 45°) = 2n, find n.
( (
)( )(
) )
EXERCISE–II
Q.1 Q.2
If tan α = p/q where α = 6β, α being an acute angle, prove that; 1 (p cosec 2 β − q sec 2 β) = p 2 +q 2 . 2 Let A1 , A2 , A3 ............ An are the vertices of a regular n sided polygon inscribed in a circle of radius R. If (A1 A2)2 + (A1 A3)2 + ......... + (A1 An)2 = 14 R2 ,15find the number of sides in the polygon.
Q.3 Q.4 Q.5
cos 3θ + cos 3φ = (cosθ + cosφ) cos(θ + φ) – (sinθ + sinφ) sin(θ + φ) 2 cos(θ − φ) − 1 Without using the surd value for sin 180 or cos 360 , prove that 4 sin 360 cos 180 = 5 sin x sin3x sin9x 1 + + = (tan27x − tanx) Show that , cos3x cos9x cos27x 2
Prove that:
5
Q.6
Let x1 =
rπ
∏ cos 11 r =1
5
and x2 =
rπ
∑ cos 11 , then show that r =1
Q.25
1 π x1 · x2 = cos ec − 1 , where Π denotes the continued product. 64 22 2π If θ = , prove that tan θ . tan 2 θ + tan 2 θ . tan 4 θ + tan 4 θ . tan θ = − 7. 7 cosx π prove that , > 8. For 0 < x < 2 sin x(cosx −sinx ) 4 3π 2π π 2π 7 7 prove that, sin α + sin 2α + sin 4α = (b) sin . sin . sin = (a) If α = 7 7 7 7 2 8 88 1 cos k Let k = 1°, then prove that ∑ = sin 2 k n =0 cos nk · cos(n + 1)k 3 Prove that the value of cos A + cos B + cos C lies between 1 & where A + B + C = π. 2 If cosA = tanB, cosB = tanC and cosC = tanA , then prove that sinA = sinB = sinC = 2 sin18°. 3 + cos x Show that ∀ x ∈ R can not have any value between − 2 2 and 2 2 . What inference sin x sin x ? can you draw about the values of 3 + cos x 5 If (1 + sin t)(1 + cos t) = . Find the value of (1 – sin t)(1 – cos t). 4 sin 8 α cos8 α 1 sin 4 α cos4 α 1 + 3 = + = Prove that from the equality follows the relation ; 3 a b a b a +b (a +b )3 . Prove that the triangle ABC is equilateral iff , cot A + cot B + cot C = 3 . Prove that the average of the numbers n sin n°, n = 2, 4, 6, ......., 180, is cot 1°. Prove that : 4 sin 27° = 5+ 5 1 / 2 − 3− 5 1 / 2 . A C B If A+B+C = π; prove that tan2 + tan2 + tan2 ≥ 1. 2 2 2 A B C 1 If A+B+C = π (A , B , C > 0) , prove that sin . sin . sin ≤ . 2 2 2 8 Show that elliminating x & y from the equations , sin x + sin y = a ; 8ab cos x + cos y = b & tan x + tan y = c gives 2 2 2 = c. a +b −4a 2 Determine the smallest positive value of x (in degrees) for which tan(x + 100°) = tan(x + 50°) tan x tan (x – 50°). x tan n n 2 Evaluate : ∑ x n − 1 n =1 2 cos n −1 2 β+ γ−α γ +α −β α+β−γ If α + β + γ = π & tan · tan · tan = 1, then prove that; 4 4 4 1 + cos α + cos β + cos γ = 0. ∀ x ∈ R, find the range of the function, f (x) = cos x (sin x + sin 2 x + sin 2 α ) ; α ∈ [0, π]
Q.1
sec2θ =
Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 Q.13
Q.14 Q.15 Q.16 Q.17 Q.18 Q.19 Q.20 Q.21
(
) (
)
(
Q.22
Q.23
Q.24
Q.2
)
EXERCISE–III
4xy is true if and only if : ( x + y) 2 (A) x + y ≠ 0 (B) x = y , x ≠ 0 (a)
Let n be an odd integer. If sin nθ = (A) b0 = 1, b1 = 3
[JEE ’96, 1] n
(C) x = y
(D) x ≠ 0 , y ≠ 0
∑ br sinr θ, for every value of θ, then :
r=0
16
(B) b0 = 0, b1 = n
(b)
(C) b0 = − 1, b1 = n (D) b0 = 0, b1 = n2 − 3n + 3 Let A0 A1 A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius . Then the product of the lengths of the line segments A0 A1, A0 A2 & A0 A4 is : 3
Q.3
3 3
(B) 3 3 (C) 3 (D) (A) 4 2 (c) Which of the following number(s) is/are rational ? [ JEE '98, 2 + 2 + 2 = 6 out of 200 ] (A) sin 15º (B) cos 15º (C) sin 15º cos 15º (D) sin 15º cos 75º θ For a positive integer n, let fn (θ) = tan (1+ sec θ) (1+ sec 2θ) (1+ sec 4θ) .... (1 + sec2nθ) Then 2 π
π
π
π
=1 (A) f2 = 1 (B) f3 = 1 (C) f4 = 1 (D) f 5 16 32 64 128 Q.4(a) Let f (θ) = sin θ (sin θ + sin 3 θ) . Then f (θ) : [ JEE 2000 Screening. 1 out of 35 ] (A) ≥ 0 only when θ ≥ 0 (B) ≤ 0 for all real θ (C) ≥ 0 for all real θ (D) ≤ 0 only when θ ≤ 0 .
(b) In any triangle ABC, prove that, cot
[JEE '99,3]
A B A B C C + cot + cot = cot cot cot . [JEE 2000] 2 2 2 2 2 2
Q.5(a) Find the maximum and minimum values of 27cos 2x · 81sin 2x. π (b) Find the smallest positive values of x & y satisfying, x − y = , cot x + cot y = 2. [REE 2000, 3] 4 π and β + γ = α then tanα equals [ JEE 2001 (Screening), 1 out of 35 ] Q.6 If α + β = 2 (A) 2(tanβ + tanγ) (B) tanβ + tanγ (C) tanβ + 2tanγ (D) 2tanβ + tanγ 1 1 Q.7 If θ and φ are acute angles satisfying sinθ = , cos φ = , then θ + φ ∈ [JEE 2004 (Screening)] 2 3 π π π 2π 2π 5π 5π (A) , (D) , π (B) , (C) , 2 3 6 3 2 3 6 Q.8 In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is (A) 4 + 2 3 (B) 6 + 4 3 (C) 12 +
Q.9
7 3 4
(D) 3 +
7 3 4
[JEE 2005 (Screening)]
π Let θ ∈ 0, and t1 = (tanθ)tanθ, t2 = (tanθ)cotθ, t3 = (cotθ)tanθ , t4 = (cotθ)cotθ, then 4 (A) t1 > t2 > t3 > t4 (B) t4 > t3 > t1 > t2 (C) t3 > t1 > t2 > t4 (D) t2 > t3 > t1 > t4 [JEE 2006, 3]
ANSWER SHEET (EXERCISE–I)
Q 5. (a) 4
(b) −1
(c) 3
(d) 4
(e) 13
5 4
(f) 3
Q 10.
56 33
Q 12. (a) ymax = 11 ; ymin = 1 (b) ymax = ; ymin = − 1, (c) 49 3 2 Q14. (a) max = 2 sin (c/2), (b) max. = sin (c/2), (c) min = 2 tan (c/2), (d) min = 2 cosec (c/2) Q 15. n = 7 Q23. 1 Q.25 n = 23
EXERCISE –II
1 1 − 2 2 , 2 2 2 1 − Q 23. Q.25 – sin 2 x 2 n −1 sin x 2 n −1
Q.2
n = 7 Q.13
Q.14
13 − 10 4
Q.22
x = 30°
1 + sin 2 α ≤ y ≤ 1 + sin 2 α
EXERCISE–III
Q.1 Q.5 Q.8
Q.2 (a) B, (b) C, (c) C
B
Q.3 A, B, C, D
π 5π (a) max. = 35 & min. = 3–5 ; (b) x = ;y= 12 6
B
Q.9
Q.6 C
B
17
Q.7 B
Q.4 (a) C
EXERCISE–IV (Objective) Part : (A) Only one correct option
(
(32π + x ) −sin3 (72π − x ) when simplified reduces to: cos(x − 2π ).tan (32π + x ) )
tan x − 2π .cos 1 .
(C) − sin x cos x (D) sin2x 4 3π 6 π 4 6 The expression 3 sin 2 − α + sin (3π + α) – 2 sin 2 + α + sin (5π + α ) is equal to (A) 0 (B) 1 (C) 3 (D) sin 4α + sin 6α If tan A & tan B are the roots of the quadratic equation x 2 − ax + b = 0, then the value of sin2 (A + B). (A) sin x cos x
2. 3.
5. 6. 7. 8. 9. 10.
11.
13. 14. 15.
16.
17.
18.
a2
a2
The greatest and least value of log (A) 2 & 1 (B) 5 & 3
2 3 3
3π < α < π, then 4 (A) 1 + cot α
2
(sin x − cos x + 3 2 ) are respectively: (C) 7 & 5
(D) 9 & 7
(B)
4 3 3
(C)
(D) none
3
1 is equal to sin2 α (B) – 1 – cot α (C) 1 – cot α (D) – 1 + cot α π x 3π If x ∈ π, then 4 cos2 − + 4 sin 4 x + sin 2 2 x is always equal to 2 4 2 (A) 1 (B) 2 (C) – 2 (D) none of these If 2 cos x + sin x = 1, then value of 7 cos x + 6 sin x is equal to (A) 2 or 6 (B) 1 or 3 (C) 2 or 3 (D) none of these 11 If cosec A + cot A = , then tan A is 2 15 117 21 44 (A) (B) (C) (D) 16 43 22 117 1 If cot α + tan α = m and – cos α = n, then cos α 2 1/3 2 1/3 (A) m (mn ) – n(nm ) = 1 (B) m(m 2n)1/3 – n(nm 2)1/3 = 1 (C) n(mn2)1/3 – m(nm 2)1/3 = 1 (D) n(m 2n)1/3 m(mn2)1/3 = 1 cos 6 x + 6 cos 4 x + 15 cos 2x + 10 The expression is equal to cos 5 x + 5 cos 3 x + 10 cos x (B) 2 cos x (C) cos2 x (D) 1 + cos x (A) cos 2x sin A cos A 3 5 If = and = , 0 < A, B < π/2, then tan A + tan B is equal to sin B cos B 2 2
If
(A)
3/ 5
2 cot α +
(B)
If sin 2θ = k, then the value of (A)
1− k2 k
(B)
tan θ 1 + tan θ
2 − k2 k
(D) ( 5 + 3 ) / 5
(C) 1
5/ 3 3
19.
a2
In a right angled triangle the hypotenuse is 2 2 times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are π π π 3π π 3π π π (A) & (B) & (C) & (D) & 3 6 8 8 5 10 4 4 1 1 cos290 ° + 3 sin250 ° = (A)
12.
a2
(C) (D) 2 (B) 2 2 a 2 +(1−b)2 a +b (b+c )2 b (1−a)2 2 2 The value of log2 [cos (α + β) + cos (α − β) − cos 2α. cos 2β] : (A) depends on α & β both (B) depends on α but not on β (C) depends on β but not on α (D) independent of both α & β. cos20°+8sin70°sin50 °sin10° is equal to: sin 2 80° (A) 1 (B) 2 (C) 3/4 (D) none If cos A = 3/4, then the value of 16cos2 (A/2) – 32 sin (A/2) sin (5A/2) is (A) – 4 (B) – 3 (C) 3 (D) 4 If y = cos2 (45º + x) + (sin x − cos x)2 then the maximum & minimum values of y are: (A) 2 & 0 (B) 3 & 0 (C) 3 & 1 (D) none π 3π 5π 17π The value of cos + cos + cos +...... + cos is equal to: 19 19 19 19 (A) 1/2 (B) 0 (C) 1 (D) none (A)
4.
(B) − sin2 x
2
cot θ 3
+
1 + cot 2 θ
is equal to
(C) k182 + 1
(D) 2 – k 2
Part : (B) May have more than one options correct 20. Which of the following is correct ? (A) sin 1° > sin 1 (B) sin 1° < sin 1 (C) cos 1° > cos 1 (D) cos 1° < cos 1 21. If 3 sin β = sin (2α + β), then tan (α + β) – 2 tan α is (A) independent of α (B) independent of β (C) dependent of both α and β (D) independent of α but dependent of β 22.
4 It is known that sin β = & 0 < β < π then the value of 5
(A) independent of α for all β in (0, π)
(B)
5 3
3 sin(α + β ) −
2 cos (α + β) cos 6π
sinα
is:
for tan β > 0
3 (7 + 24 cot α ) for tan β < 0 (D) none 15 If the sides of a right angled triangle are {cos2α + cos2β + 2cos(α + β)} and {sin2α + sin2β + 2sin(α + β)}, then the length of the hypotenuse is: α−β α +β (A) 2[1+cos(α − β)] (B) 2[1 − cos(α + β)] (C) 4 cos2 (D) 4sin2 2 2 If x = sec φ − tan φ & y = cosec φ + cot φ then: y+1 1+ x y −1 (A) x = y − 1 (B) y = 1 − x (C) x = y + 1 (D) xy + x − y + 1 = 0 (a + 2) sin α + (2a – 1) cos α = (2a + 1) if tan α = 2a 2a 4 3 (A) (B) (C) 2 (D) 2 3 4 a +1 a −1 2b If tan x = , (a ≠ c) a−c y = a cos2x + 2b sin x cos x + c sin2x z = a sin2x – 2b sin x cos x + c cos2x, then (A) y = z (B) y + z = a + c (C) y – z = a – c (D) y – z = (a – c)2 + 4b2
(C)
23.
24.
25.
26.
n
27.
28.
n
cos A + cos B sin A + sinB + sin A − sinB cos A − cos B A −B A −B (B) 2 cotn : n is even (C) 0 : n is odd (A) 2 tann (D) none 2 2 6 6 2 The equation sin x + cos x = a has real solution if 1 1 1 1 (D) a ∈ , 1 (C) a ∈ − (A) a ∈ (–1, 1) (B) a ∈ − 1, − 2 2 2 2
EXERCISE–IV (Subjective) 1. 2. 3.
4. 5. 6.
7. 8. 9. 10. 11.
The minute hand of a watch is 1.5 cm long. How far does its tip move in 50 minutes? (Use π = 3.14). If the arcs of the same length in two circles subtend angles 75° and 120°at the centre, find the ratio of their radii. Sketch the following graphs : x (i) y = 3 sin 2x (ii) y = 2 tan x (iii) y = sin 2 3π 3π − θ + cot (2π + θ) = 1. + θ cos (2π + θ) cot Prove that cos 2 2 θ 9θ 5θ – cos 3 θ cos = sin 5 θ sin . 2 2 2 3 3π x x If tan x = , π < x < , find the value of sin and cos . 4 2 2 2 2 α −π 1 − cot 4 9α + cos α cot 4α prove that = cosec 4α. sec 2 π α − 2 1 + cot 2 4 Prove that, sin 3 x. sin3 x + cos 3 x. cos3 x = cos3 2 x. p 1 If tan α = where α = 6 β, α being an acute angle, prove that; (p cosec 2 β − q sec 2 β) = p 2 + q 2 . q 2 tan α + tan γ sin 2α + sin 2γ If tan β = 1 + tan α. tan γ , prove that sin 2β = 1 + sin 2α. sin 2γ .
Prove that cos 2 θ cos
(i)
(ii)
1° = 2 + 2− 3− 6 . 2
tan 142
cot 7
1° 1° or tan 82 = 2 2
Show that:
( 3 + 2 )( 2 +1) 19
(iii)
or
2+ 3+ 4+ 6
(
4 sin 27° = 5 + 5
) − (3 − 5 ) 1/ 2
1/ 2
12. 13.
14. 15.
16. 17.
18. 19.
20. 21. 22.
Prove that, tan α + 2 tan 2α + 4 tan 4α + 8 cot 8 α = cot α. −3 If cos (β − γ) + cos (γ − α) + cos (α − β) = , prove that 2 cos α + cos β + cos γ = 0, sin α + sin β + sin γ = 0. 1 sin8 α cos 8 α sin4 α cos 4 α 1 + = + = follows the relation a3 b3 a b a+b (a + b)3 Prove that: cosec θ + cosec 2 θ + cosec 22 θ +... + cosec 2 n − 1θ = cot (θ/2) − cot 2n − 1 θ. Hence or 4π 8π 16π 32π otherwise prove that cosec + cosec + cosec + cosec =0 15 15 15 15 1 1 1 Let A1, A2,......, An be the vertices of an n−sided regular polygon such that; A A = A A + A A . 1 2 1 3 1 4 Find the value of n. If A + B + C = π, then prove that 1 A B C A B C ≥1 (ii) sin . sin . sin ≤ . (i) tan² + tan² + tan² 8 2 2 2 2 2 2 3 (iii) cos A + cos B + cos C ≤ 2 ax sin θ by cos θ ax by + = a2 – b2, – = 0. Show that (ax)2/3 + (by)2/3 = (a2 – b2) 2/3 If 2 cos θ sin θ cos θ sin2 θ n n n n If Pn = cos θ + sin θ and Q n = cos θ – sin θ, then show that Pn – Pn – 2 = – sin2θ cos2θ Pn – 4 Q n – Q n – 2 = – sin2θ cos2θ Q n – 4 and hence show that P4 = 1 – 2 sin2θ cos2θ , Q 4 = cos2θ – sin2θ If sin (θ + α) = a & sin (θ + β) = b (0 < α, β, θ < π/2) then find the value of cos2 (α − β) − 4 ab cos(α − β) If A + B + C = π, prove that tan B tan C + tan C tan A + tan A tan B = 1 + sec A. sec B. sec C. If tan2α + 2tanα. tan2β = tan2β + 2tanβ. tan2α, then prove that each side is equal to 1 or tan α = ± tan β.
Prove that from the equality
EXERCISE–IV
EXERCISE–V
1. D
2. B
3. A
4. D
5. B
6. C
7. B
8. A
9. B
10. B
11. B
12. B
13. B
14. A
15. C
16. A
17. B
18. D
19. B
20. BC
21. AB 22. BC 23. AC 24. BCD
1. 7.85 cm
6. sin
x = 2
25. BD 26. BC 16. n = 7
27. BC 28. BD
20
2. r 1 : r2 = 8 : 5 3 10
and cos
x =– 2
20. 1 − 2a2 − 2b2
1 10
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: MATHEMATICS TOPIC: 2 XI M 2. Trigonometric Equations Index: 1. Key Concepts 2. Exercise I to III 3. Answer Key 4. Assertion and Reasons 5. 34 Yrs. Que. from IIT-JEE 6. 10 Yrs. Que. from AIEEE
1
1. 2.
Trigonometric Equation : An equation involving one or more trigonometric ratios of an unknown angle is called a trigonometric equation.
Solution of Trigonometric Equation :
A solution of trigonometric equation is the value of the unknown angle that satisfies the equation. π 3π 9π 11π 1 e.g. if sinθ = , , , ........... ⇒ θ= , 4 4 4 4 2 Thus, the trigonometric equation may have infinite number of solutions (because of their periodic nature) and can be classified as : (i) Principal solution (ii) General solution. Principal solutions: 2 .1 The solutions of a trigonom etric equation which lie in the interv al [0, 2π) are called Principal solutions. 1 e.g Find the Principal solutions of the equation sinx = . 2 Solution.
1 2 ∵ there exists two values π 5π 1 i.e. and which lie in [0, 2π) and whose sine is 6 6 2 π 1 ∴ Principal solutions of the equation sinx = are , 6 2 General Solution :
∵
2 .2
sinx =
5π Ans. 6
The expression involving an integer 'n' which gives all solutions of a trigonometric equation is called General solution. General solution of some standard trigonometric equations are given below.
3.
General Solution of Some Standard Trigonometric Equations : (i)
If sin θ = sin α
⇒ θ = n π + (−1)n α
(ii)
If cos θ = cos α
⇒ θ = 2nπ ± α
(iii)
If tan θ = tan α
⇒ θ = nπ + α
(iv)
If sin² θ = sin² α
⇒ θ = n π ± α, n ∈ Ι.
(v)
If cos² θ = cos² α
⇒ θ = n π ± α, n ∈ Ι.
(vi) If tan² θ = tan² α Some Important deductions : ⇒ (i) sinθ = 0 (ii)
Trigonometric equations which can be solved by transforming a sum or difference of trigonometric ratios into their product. Solved Example # 8 Solve cos3x + sin2x – sin4x = 0 Solution. ⇒ ⇒ ⇒ ⇒
∴
cos3x + sin2x – sin4x = 0 cos3x – 2cos3x.sinx = 0 cos3x = 0 π 3x = (2n + 1) , n ∈ Ι 2 π x = (2n + 1) , n ∈ Ι 6 solution of given equation is π ,n∈Ι or (2n + 1) 6
Type - 4 Trigonometric equations which can be solved by transforming a product of trigonometric ratios into their sum or difference. Solved Example # 9 Solve Solution. ∵ ⇒ ⇒ ⇒ ⇒ ⇒
∴
Type - 5
sin5x.cos3x = sin6x.cos2x sin5x.cos3x = sin6x.cos2x ⇒ sin8x + sin2x = sin8x + sin4x ⇒ 2sin2x.cos2x – sin2x = 0 ⇒ sin2x = 0 or 2cos2x – 1 = 0 1 2x = nπ, n ∈ Ι or cos2x = 2 π nπ x= , n ∈ Ι or 2x = 2nπ ± , n ∈ Ι 3 2 π ⇒ x = nπ ± , n ∈ Ι 6 Solution of given equation is π nπ ,n∈Ι or nπ π± ,n∈Ι 6 2
Trigonometric Equations of the form a sinx + b cosx = c, where a, b, c ∈ R, can be solved by dividing both sides of the equation by
a2 + b2 .
5
Solve sinx + cosx =
2
Solution. 2
∵ Here
sinx + cosx = a = 1, b = 1.
∴
divide both sides of equation (i) by 1 1 sinx . + cosx. =1 2 2 π π sinx.sin + cosx.cos = 1 4 4 π cos x − = 1 4
⇒ ⇒ ⇒ ⇒
∴
..........(i) 2 , we get
π = 2nπ, n ∈ Ι 4 π ,n∈Ι x = 2nπ + 4 Solution of given equation is π 2nπ π+ ,n∈Ι Ans. 4
x–
Note : Trigonometric equation of the form a sinx + b cosx = c can also be solved by changing sinx and cosx into their corresponding tangent of half the angle. Solved Example # 11 Solve 3cosx + 4sinx = 5 Solution. ∵
∵
∴ ⇒
Let ∴
⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒
3cosx + 4sinx = 5 .........(i) x 2 x 2 tan 1 − tan 2 2 cosx = & sinx = x x 1 + tan 2 1 + tan 2 2 2 equation (i) becomes x 2 x 2 tan 1 − tan 2 2 =5 +4 3 ........(ii) 2 x 2 x 1 + tan 1 + tan 2 2 x tan =t 2 equation (ii) becomes 1− t2 2t + 4 =5 3 2 2 1+ t 1+ t 4t2 – 4t + 1 = 0 (2t – 1)2 = 0 1 x t= ∵ t = tan 2 2 x 1 tan = 2 2 x 1 tan = tanα, where tanα = 2 2 x = nπ + α 2 1 α where α = tan –1 , n ∈ Ι x = 2nπ π + 2α 2
Self Practice Problems : 1.
Solve
3 cosx + sinx = 2 6
Ans.
Page : 6 of 15 TRIG. EQUATIONS
Solved Example # 10
x =0 2 π 2nπ + , n ∈ Ι 6
Solve
sinx + tan
Ans.
(1)
(2)
x = 2nπ, n ∈ Ι
Type - 6
Trigonometric equations of the form P(sinx ± cosx, sinx cosx) = 0, where p(y, z) is a polynomial, can be solved by using the substitution sinx ± cosx = t. Solved Example # 12 Solve sinx + cosx = 1 + sinx.cosx Solution. ∵ Let ⇒
divide both sides of equation (ii) by 2 , we get 1 1 1 ⇒ sinx. + cosx. = 2 2 2 ⇒ ⇒
(i)
(ii)
π π cos x − = cos 4 4 π π x– = 2nπ ± 4 4 if we take positive sign, we get π ,n∈Ι Ans. x = 2nπ π+ 2 if we take negative sign, we get x = 2nπ Ans. π, n ∈ Ι
Self Practice Problems: 1.
Solve
sin2x + 5sinx + 1 + 5cosx = 0
2.
Solve
3cosx + 3sinx + sin3x – cos3x = 0
3.
Solve
(1 – sin2x) (cosx – sinx) = 1 – 2sin2x. π π (1) nπ – , n ∈ Ι (2) nπ – , n ∈ Ι 4 4 π (3) 2nπ + , n ∈ Ι or 2nπ, n ∈ Ι or 2
Ans.
Type - 7
nπ +
π ,n∈Ι 4
Trigonometric equations which can be solved by the use of boundness of the trigonometric ratios sinx and cosx. Solved Example # 13 x x Solve sinx cos − 2 sin x + 1 + sin − 2 cos x cos x = 0 4 4 Solution. x x .......(i) ∵ sinx cos − 2 sin x + 1 + sin − 2 cos x cos x = 0 4 4 ⇒ ⇒ ⇒
x x – 2sin2x + cosx + sin .cosx – 2cos2x = 0 4 4 x x sin x. cos + sin . cos x – 2 (sin2x + cos2x) + cosx = 0 4 4
sinx.cos
sin
5x + cosx = 2 4
7
........(ii)
Page : 7 of 15 TRIG. EQUATIONS
2.
and
cosx = 1
and
x = 2m π, m ∈ Ι
and
x = 2m π, m ∈ Ι
........(iv)
– 4, p ∈ Ι
∴
general solution of given equation can be obtained by substituting either m = 4p – 3 in equation (iv) or n = 5p – 4 in equation (iii)
∴
general solution of equation (i) is (8p – 6)π π, p ∈ Ι Ans.
Self Practice Problems : 1.
Solve
sin3x + cos2x = – 2
2.
Solve
3 sin 5 x − cos 2 x − 3 = 1 – sinx π (1) (4p – 3) , p ∈ Ι (2) 2
Ans.
2m π +
π , m ∈Ι 2
SHORT REVISION TRIGONOMETRIC EQUATIONS & INEQUATIONS THINGS TO REMEMBER : π π
2.
If sin θ = sin α ⇒ θ = n π + (−1)n α where α ∈ − , , n ∈ I . 2 2 If cos θ = cos α ⇒ θ = 2 n π ± α where α ∈ [0 , π] , n ∈ I .
3.
If tan θ = tan α ⇒ θ = n π + α where α ∈ − π , π , n ∈ I .
4.
If sin² θ = sin² α ⇒ θ = n π ± α.
5.
cos² θ = cos² α ⇒ θ = n π ± α.
6. 7.
tan² θ = tan² α ⇒ θ = n π ± α. [ Note : α is called the principal angle ] TYPES OF TRIGONOMETRIC EQUATIONS :
1.
2
2
(a)
Solutions of equations by factorising . Consider the equation ; (2 sin x − cos x) (1 + cos x) = sin² x ; cotx – cosx = 1 – cotx cosx
(b)
Solutions of equations reducible to quadratic equations. Consider the equation :
(c)
3 cos² x − 10 cos x + 3 = 0 and 2 sin2x + 3 sinx + 1 = 0 Solving equations by introducing an Auxilliary argument . Consider the equation :
(d)
sin x + cos x = 2 ; 3 cos x + sin x = 2 ; secx – 1 = ( 2 – 1) tanx Solving equations by Transforming a sum of Trigonometric functions into a product. Consider the example : cos 3 x + sin 2 x − sin 4 x = 0 ; sin2x + sin22x + sin23x + sin24x = 2 ; sinx + sin5x = sin2x + sin4x
(e)
Solving equations by transforming a product of trigonometric functions into a sum. 8
Page : 8 of 15 TRIG. EQUATIONS
Now equation (ii) will be true if 5x =1 sin 4 π 5x ⇒ = 2nπ + , n ∈ Ι 2 4 (8n + 2)π ⇒ x = ,n∈Ι ........(iii) 5 Now to find general solution of equation (i) (8n + 2)π = 2m π 5 ⇒ 8n + 2 = 10m 5m − 1 ⇒ n= 4 if m=1 then n=1 if m=5 then n=6 ......... ......... ......... ......... ......... ......... if m = 4p – 3, p ∈ Ι then n = 5p
sin 5 x . cos 3 x = sin 6 x .cos 2 x ; 8 cosx cos2x cos4x = (f)
sin 6 x ; sin3θ = 4sinθ sin2θ sin4θ sin x
Solving equations by a change of variable : (i) Equations of the form of a . sin x + b . cos x + d = 0 , where a , b & d are real numbers & a , b ≠ 0 can be solved by changing sin x & cos x into their corresponding tangent of half the angle. Consider the equation 3 cos x + 4 sin x = 5. (ii)
Many equations can be solved by introducing a new variable . eg. the equation sin4 2 x + cos4 2 x = sin 2 x . cos 2 x changes to
1 2
2 (y + 1) y − = 0 by substituting , sin 2 x . cos 2 x = y.. Solving equations with the use of the Boundness of the functions sin x & cos x or by making two perfect squares. Consider the equations : x x sin x cos − 2 sin x + 1+ sin − 2cos x . cos x = 0 ; 4 4 4 11 sin2x + 2tan2x + tanx – sinx + =0 3 12 TRIGONOMETRIC INEQUALITIES : There is no general rule to solve a Trigonometric inequations and the same rules of algebra are valid except the domain and range of trigonometric functions should be kept in mind. (g)
8.
x 1 Consider the examples : log 2 sin < – 1 ; sin x cos x + < 0 ; 5 − 2 sin 2 x ≥ 6 sin x − 1 2 2
EXERCISE–I 1 52
1 1 + log15 cos x + log 5 (sin x ) 2 = 15 2 +5
Q.1
Solve the equation for x,
Q.2
Find all the values of θ satisfying the equation; sin θ + sin 5 θ = sin 3 θ such that 0 ≤ θ ≤ π.
Q.3
Find all value of θ, between 0 & π, which satisfy the equation; cos θ . cos 2 θ . cos 3 θ = 1/4.
Q.4
Solve for x , the equation
Q.5
Determine the smallest positive value of x which satisfy the equation,
Q.6
π 2 sin 3 x + = 4
Q.7
Find the general solution of the trigonometric equation 3
Q.8
Find all values of θ between 0° & 180° satisfying the equation; cos 6 θ + cos 4 θ + cos 2 θ + 1 = 0 .
Q.9
Find the solution set of the equation, log −x 2 −6x (sin 3x + sin x) = log −x 2 −6x (sin 2x).
13 − 18 tanx = 6 tan x – 3, where – 2π < x < 2π.
1 + 8 sin 2 x . cos2 2 x
1 + sin 2 x − 2 cos 3 x = 0 .
1 + log 3 (cos x + sin x ) 2
−2
log 2 (cos x − sin x )
= 2.
10
10
Q.10 Find the value of θ, which satisfy 3 − 2 cosθ − 4 sinθ − cos 2θ + sin 2θ = 0. Q.11
Find the general solution of the equation, sin πx + cos πx = 0. Also find the sum of all solutions in [0, 100].
Q.12 Find the least positive angle measured in degrees satisfying the equation sin3x + sin32x + sin33x = (sinx + sin2x + sin3x)3. 9
Page : 9 of 15 TRIG. EQUATIONS
Consider the equation :
Q.14 Prove that the equations (a) have no solution. Q.15 (a) (b) (c)
sin x · sin 2x · sin 3x = 1
(b)
sin x · cos 4x · sin 5x = – 1/2
Let f (x) = sin6x + cos6x + k(sin4x + cos4x) for some real number k. Determine all real numbers k for which f (x) is constant for all values of x. all real numbers k for which there exists a real number 'c' such that f (c) = 0. If k = – 0.7, determine all solutions to the equation f (x) = 0.
Q.16 If α and β are the roots of the equation, a cos θ + b sin θ = c then match the entries of column-I with the entries of column-II. Column-I Column-II 2b (P) (A) sin α + sin β a+c c−a (B) sin α . sin β (Q) c+ a 2bc α β (C) tan + tan (R) 2 2 2 a +b 2 (D)
tan
α 2
. tan
β 2
=
(S)
c 2 −a 2 a 2 +b 2
Q.17 Find all the solutions of, 4 cos2x sin x − 2 sin2x = 3 sin x. Q.18 Solve for x, (− π ≤ x ≤ π) the equation; 2 (cos x + cos 2 x) + sin 2 x (1 + 2 cos x) = 2 sin x. Q.19 Solve the inequality sin2x >
2 sin2x + (2 –
2 )cos2x.
Q.20 Find the set of values of 'a' for which the equation, sin4 x + cos4 x + sin 2x + a = 0 possesses solutions. Also find the general solution for these values of 'a'. Q.21 Solve: tan22x + cot22x + 2 tan 2x + 2 cot 2x = 6. Q.22 Solve: tan2x . tan23x . tan 4x = tan2x − tan23x + tan 4x. Q.23 Find the set of values of x satisfying the equality 2 cos 7 x π 3π > 2cos 2 x . sin x − – cos x + = 1 and the inequality cos 3 + sin 3 4 4
Q.24 Let S be the set of all those solutions of the equation, (1 + k)cos x cos (2x − α) = (1 + k cos 2x) cos(x − α) which are independent of k & α. Let H be the set of all such solutions which are dependent on k & α. Find the condition on k & α such that H is a non-empty set, state S. If a subset of H is (0, π) in which k = 0, then find all the permissible values of α. Q.25 Solve for x & y,
x cos 3 y + 3x cos y sin 2 y = 14 x sin 3 y + 3x cos 2 y sin y = 13
Q.26 Find the value of α for which the three elements set S = {sin α, sin 2α, sin 3α} is equal to the three element set T = {cos α, cos 2α, cos 3α}. Q.27 Find all values of 'a' for which every root of the equation, a cos 2x + a cos 4x + cos 6x = 1 10
is also a root of the equation, sin x cos 2 x = sin 2x cos 3x −
1 sin 5x , and conversely, every root 2
Page : 10 of 15 TRIG. EQUATIONS
Q.13 Find the general values of θ for which the quadratic function cos θ + sin θ is the square of a linear function. (sinθ) x2 + (2cosθ)x + 2
Q.28 Solve the equations for 'x' given in column-I and match with the entries of column-II. Column-I Column-II (A)
cos 3x . cos3 x + sin 3x . sin3 x = 0
(P)
nπ ±
π 3
(B)
sin 3α = 4 sin α sin(x + α) sin(x − α)
(Q)
nπ +
π , n∈I 4
(R)
nπ π , n∈I + 4 8
(S)
nπ 2
where α is a constant ≠ nπ. (C)
| 2 tan x – 1 | + | 2 cot x – 1 | = 2.
(D)
sin10x + cos10x =
29 cos42x. 16
±
π 4
EXERCISE–II Q.1 Q.2
Solve the following system of equations for x and y [REE ’2001(mains), 3] (cos ec 2 x − 3 sec 2 y) ( 2 cos ecx + 3 |sec y |) 5 = 1 and 2 = 64. The number of integral values of k for which the equation 7cosx + 5sinx = 2k + 1 has a solution is (A) 4 (B) 8 (C) 10 (D) 12 [JEE 2002 (Screening), 3]
Q.3
cos(α – β) = 1 and cos(α + β) = 1/e, where α, β ∈ [– π, π], numbers of pairs of α, β which satisfy both the equations is (A) 0 (B) 1 (C) 2 (D) 4 [JEE 2005 (Screening)]
Q.4
If 0 < θ < 2π, then the intervals of values of θ for which 2sin2θ – 5sinθ + 2 > 0, is
Q.5
π 5π π 5π π π 5π 41π , π [JEE 2006, 3] (A) 0, ∪ , 2π (B) , (C) 0, ∪ , (D) 8 6 6 6 8 6 6 48 The number of solutions of the pair of equations 2 sin2θ – cos2θ = 0 2 cos2θ – 3 sin θ = 0 in the interval [0, 2π] is (A) zero (B) one (C) two (D) four [JEE 2007, 3]
5π 1 π Q.10 θ = 2 n π or 2 n π + ; n ∈ I Q.11 x = n – , n ∈ I; sum = 5025Q.12 2 3 4 π or (2n+1)π – tan–12 , n ∈ I 4
Q.16 (A) R; (B) S; (C) P; (D) Q Q.17 Q.18
±π −π , ,± π 3 2
Q.19
nπ +
Q.15 (a) –
3 ; (b) k ∈ 2
nπ ; n π + (−1)n
π π < x < nπ + 8 4 11
π 10
1 nπ π − 1, − 2 ; (c) x = 2 ± 6
or n π + (−1)n
3π − 10
72°
Page : 11 of 15 TRIG. EQUATIONS
of the second equation is also a root of the first equation.
[
Q.21 x = Q.22
(
1 n π + (− 1) n sin −1 1 − 2 a + 3 2 nπ 4
+ (−1)n
π 8
or
nπ 4
)]
3 1
where n ∈ I and a ∈ − , 2 2
π 24
+ (−1)n+1
(2 n + 1) π , k π , where n , k ∈ I 4
3π , n ∈I 4 (i) k sin α ≤ 1 (ii) S = n π , n ∈ I (iii) α ∈ (− m π , 2 π − m π) m ∈ I
Q.23 x = 2nπ + Q.24
Q.25 x = ± 5 5 & y = n π + tan−1
1 2
Q.26
Q.27 a = 0 or a < − 1
Q.28
nπ π + 2 8 (A) S; (B) P; (C) Q; (D) R
EXERCISE–II Q.1 Q.2
π π x = nπ + (–1)n and y = mπ + where m & n are integers. 6 6 B Q.3 D Q.4 A Q.5 C
Part : (A) Only one correct option 1.
The solution set of the equation 4sinθ .cosθ – 2cosθ – 2 3 sinθ + 3 = 0 in the interval (0, 2π) is 3 π 7π (A) , 4 4
2.
2π , n∈Ι 3
(C) nπ or m π ±
5.
π where n, m ∈ Ι 3
If 20 sin2 θ + 21 cos θ − 24 = 0 &
(A) 3 4.
π 5π 3π (C) , π, , 4 3 3
π 5 π 11π , (D) , 6 6 6
All solutions of the equation, 2 sinθ + tanθ = 0 are obtained by taking all integral values of m and n in: (A) 2nπ +
3.
π 5π (B) , 3 3
(B)
(B) nπ or 2m π ±
2π where n, m ∈ Ι 3
(D) nπ or 2m π ±
π where n, m ∈ Ι 3
7π θ < θ < 2π then the values of cot is: 4 2
15 3
(C) −
15 3
The general solution of sinx + sin5x = sin2x + sin4x is: (A) 2 nπ ; n ∈ Ι (B) nπ ; n ∈ Ι (C) nπ/3 ; n ∈ Ι A triangle ABC is such that sin(2A + B) =
(D) − 3 (D) 2 nπ/3 ; n ∈ Ι
1 . If A, B, C are in A.P. then the angle A, B, C are 2
respectively. (A) 6.
7.
5π π π , , 12 4 3
(B)
π π 5π , , 4 3 12
The maximum value of 3sinx + 4cosx is (A) 3 (B) 4
(C)
π π 5π , , 3 4 12
(C) 5
If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of the equation (A) x 2 − 6x + 1 = 0 (B) 6x 2 − x − 1 = 0 (C) 6x 2 + x + 1 = 0 12
(D)
π 5π π , , 3 12 4
(D) 7 (D) x 2 − x + 6 = 0
Page : 12 of 15 TRIG. EQUATIONS
Q.20
sin 3 θ − cos 3 θ cos θ − − 2 tan θ cot θ = − 1 if: sin θ − cos θ 1 + cot 2 θ
(A) θ ∈ 0 ,
π 2
π , π 2
(C) θ ∈ π ,
(B) θ ∈
3π 2
3π , 2π 2
(D) θ ∈
9.
The number of integral values of a for which the equation cos 2x + a sin x = 2a − 7 possesses a solution is (A) 2 (B) 3 (C) 4 (D) 5
10.
The principal solution set of the equation, 2 cos x = 2 + 2 sin 2 x is
π 13 π (A) , 8 8 11.
π 13 π (B) , 4 8
π 13 π (C) , 4 10
π 13 π (D) 8 , 10
The number of all possible triplets (a1, a2, a3) such that : a1 + a2 cos 2x + a3 sin2x = 0 for all x is (A) 0 (B) 1 (C) 2 (D) infinite
12.
nπ , n ∈ N, then greatest value of n is If 2tan2x – 5 secx – 1 = 0 has 7 different roots in 0, 2 (A) 8 (B) 10 (C) 13 (D) 15
13.
The solution of |cosx| = cosx – 2sinx is (A) x = nπ, n ∈ Ι (C) x = nπ + (–1) n
(B) x = nπ + π , n ∈Ι 4
π ,n∈Ι 4
(D) (2n + 1)π +
π ,n∈Ι 4
14.
The arithmetic mean of the roots of the equation 4cos3x – 4cos2x – cos(π + x) – 1 = 0 in the interval [0, 315] is equal to (A) 49π (B) 50π (C) 51π (D) 100π
15.
Number of solutions of the equation cos 6x + tan2 x + cos 6x . tan2 x = 1 in the interval [0, 2π] is : (B) 5 (C) 6 (D) 7 (A) 4
Part : (B) May have more than one options correct 16.
sinx − cos2x − 1 assumes the least value for the set of values of x given by: (A) x = nπ + (−1) n+1 (π/6) , n ∈ Ι (B) x = nπ + (−1)n (π/6) , n ∈ Ι n (C) x = nπ + (−1) (π/3), n ∈ Ι (D) x = nπ − (−1) n ( π/6) , n ∈ Ι
17.
cos4x cos8x − cos5x cos9x = 0 if (A) cos12x = cos 14 x (C) sinx = 0
18.
The equation 2sin (A) sin2x = 1
19.
20.
(B) sin13 x = 0 (D) cosx = 0
x x . cos2x + sin2x = 2 sin . sin2x + cos2x has a root for which 2 2 1 1 (B) sin2x = – 1 (C) cosx = (D) cos2x = – 2 2
sin2x + 2 sin x cos x − 3cos2x = 0 if (A) tan x = 3 (C) x = nπ + π/4, n ∈ Ι
(B) tanx = − 1 (D) x = nπ + tan−1 (−3), n ∈ Ι
sin2x − cos 2x = 2 − sin 2x if (A) x = nπ/2, n ∈ Ι (C) x = (2n + 1) π/2, n ∈ Ι
(B) x = nπ − π/2, n ∈ Ι (D) x = nπ + ( −1)n sin−1 (2/3), n ∈ Ι
13
Page : 13 of 15 TRIG. EQUATIONS
8.
Page : 14 of 15 TRIG. EQUATIONS
1.
Solve
cot θ = tan8θ
2.
Solve
x x cot – cosec = cotx 2 2
3.
Solve
1 cotθ + 1 = 0. cot 2θ + 3 + 3
4.
Solve
cos2θ + 3 cosθ = 0.
5.
Solve the equation: sin 6x = sin 4x − sin 2x .
6.
Solve: cos θ + sin θ = cos 2 θ + sin 2 θ .
7.
Solve
4 sin x . sin 2x . sin 4x = sin 3x .
8.
Solve
sin2nθ – sin2(n – 1)θ = sin2θ, where n is constant and n ≠ 0, 1
9.
Solve
tanθ + tan2θ +
10.
Solve: sin3 x cos 3 x + cos3 x sin 3 x + 0.375 = 0
11.
Solve the equation,
12.
Solve the equation: sin 5x = 16 sin5 x .
13.
If tan θ + sin φ =
14.
Solve for x , the equation
15.
Find the general solution of sec 4 θ − sec 2 θ = 2 .
16.
Solve the equation
17.
Solve for x: 2 sin 3 x +
18.
Solve the equation for 0 ≤ θ ≤ 2 π; sin 2θ + 3 cos2θ
19.
Solve: tan2 x . tan2 3 x . tan 4 x = tan2 x − tan2 3 x + tan 4 x .
20.
Find the values of x, between 0 & 2 π, satisfying the equation; cos 3x + cos 2x = sin
3 tanθ tan2θ =
3.
sin 3 x − cos 3 x cos x 2 2 = . 3 2 + sin x
7 3 & tan² θ + cos² φ = then find the general value of θ & φ . 4 2
13 − 18 tan x = 6 tan x − 3, where − 2 π < x < 2 π .
3 sin x − cos x = cos² x . 2
π = 4
1 + 8 sin 2 x . cos 2 2 x .
(
)
2
14
π − 5 = cos − 2θ . 6
3x x + sin . 2 2
Solve: cos
22.
Solve the equation, sin2 4 x + cos2 x = 2 sin 4 x cos4 x .
EXERCISE # 1
1 π n + , n∈Ι 3 3
9. 1. D
2. B
3. D
4. C
5. B
6. C
7. B
8. B
9. D
10. A
11. D
12. D
13. D
14. C
15. D
16. AD 17. ABC
10. x =
nπ π + ( − 1)n + 1 , n∈Ι 4 24
18. ABCD 19. CD 11. x = (4 n + 1)
20. BC
π ,n∈Ι 2
EXERCISE # 2 12. x = n π ; x = n π ± 1.
1 π n + , n ∈ Ι 2 9
13. θ = n π + 2. x = 4nπ ±
3. θ = nπ –
2π ,n∈Ι 3
π ,n∈Ι 3
or nπ –
π ,n∈Ι 6
17 − 3 , n ∈Ι 4. 2nπ ± α where α = cos–1 4 nπ π , n ∈ Ι or n π ± , n ∈ Ι 4 6
2nπ π 6. 2 n π, n ∈ Ι or + , n∈Ι 3 6
17. (24 + 1)
18. θ =
π , n∈ Ι 3
π π , ∈ Ι or x = (24k – 7) , k∈Ι 12 12
7 π 19 π , 12 12
π 5π 9 π 13 π , ,π, , 7 7 7 7
21. φ
22. x = (2 n + 1)
15
or 2 n π ±
(2 n + 1) π , k π, where n, k ∈ Ι 4
nπ π ± , n ∈Ι 3 9
1 π mπ , m ∈ Ι or m + ,m ∈Ι 2 n n −1
2 3
2nπ π π ± or 2nπ ± , n ∈ Ι 5 10 2
20.
8. m π, m ∈ Ι or
π π , φ = n π + (−1)n , n ∈ I 4 6
16. x = (2 n + 1)π, , n ∈ Ι
19. 7. x = n π, n ∈ Ι or
π ,n∈Ι 6
14. α − 2 π; α − π, α, α + π, where tan α =
15.
5.
Page : 15 of 15 TRIG. EQUATIONS
2x cos 6 x = − 1 . 3
21.
π , n∈I 2
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: MATHEMATICS TOPIC: 3 XI M 3. Properties of Triangle Index: 1. Key Concepts 2. Exercise I to V 3. Answer Key 4. Assertion and Reasons 5. 34 Yrs. Que. from IIT-JEE 6. 10 Yrs. Que. from AIEEE
1
Properties & Solution of Triangle 1. Sine Rule: In any triangle ABC, the sines of the angles are proportional to the opposite sides i.e. a b c = = . sin A sin B sin C A −B cos 2 . C sin 2 A −B cos a+b 2 = . We have to prove C c sin 2 From sine rule, we know that a b c = = = k (let) sin A sin B sin C a = k sinA, b = k sinB and c = k sinC a+b L.H.S. = c
Example :
a+b In any ∆ABC, prove that = c
Solution.
∵ ∵ ⇒ ∵
k (sin A + sin B ) = k sin C
C A −B cos 2 2 = C C sin cos 2 2 = R.H.S. Hence L.H.S. = R.H.S. Proved In any ∆ABC, prove that (b2 – c2) cot A + (c 2 – a2) cot B + (a2 – b2) cot C = 0 ∵ We have to prove that (b2 – c 2) cot A + (c2 – a2) cot B + (a2 – b2) cot C = 0 ∵ from sine rule, we know that a = k sinA, b = k sinB and c = k sinC ∴ (b2 – c 2) cot A = k 2 (sin2B – sin2C) cot A ∵ sin2B – sin2C = sin (B + C) sin (B – C) ∴ (b2 – c2) cot A = k2 sin (B + C) sin (B – C) cotA cos A ∴ (b2 – c 2) cot A = k2 sin A sin (B – C) sin A = – k2 sin (B – C) cos (B + C) cos
Example : Solution.
k2 [2sin (B – C) cos (B + C)] 2 k2 ⇒ (b2 – c2) cot A = – [sin 2B – sin 2C] 2 k2 Similarly (c2 – a2) cot B = – [sin 2C – sin 2A] 2 k2 and (a2 – b2) cot C = – [sin 2A – sin 2B] 2 adding equations (i), (ii) and (iii), we get (b2 – c 2) cot A + (c2 – a2) cot B + (a2 – b2) cot C = 0 Self Practice Problems In any ∆ABC, prove that A A 1. a sin + B = (b + c) sin . 2 2
A +B A −B sin cos 2 2 = C C sin cos 2 2 A −B cos 2 = C sin 2
∵
B+C=π–A
∵
cosA = – cos(B + C)
=–
2.
a 2 sin(B − C) b 2 sin(C − A ) c 2 sin( A − B) + + =0 sin B + sin C sin C + sin A sin A + sin B
or a² = b² + c² − 2bc cos A = b2 + c2 + 2bc cos (B + C)
c2 + a 2 − b 2 a 2 + b 2 − c2 (iii) cos C = 2 ca 2a b In a triangle ABC if a = 13, b = 8 and c = 7, then find sin A.
(ii) cos B = Example : Solution.
*Example : Solution.
64 + 49 − 169 b2 + c 2 − a2 = 2 .8 .7 2bc 2π 1 ⇒ cosA = – ⇒ A= 3 2 2π 3 = Ans. ∴ sinA = sin 3 2 In a ∆ ABC, prove that a(b cos C – c cos B) = b2 – c 2 ∵ We have to prove a (b cosC – c cosB) = b2 – c 2. ∵ from cosine rule we know that
∵
cosA =
a 2 + b2 − c 2 & 2ab a 2 + b 2 − c 2 L.H.S. = a b 2ab
cosC ∴
=
a2 + c 2 − b 2 2ac 2 2 2 − c a + c − b 2ac
cos B =
a2 + b2 − c 2 (a 2 + c 2 − b 2 ) – 2 2 = (b2 – c2) Hence L.H.S. = R.H.S. Proved =
Example : Solution.
= R.H.S.
a b c a If in a ∆ABC, ∠A = 60° then find the value of 1 + + 1 + − . c c b b ∵ ∠A = 60° a b c a c +a+b b+c −a 1 + + 1 + − = ∵ c c b b c b
a b c a 1 + + 1 + − = 3 Ans. c c b b Self Practice Problems :
∴
a 2 + ab + b 2 , then prove that the greatest angle is 120°. A a(cosB + cosC) = 2(b + c) sin2 . 2
1.
The sides of a triangle ABC are a, b,
2.
In a triangle ABC prove that
3.
Projection Formula:
(i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA Example : In a triangle ABC prove that a(b cosC – c cosB) = b2 – c2 Solution. ∵ L.H.S. = a (b cosC – c cosB) = b (a cosC) – c (a cosB) ............(i) ∵ From projection rule, we know that b = a cosC + c cosA ⇒ a cosC = b – c cosA ⇒ a cosB = c – b cosA & c = a cosB + b cosA Put values of a cosC and a cosB in equation (i), we get L.H.S. = b (b – ccos A) – c(c – b cos A) = b2 – bc cos A – c2 + bc cos A = b2 – c 2 = R.H.S. Hence L.H.S. = R.H.S. Proved Note: We have also proved a (b cosC – ccosB) = b2 – c 2 by using cosine – rule in solved *Example. Example : In a ∆ABC prove that (b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c. 3
Solution.
∵
L.H.S. = (b + c) cos A (c + a) cos B + (a + B) cos C
= = = = Hence L.H.S. =
b cos A + c cos A + c cos B + a cos B + a cos C + b cos C (b cos A + a cos B) + (c cos A + a cos C) + (c cos B + b cos C) a+b+c R.H.S. R.H.S. Proved
Self Practice Problems
1.
In a ∆ABC, prove that B 2 C + c cos 2 = a + b + c. 2 b cos 2 2
2.
cos B c − b cos A = . cos C b − c cos A
3.
cos A cos B cos C a2 + b2 + c 2 + + = . c cos B + b cos C a cos C + c cos A a cos B + b cos A 2abc
4. Napier’s Analogy - tangent rule: B−C = 2 A−B (iii) tan = 2 Example : Find the
(i) tan
Solution.
c −a B A C−A b−c cot (ii) tan = cot c +a 2 2 2 b+c a −b C cot a +b 2 unknown elements of the ∆ABC in which a = 3 + 1, b = 3 – 1, C = 60°.
∵
a = 3 + 1, b = 3 – 1, C = 60° A + B + C = 180°
∴
A + B = 120°
∵
From law of tangent, we know that
∵
.......(i)
a−b C A −B tan = cot a + b 2 2
= =
( 3 + 1) − ( 3 − 1)
cot 30°
( 3 + 1) + ( 3 − 1) 2 2 3
cot 30°
A −B =1 tan 2 π A −B ∴ = 45° = 4 2 ⇒ A – B = 90° From equation (i) and (ii), we get A = 105° and B = 15° Now,
⇒
∵
From sine-rule, we know that
∴
c=
.......(ii)
a b c = = sin A sin B sin C
a sin C ( 3 + 1) sin 60° = sin A sin105°
3 2 3 +1
( 3 + 1)
=
∵
sin105° =
3 +1 2 2
2 2
⇒
c=
∴ c= Self Practice Problem 1.
6 6 , A = 105°, B = 15°
In a ∆ABC if b = 3, c = 5 and cos (B – C) = Ans.
Ans. 7 A , then find the value of tan . 25 2
1 3 4
2.
A B C B −C C−A A −B If in a ∆ABC, we define x = tan tan , y = tan tan and z = tan tan 2 2 2 2 2 2 then show that x + y + z = – xyz.
5. Trigonometric Functions of Half Angles: (i)
sin
(s − c) (s − a ) (s − b) (s − c) A B C ; sin = ; sin = = ca b c 2 2 2
(ii)
cos
s (s − b) s (s − a ) A B C = ; cos = ; cos = ca bc 2 2 2
(iii)
tan
A = 2
(iv)
sin A =
(s − a ) (s − b) ab
s (s − c) ab
(s − b) (s − c) ∆ a+b+c = where s = is semi perimetre of triangle. s (s − a ) s (s − a ) 2
2 bc
s(s − a )(s − b)(s − c) =
2∆ bc
6. Area of Triangle (∆) ∆=
1 1 1 ab sin C = bc sin A = ca sin B = s (s − a ) (s − b) (s − c) 2 2 2
Example :
In a ∆ABC if a, b, c are in A.P. then find the value of tan
Solution.
∵
∴ ∴ ∵ ⇒ ∵
∴ ∴ ⇒
tan
∆ A = s(s − a) 2
and tan
A C . tan . 2 2
∆ C = s(s − c ) 2
∆2 A C . tan = 2 s ( s − a)(s − c ) 2 2 s −b b A C tan . tan = =1– s s 2 2 it is given that a, b, c are in A.P. 2b = a + c a+b+c 3b s= = 2 2 b 2 = put in equation (i) s 3 2 A C tan . tan =1– 3 2 2 1 A C . tan = Ans. tan 3 2 2 tan
∵
∆ 2 = s (s – a) (s – b) (s – c)
........(i)
Example :
In a ∆ABC if b sinC(b cosC + c cosB) = 42, then find the area of the ∆ABC.
Solution.
∵ ∵
∵
∴ Example : Solution.
b sinC (b cosC + c cosB) = 42 From projection rule, we know that a = b cosC + c cosB put in (i), we get ab sinC = 42 1 ∆= ab sinC 2 ∆ = 21 sq. unit Ans.
........(i) given ........(ii)
C A B In any ∆ABC prove that (a + b + c) tan + tan = 2c cot . 2 2 2 B A ∵ L.H.S. = (a + b + c) tan + tan 2 2 A 2
=
(s − b)(s − c ) s(s − a)
and tan
B = 2
(s − a)(s − c ) s(s − b)
∵
tan
∴
(s − b)(s − c ) (s − a)(s − c ) + L.H.S. = (a + b + c) s(s − a) s(s − b)
= 2s
s−c s
s−b s−a + s − a s − b 5
=2
=2 = 2c
s−b+s−a s(s − c ) (s − a)(s − b )
∵
2s= a + b + c
∴
2s – b – a = c
∵
cot
c s(s − c ) (s − a )(s − b )
s(s − c ) (s − a)(s − b)
= 2c cot = R.H.S. Hence L.H.S. = R.H.S.
C = 2
s(s − c ) (s − a)(s − b)
C 2
Proved
7. m - n Rule:
(m + n) cot θ = m cot α − n cot β = n cot B − m cot C Example :
If the median AD of a triangle ABC is perpendicular to AB, prove that tan A + 2tan B = 0.
Solution.
From the figure, we see that θ = 90° + B (as θ is external angle of ∆ABD)
Now if we apply m-n rule in ∆ABC, we get (1 + 1) cot (90 + B) = 1. cot 90° – 1.cot (A – 90°) ⇒ – 2 tan B = cot (90° – A) ⇒ – 2 tan B = tan A ⇒ tan A + 2 tan B = 0 Hence proved. Example :
The base of a triangle is divided into three equal parts. If t 1, t2, t3 be the tangents of the angles subtended by these parts at the opposite vertex, prove that 1 1 1 1 1 4 1 + 2 = + + . t2 t1 t 2 t 2 t 3
Solution.
Let point D and E divides the base BC into three equal parts i.e. BD = DE = DC = d (Let) and let α, β and γ be the angles subtended by BD, DE and EC respectively at their opposite vertex. ⇒ t 1 = tanα, t2 = tanβ and t3 = tanγ Now in ∆ABC ∵ BE : EC = 2d : d = 2 : 1 ∴ from m-n rule, we get (2 + 1) cotθ = 2 cot (α + β) – cotγ ⇒ 3cotθ = 2 cot (α + β) – cotγ .........(i) again ∵ in ∆ADC ∵ DE : EC = x : x = 1 : 1 ∴ if we apply m-n rule in ∆ADC, we get (1 + 1) cotθ = 1. cotβ – 1 cotγ 2cotθ = cotβ – cotγ .........(ii) from (i) and (ii), we get 2 cot(α + β) − cot γ 3 cot θ = cot β − cot γ 2 cot θ ⇒ 3cotβ – 3cotγ = 4cot (α + β) – 2 cotγ ⇒ 3cotβ – cotγ = 4 cot (α + β) cot α. cot β − 1 ⇒ 3cotβ – cotγ = 4 cot β + cot α 2 ⇒ 3cot β + 3cotα cotβ – cotβ cotγ – cotα cotγ = 4 cotα cotβ – 4 ⇒ 4 + 3cot2β = cotα cotβ + cotβ cotγ + cotα cotγ ⇒ 4 + 4cot2β = cotα cotβ + cotα cotγ + cotβ cotγ + cot 2β ⇒ 4(1 + cot 2β) = (cotα + cotβ) (cotβ + cotγ) 1 1 1 1 1 6 + + ⇒ 4 1 + 2 = tan α tan β tan β tan γ tan β
1 1 1 1 1 4 1 + 2 = + + t1 t 2 t 2 t 3 t2 Self Practice Problems : ⇒
1.
Hence proved
1
In a ∆ABC, the median to the side BC is of length
11 − 6 3 30° and 45°. Prove that the side BC is of length 2 units.
and it divides angle A into the angles of
8. Radius of Circumcirlce : R=
c a b a bc = = = 2 sinA 2 sinB 2 sinC 4∆ s R
Example :
In a ∆ABC prove that sinA + sinB + sinC =
Solution.
In a ∆ABC, we know that a b c = = = 2R sin A sin B sin C a b c ∴ sin A = , sinB = and sinC = . 2R 2R 2R a+b+c ∴ sinA + sinB + sinC = ∵ a + b + c = 2s 2R 2s s = ⇒ sinA + sinB + sinC = . 2R R In a ∆ABC if a = 13 cm, b = 14 cm and c = 15 cm, then find its circumradius. abc .......(i) ∵ R= 4∆ ∵ ∆ = s(s − a )(s − b)(s − c )
Example : Solution.
a+b+c = 21 cm 2 ∴ ∆ = 21.8.7.6 = 7 2.4 2.3 2 ⇒ ∆ = 84 cm 2 13 .14.15 65 ∴ R= = cm 4.84 8 65 cm. ∴ R= 8 A B C In a ∆ABC prove that s = 4R cos . cos . cos . 2 2 2 In a ∆ABC,
∵
Example : Solution.
s(s − a ) s(s − b) B C = = , cos and cos bc ca 2 2 A B C ∵ R.H.S. = 4R cos . cos . cos . 2 2 2 s( s − a)(s − b)(s − c ) abc = .s ∵ (abc )2 ∆ = s = L.H.S. Hence R.H.L = L.H.S. proved 1 1 1 1 4R In a ∆ABC, prove that + + – = . s−a s −b s−c s ∆ 4R 1 1 1 1 + + – = s−a s −b s−c s ∆ 1 1 1 1 + − + ∵ L.H.S. = s−a s−b s−c s
∵
Example : Solution.
s=
cos
A = 2
2s − a − b (s − s + c ) + ( s − a)(s − b) s( s − c ) c c + = ( s − a)(s − b) s(s − c ) =
∵
s( s − c ) abc and R = ab 4∆
∆=
s(s − a)(s − b)(s − c )
2s = a + b + c
2s2 − s(a + b + c ) + ab s(s − c ) + ( s − a )(s − b) =c =c ∆2 s(s − a)(s − b)(s − c ) 7
If α, β, γ are the distances of the vertices of a triangle from the corresponding points of contact with the αβ y incircle, then prove that = r2 α+β+y
9. Radius of The Incircle : ∆ s a sin B2 sin C2 (iii) r = cos A2
A B C = (s − b) tan = (s − c) tan 2 2 2 A B C (iv) r = 4R sin sin sin 2 2 2
(ii) r = (s − a) tan
(i) r =
& so on
10. Radius of The Ex- Circles : A B C ∆ ; ∆ ; ∆ r2 = r3 = (ii) r1 = s tan ; r2 = s tan ; r 3 = s tan 2 2 2 s−a s−b s−c a cos B2 cos C2 A B C (iii) r1 = & so on (iv) r 1 = 4 R sin . cos . cos 2 2 2 cos A2 Example : In a ∆ABC, prove that r1 + r2 + r3 – r = 4R = 2a cosecA
(i) r1 =
Solution.
∵
L.H.S
= r1 + r2 + r 3 – r ∆ ∆ ∆ ∆ = + + – s−a s −b s−c s 1 1 1 1 + − +∆ =∆ s−a s−b s−c s s − b + s − a s − s + c = ∆ (s − a)(s − b) + s(s − c ) c c + =∆ (s − a)(s − b) s(s − c ) s(s − c ) + (s − a)(s − b) = c∆ s(s − a)(s − b )(s − c ) 2s 2 − s(a + b + c ) + ab = c∆ ∆2 abc = ∆
= 4R = 2acosecA
∵
a + b + c = 2s
∵ ∵
R=
abc 4∆
a = 2R = acosecA sin A
Example :
= R.H.S. Hence L.H.S. = R.H.S. proved If the area of a ∆ABC is 96 sq. unit and the radius of the escribed circles are respectively 8, 12 and 24. Find the perimeter of ∆ABC.
Solution.
∵ ∵ ∵ ∵
∴
∆ = 96 sq. unit r1 = 8, r2 = 12 and r3 = 24 ∆ ⇒ s – a = 12 r1 = s−a ∆ r2 = ⇒ s–b=8 s−b ∆ r3 = ⇒ s–c=4 s−c 8 adding equations (i), (ii) & (iii), we get
.........(i) .........(ii) .........(iii)
∴
3s – (a + b + c) = 24 s = 24 perimeter of ∆ABC = 2s = 48 unit.
Self Practice Problems
In a ∆ABC prove that 1.
r 1r2 + r2r3 + r3r1 = s2
2.
rr1 + rr 2 + rr 3 = ab + bc + ca – s2
3.
If A, A1, A2 and A3 are the areas of the inscribed and escribed circles respectively of a ∆ABC, then prove 1 1 1 1 = that + + . A A1 A2 A3
4.
c r1 − r r2 − r + = r . a b 3
11. Length of Angle Bisectors, Medians & Altitudes :
(i) Length of an angle bisector from the angle A = β a =
2 bc cos A 2 b+c
;
1 2 b2 + 2 c2 − a 2 2 2∆ & (iii) Length of altitude from the angle A = Aa = a 3 2 2 2 NOTE : ma + m b + m c = (a2 + b2 + c2) 4 (ii) Length of median from the angle A = m a =
Example :
AD is a median of the ∆ABC. If AE and AF are medians of the triangles ABD and ADC respectively, and AD = m 1, AE = m 2 , AF = m 3 , then prove that m 22 + m 32 – 2m 12 =
Solution.
In ∆ABC 1 (2b2 + 2c2 – a2) = m 12 AD2 = 4 1 a2 (2c2 + 2AD2 – ) ∵ In ∆ABD, AE2 = m 22 = 4 4 2 1 2AD2 + 2b 2 − a Similarly in ∆ADC, AF 2 = m 32 = 4 4 by adding equations (ii) and (iii), we get
In a ∆ABC a = 5, b = 4, c = 3. ‘G’ is the centroid of triangle, then find circumradius of ∆GAB. 5 Ans. 13 12
12. The Distances of The Special Points from Vertices and Sides of Triangle: (i)
Circumcentre (O)
:
OA = R & Oa = R cos A
(ii)
Incentre (I)
:
IA = r cosec
(iii)
Excentre (I1)
:
(iv)
Orthocentre (H)
:
HA = 2R cos A & Ha = 2R cos B cos C
(v)
Centroid (G)
:
GA =
Example :
Solution.
A & Ia = r 2 A & I 1a = r1 I1 A = r1 cosec 2
1 2∆ 2b2 +2c2 −a 2 & Ga = 3 3a
If x, y and z are respectively the distances of the vertices of the ∆ABC from its orthocentre, then prove that abc a c b (i) + + = (ii) x y + z = 2(R + r) xyz x z y ∵ x = 2R cosA, y = 2R cosB, z = 2R cosC and and a = 2R sinA, b = 2R sinB, c = 2R sinC a c b ∴ + + = tanA + tan B + tan C .........(i) x z y & ∵ ∴
∵ ∵
∴
∴
abc ........(ii) xyz = tanA. tanB. tanC We know that in a ∆ABC Σ tanA = Π tanA From equations (i) and (ii), we get abc a c b + + = xyz x z y x + y + z = 2R (cosA + cosB + cosC) A B C in a ∆ABC cosA + cosB + cosC = 1 + 4sin sin sin 2 2 2 A B C x + y + z = 2R 1 + 4 sin . sin . sin 2 2 2 A B C = 2 R + 4R sin . sin . sin 2 2 2 x + y + z = 2(R + r)
r = 4R sin
∵
B C A sin sin 2 2 2
Self Practice Problems A B C tan tan . 2 2 2
1.
If Ι be the incentre of ∆ABC, then prove that ΙA . ΙB . ΙC = abc tan
2.
If x, y, z are respectively be the perpendiculars from the circumcentre to the sides of ∆ABC, then prove abc a c b that + + = . 4 xyz x z y
13. Orthocentre and Pedal Triangle: The triangle KLM which is formed by joining the feet of the altitudes is called the Pedal Triangle. (i) Its angles are π − 2A, π − 2B and π − 2C. (ii) Its sides are a cosA = R sin 2A, b cosB = R sin 2B and c cosC = R sin 2C (iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal.
14. Excentral Triangle: The triangle formed by joining the three excentres Ι1, Ι2 and Ι 3 of ∆ ABC is called the excentral or excentric triangle. (i) ∆ ABC is the pedal triangle of the ∆ Ι1 Ι 2 Ι 3. (ii) Its angles are
π C π A π B − , − & − . 2 2 2 2 2 2
10
(iii)
(iv)
(v)
A , 2 B C & 4 R cos . 4 R cos 2 2 A Ι Ι1 = 4 R sin ; 2 B C Ι Ι2 = 4 R sin ; Ι Ι 3 = 4 R sin . 2 2 Its sides are 4 R cos
Incentre Ι of ∆ ABC is the orthocentre of the excentral ∆ Ι 1 Ι 2 Ι 3.
15. Distance Between Special Points : (i) Distance between circumcentre and orthocentre OH2 = R2 (1 – 8 cosA cos B cos C) (ii) Distance between circumcentre and incentre A B C OΙ2 = R2 (1 – 8 sin sin sin ) = R2 – 2Rr 2 2 2 (iii) Distance between circumcentre and centroid 1 OG2 = R2 – (a2 + b2 + c2) 9 In Ι is the incentre and Ι1, Ι 2, Ι3 are the centres of escribed circles of the ∆ABC, prove that Example : 2 (ii) ΙΙ12 + Ι 2Ι32 = ΙΙ22 + Ι3Ι 12 = ΙΙ 32 + Ι1Ι 22 (i) ΙΙ 1. ΙΙ2 . ΙΙ3 = 16R r Solution. (i) ∵ We know that A B C ΙΙ1 = a sec , ΙΙ2 = b sec and ΙΙ3 = c sec 2 2 2 C A B ∵ Ι 1Ι2 = c. cosec , Ι2 Ι3 = a cosec and Ι 3Ι1 = b cosec 2 2 2 A B C ∵ ΙΙ1 . ΙΙ 2 . ΙΙ3 = abc sec sec .sec ........(i) 2 2 2 ∵ a = 2R sin A, b = 2R sinB and c = 2R sinC ∴ equation (i) becomes A B C ∵ ΙΙ1. ΙΙ 2 . ΙΙ3 = (2R sin A) (2R sin B) (2R sinC) sec sec sec 2 2 2
∴
A A B B C C 2 sin cos 2 sin cos 2 sin cos 2 2 2 2 2 2 = 8R3 . C A B cos . cos . cos 2 2 2 A B C A B C = 64R3 sin sin sin ∵ r = 4R sin sin sin 2 22 2 2 2 2 ΙΙ1 . ΙΙ2 . ΙΙ 3 = 16R r Hence Proved
ΙΙ1 + Ι 2Ι3 = ΙΙ2 + Ι3Ι1 = ΙΙ3 + Ι1Ι2 2
(ii)
2
2
2
2
2
a2 A A + a2 cosec2 = A A 2 2 sin2 cos 2 2 2 A A 16 R 2 sin2 . cos2 A A 2 2 2 2 2 ∵ a = 2 R sinA = 4R sin cos ∴ ΙΙ 1 + Ι 2Ι3 = = 16R 2 A 2 A 2 2 sin . cos 2 2 2 Similarly we can prove ΙΙ22 + Ι3Ι 12 = ΙΙ32 + Ι1Ι22 = 16R Hence ΙΙ12 + Ι 2Ι32 = ΙΙ22 + Ι3Ι12 = ΙΙ32 + Ι1Ι22 Self Practice Problem : π 1. In a ∆ ABC, if b = 2 cm, c = 3 cm and ∠A = , then find distance between its circumcentre and 6 incentre.
∵
Ans.
ΙΙ1 + Ι2Ι3 = a2 sec2 2
2
2 − 3 cm
11
SHORT REVISION SOLUTIONS OF TRIANGLE I.
SINE FORMULA :
In any triangle ABC ,
II.
COSINE FORMULA :
(i) cos A =
a b c . = = sin A sin B sin C
b 2 +c 2 −a 2 2bc
or a² = b² + c² − 2bc. cos A
c 2 +a 2 −b 2 2ca (i) a = b cos C + c cos B
a 2 +b 2 −c 2 2ab (ii) b = c cos A + a cos C
(iii) cos C =
(ii) cos B = III.
IV.
V.
VI.
VII.
PROJECTION FORMULA :
(iii) c = a cos B + b cos A A B−C b−c NAPIER’S ANALOGY − TANGENT RULE : (i) tan = cot b+ c 2 2 C− A c−a B C A− B a −b (ii) tan = cot (iii) tan = cot a +b 2 2 c+a 2 2 TRIGONOMETRIC FUNCTIONS OF HALF ANGLES :
(s−c)(s−a ) C ; sin = ca 2
(s−a )(s−b) ab
A = 2
(s−b)(s−c) B ; sin = bc 2
cos
A = 2
s(s−a ) B ; cos = bc 2
(iii)
tan
A = 2
∆ (s−b)(s−c) a + b+c = where s = & ∆ = area of triangle. s(s−a ) s(s−a ) 2
(iv)
Area of triangle = s(s−a )(s−b)(s−c) .
(i)
sin
(ii)
s(s−b) C ; cos = ca 2
s(s−c) ab
M −N
RULE : In any triangle , (m + n) cot θ = m cot α − n cot β = n cot B − m cot C 1 2
ab sin C =
1 2
bc sin A =
1 2
ca sin B = area of triangle ABC.
a b c = = = 2R sin A sin B sin C a bc Note that R = 4 ∆ ; Where R is the radius of
circumcircle & ∆ is area of triangle
VIII. Radius of the incircle ‘r’ is given by: (a) r = (c) r =
∆ a +b+c where s = 2 s a sin B2 sin C2 & so on cos A2
(b) r = (s − a) tan (d) r = 4R sin
Radius of the Ex− circles r1 , r2 & r3 are given by :
(a)
r1 = (c)
r1 =
a cos B2 cos C2 cos A2
r2 = 4 R sin X.
B 2
(b)
. cos
A 2
. cos
r1 = s tan
(d)
& so on C 2
= (s − b) tan
= (s − c) tan
A 2
. cos
C 2
. cos
r3 = 4 R sin
B 2
. cos
A 2
C 2
. cos
LENGTH OF ANGLE BISECTOR & MEDIANS : If ma and βa are the lengths of a median and an angle bisector from the angle A then, 12
C 2
C A B ; r2 = s tan ; r3 = s tan 2 2 2
r1 = 4 R sin
;
B 2
A B C sin sin 2 2 2
IX.
∆ ∆ ∆ ; r2 = ; r3 = s−c s−b s−a
A 2
B 2
;
ma =
1 2
2 b 2 + 2 c 2 − a 2 and βa =
Note that m2a + m2b + m2c = XI.
− − − − XII
− − −
2 bc cos A 2 b+c
3 2 (a + b2 + c2) 4
ORTHOCENTRE AND PEDAL TRIANGLE : The triangle KLM which is formed by joining the feet of the altitudes is called the pedal triangle. the distances of the orthocentre from the angular points of the ∆ ABC are 2 R cosA , 2 R cosB and 2 R cosC the distances of P from sides are 2 R cosB cosC, 2 R cosC cosA and 2 R cosA cosB the sides of the pedal triangle are a cosA (= R sin 2A), b cosB (= R sin 2B) and c cosC (= R sin 2C) and its angles are π − 2A, π − 2B and π − 2C. circumradii of the triangles PBC, PCA, PAB and ABC are equal . EXCENTRAL TRIANGLE : The triangle formed by joining the three excentres I1, I2 and I3 of ∆ ABC is called the excentral or excentric triangle. Note that : Incentre I of ∆ ABC is the orthocentre of the excentral ∆ I1I2I3 . ∆ ABC is the pedal triangle of the ∆ I1I2I3 . the sides of the excentral triangle are A C B , 4 R cos and 4 R cos 2 2 2 π A π B and its angles are , and π − C . − − 2 2 2 2 2 2 C B A I I1 = 4 R sin ; I I2 = 4 R sin ; I I3 = 4 R sin . 2 2 2
4 R cos
−
XIII. THE DISTANCES BETWEEN THE SPECIAL POINTS : The distance between circumcentre and orthocentre is = R . 1 − 8 cos A cos B cos C (a) (b)
The distance between circumcentre and incentre is = R 2 − 2 R r
(c) XIV.
The distance between incentre and orthocentre is 2 r 2 − 4 R 2 cos A cos B cos C Perimeter (P) and area (A) of a regular polygon of n sides inscribed in a circle of radius r are given by π 1 2π P = 2nr sin and A = nr2 sin 2 n n Perimeter and area of a regular polygon of n sides circumscribed about a given circle of radius r is given by π π P = 2nr tan and A = nr2 tan n n
EXERCISE–I Q.1
With usual notations, prove that in a triangle ABC: b−c c−a a −b + + =0 r3 r1 r2
Q.2
a cot A + b cot B + c cot C = 2(R + r)
Q.3
r3 r1 r2 3 + + = (s − b) (s − c) (s − c) (s − a ) (s − a ) (s − b) r
Q.4
r1 − r r2 − r c + = a b r3
Q.5
abc A B C cos cos cos = ∆ s 2 2 2
Q.6
(r1 + r2)tan
Q.7
(r1− r) (r2− r)(r3− r) = 4 R r2
Q.8 (r + r1)tan 13
C C = (r3 − r) cot = c 2 2
B−C C−A A−B +(r + r2)tan +(r + r3) tan =0 2 2 2
Q.9
1 1 1 1 a 2 + b2 + c2 + + + = r 2 r12 r2 2 r32 ∆2
Q.10 (r3+ r1) (r3+ r2) sin C = 2 r3 r2 r3 + r3r1 + r1r2
Q.11
1 1 1 1 + + = bc ca ab 2Rr
Q.12
Q.13
bc − r2 r3 ca − r3r1 ab − r1r2 = = =r r3 r1 r2
1 1 1 1 1 1 4R − − − = 2 2 r r1 r r2 r r3 r s 2
Q.14
Q.15 Rr (sin A + sin B + sin C) = ∆
Q.16
1 1 1 1 41 1 1 + + + = + + r r r r r r1 r2 r3 1 2 3 2R cos A = 2R + r – r1
A B C s2 a 2 + b2 + c2 + cot + cot = Q.18 cot A + cot B + cot C = 2 2 2 ∆ 4∆ Given a triangle ABC with sides a = 7, b = 8 and c = 5. If the value of the expression
Q.17 cot Q.19
(∑ sin A ) ∑ cot A can be expressed in the form qp where p, q ∈ N and qp is in its lowest form find 2 the value of (p + q). Q.20 If r1 = r + r2 + r3 then prove that the triangle is a right angled triangle. Q.21 If two times the square of the diameter of the circumcircle of a triangle is equal to the sum of the squares of its sides then prove that the triangle is right angled. Q.22 In acute angled triangle ABC, a semicircle with radius ra is constructed with its base on BC and tangent to the other two sides. rb and rc are defined similarly. If r is the radius of the incircle of triangle ABC then 1 1 1 2 prove that, = + + . ra rb rc r Q.23 Given a right triangle with ∠A = 90°. Let M be the mid-point of BC. If the inradii of the triangle ABM and ACM are r1 and r2 then find the range of r1 r2 . Q.24 If the length of the perpendiculars from the vertices of a triangle A, B, C on the opposite sides are p1, p2, p3 then prove that
1 1 1 1 1 1 1 + + = = + + . p1 p2 p3 r1 r r2 r3
a b b c c a bc ca ab Q.25 Prove that in a triangle r + r + r = 2R b + a + c + b + a + c − 3 . 1 2 3
EXERCISE–II Q.1 Q.2
b+c c+a a+b = = ; then prove that, cos A = cos B = cos C . 11 12 13 7 19 25 A b−c For any triangle ABC , if B = 3C, show that cos C = b + c & sin = . 2 2c 4c
With usual notation, if in a ∆ ABC,
π 3 · l (AB) and ∠ DBC = . Determine the ∠ABC. 2 4
Q.3
In a triangle ABC, BD is a median. If l (BD) =
Q.4
ABCD is a trapezium such that AB , DC are parallel & BC is perpendicular to them. If angle ADB = θ , BC = p & CD = q , show that AB =
Q.5
(p 2 + q 2 ) sinθ . p cos θ + q sin θ
If sides a, b, c of the triangle ABC are in A.P., then prove that A B C cosec 2A; sin2 cosec 2B; sin2 cosec 2C are in H.P.. sin2 2 2 2 14
Q.6
Find the angles of a triangle in which the altitude and a median drawn from the same vertex divide the angle at that vertex into 3 equal parts.
Q.7
In a triangle ABC, if tan
Q.8
ABCD is a rhombus. The circumradii of ∆ ABD and ∆ ACD are 12.5 and 25 respectively. Find the area of rhombus. cot C In a triangle ABC if a2 + b2 = 101c2 then find the value of . cot A + cot B
Q.9
A B C , tan , tan are in AP. Show that cos A, cos B, cos C are in AP. 2 2 2
Q.10 The two adjacent sides of a cyclic quadrilateral are 2 & 5 and the angle between them is 60°. If the area of the quadrilateral is 4 3 , find the remaining two sides. Q.11
If I be the in−centre of the triangle ABC and x, y, z be the circum radii of the triangles IBC, ICA & IAB, show that 4R3 − R (x2 + y2 + z2) − xyz = 0.
Q.12 Sides a, b, c of the triangle ABC are in H.P. , then prove that cosec A (cosec A + cot A) ; cosec B (cosec B + cot B) & cosec C (cosec C + cot C) are in A.P. Q.13 In a ∆ ABC, (i) (iii) tan2
a b = cos A cos B
(ii) 2 sin A cos B = sin C
A A C + 2 tan tan − 1 = 0, prove that (i) ⇒ (ii) ⇒ (iii) ⇒ (i). 2 2 2
Q.14 The sequence a1, a2, a3, ........ is a geometric sequence. The sequence b1, b2, b3, ........ is a geometric sequence. b1 = 1;
b2 =
4
7 − 28 + 1; 4
a1 =
4
28 and
∞
1
∑a n =1
n
∞
= ∑ bn n =1
If the area of the triangle with sides lengths a1, a2 and a3 can be expressed in the form of p q where p and q are relatively prime, find (p + q). Q.15 If p1 , p2 , p3 are the altitudes of a triangle from the vertices A , B , C & ∆ denotes the area of the 1 1 1 2ab 2 C triangle , prove that p + p − p = (a + b + c)∆ cos 2 . 1 2 3 Q.16 The triangle ABC (with side lengths a, b, c as usual) satisfies log a2 = log b2 + log c2 – log (2bc cosA). What can you say about this triangle? Q.17 With reference to a given circle, A1 and B1 are the areas of the inscribed and circumscribed regular polygons of n sides, A2 and B2 are corresponding quantities for regular polygons of 2n sides. Prove that (1) A2 is a geometric mean between A1 and B1. (2) B2 is a harmonic mean between A2 and B1. Q.18 The sides of a triangle are consecutive integers n, n + 1 and n + 2 and the largest angle is twice the smallest angle. Find n. Q.19 The triangle ABC is a right angled triangle, right angle at A. The ratio of the radius of the circle circumscribed
(
)
to the radius of the circle escribed to the hypotenuse is, 2 : 3 + 2 . Find the acute angles B & C. Also find the ratio of the two sides of the triangle other than the hypotenuse.
15
Q.20 ABC is a triangle. Circles with radii as shown are drawn inside the triangle each touching two sides and the incircle. Find the radius of the incircle of the ∆ABC. Q.21 Line l is a tangent to a unit circle S at a point P. Point A and the circle S are on the same side of l, and the distance from A to l is 3. Two tangents from point A intersect line l at the point B and C respectively. Find the value of (PB)(PC). Q.22 Let ABC be an acute triangle with orthocenter H. D, E, F are the feet of the perpendiculars from A, B, and C on the opposite sides. Also R is the circumradius of the triangle ABC. Given (AH)(BH)(CH) = 3 and (AH)2 + (BH)2 + (CH)2 = 7. Find (a) the ratio
∏ cos A , ∑ cos 2 A
(b) the product (HD)(HE)(HF)
(c) the value of R.
EXERCISE–III Q.1
The radii r1, r2, r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm, find the lengths of its sides. [REE '99, 6]
Q.2(a) In a triangle ABC , Let ∠ C = 2(r + R) is equal to: (A) a + b (b)
Q.3
Q.4
Q.5
Q.6
(B) b + c
In a triangle ABC , 2 a c sin (A) a2 + b2 − c2
1 (A − B + C) = 2
(B) c2 + a2 − b2
(C) c + a
(D) a + b + c
(C) b2 − c2 − a2
(D) c2 − a2 − b2 [JEE '2000 (Screening) 1 + 1]
Let ABC be a triangle with incentre ' I ' and inradius ' r ' . Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA & AB respectively . If r1 , r2 & r3 are the radii of circles inscribed in the quadrilaterals AFIE , BDIF & CEID respectively, prove that r r1 r r1 r2 r3 + 2 + 3 = . [JEE '2000, 7] r − r1 r − r2 r − r3 (r − r1 )(r − r2 )(r − r3 ) 1 If ∆ is the area of a triangle with side lengths a, b, c, then show that: ∆ < (a + b + c)abc 4 Also show that equality occurs in the above inequality if and only if a = b = c. [JEE ' 2001] Which of the following pieces of data does NOT uniquely determine an acute–angled triangle ABC (R being the radius of the circumcircle)? (A) a, sinA, sinB (B) a, b, c (C) a, sinB, R (D) a, sinA, R [ JEE ' 2002 (Scr), 3 ] If In is the area of n sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that On In = 2
Q.7
π . If ' r ' is the inradius and ' R ' is the circumradius of the triangle, then 2
2 1 + 1 − 2 I n n
[JEE 2003, Mains, 4 out of 60]
The ratio of the sides of a triangle ABC is 1 : 3 : 2. The ratio A : B : C is (A) 3 : 5 : 2
(B) 1 : 3 : 2
(D) 1 : 2 : 3 [JEE 2004 (Screening)] Q.8(a) In ∆ABC, a, b, c are the lengths of its sides and A, B, C are the angles of triangle ABC. The correct relation is B−C A = a cos (A) ( b − c) sin 2 2
(C) 3 : 2 : 1
A B−C ( b − c) cos = a sin (B) 16 2 2
B+C A = a cos (C) ( b + c) sin 2 2
A B+C (D) (b − c) cos = 2a sin 2 2 [JEE 2005 (Screening)] (b) Circles with radii 3, 4 and 5 touch each other externally if P is the point of intersection of tangents to these circles at their points of contact. Find the distance of P from the points of contact. [JEE 2005 (Mains), 2]
Q.9(a) Given an isosceles triangle, whose one angle is 120° and radius of its incircle is 3 . Then the area of triangle in sq. units is (A) 7 + 12 3
(C) 12 + 7 3
(B) 12 – 7 3
(D) 4π
[JEE 2006, 3] (b) Internal bisector of ∠A of a triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of ∆ABC then (A) AE is HM of b and c (C) EF =
(B) AD =
4bc A sin b+c 2
2bc A cos b+c 2
(D) the triangle AEF is isosceles
[JEE 2006, 5]
Q.10 Let ABC and ABC′ be two non-congruent triangles with sides AB = 4, AC = AC′ = 2 2 and angle B = 30°. The absolute value of the difference between the areas of these triangles is [JEE 2009, 5]
EXERCISE–I Q.19 107
Q.23
1 , 2 2
Q.3
Q.6
π/6, π/3, π/2
EXERCISE–II 120°
Q.14 9 Q.20 r = 11
Q.8
Q.16 triangle is isosceles Q.21 3
Q.9
400
Q.18 4
Q.22 (a)
50 Q.10
Q.19 B =
5π 12
;C=
3 cms & 2 cms π 12
;
b = 2+ 3 c
9 3 3 , (b) 3 , (c) 14R 2 8R
EXERCISE–III Q.1 6, 8, 10 cms Q.2 (a) A, (b) B Q.9 (a) C, (b) A, B, C, D Q.10 4
Q.5 D
Q.7
D
Q.8
(a) B; (b) 5
P. T. O.
17
Part : (A) Only one correct option 1.
2.
In a triangle ABC, (a + b + c) (b + c − a) = k. b c, if : (B) k > 6 (C) 0 < k < 4 (A) k < 0 In a ∆ABC, A =
2π 9 3 , b – c = 3 3 cm and ar ( ∆ABC) = cm 2. Then a is 3 2
(A) 6 3 cm
3.
(D) k > 4
(B) 9 cm
If R denotes circumradius, then in ∆ ABC,
(A) cos (B – C)
(B) sin (B – C)
(C) 18 cm
b2 − c 2 is equal to 2a R (C) cos B – cos C
(D) none of these
(D) none of these
4.
If the radius of the circumcircle of an isosceles triangle PQR is equal to PQ (= PR), then the angle P is π π 2π π (D) (A) (B) (C) 6 3 3 2
5.
In a ∆ ABC, the value of (A)
6.
r R
acosA + bcosB + ccosC is equal to: a+b+c
(B)
R 2r
In a right angled triangle R is equal to s+r s−r (A) (B) 2 2
(C)
R r
(C) s – r
(D)
2r R
(D)
s+r a
7.
In a ∆ABC, the inradius and three exradii are r, r1, r2 and r3 respectively. In usual notations the value of r. r1. r2. r3 is equal to abc (A) 2∆ (D) none of these (B) ∆ 2 (C) 4R
8.
In a triangle if r1 > r2 > r3, then (B) a < b < c (A) a > b > c
9.
1 1 With usual notation in a ∆ ABC r + r 1 2 where 'K' has the value equal to: (A) 1 (B) 16
(C) a > b and b < c 1 1 + r 2 r3
(D) a < b and b > c
1 1 KR 3 + = , r 2 a b2c 2 3 r1
(C) 64
(D) 128
10.
The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to: (A) ∆ (B) 2 ∆ (C) 3 ∆ (D) 4 ∆
11.
In a triangle ABC, right angled at B, the inradius is: AB + BC − AC AB + AC − BC AB + BC + AC (A) (B) (C) (D) None 2 2 2 The distance between the middle point of BC and the foot of the perpendicular from A is :
12.
(A)
13.
− a2 + b2 + c 2 2a
(B)
b2 − c 2 2a
(C)
b2 + c 2 bc
(D) none of these
In a triangle ABC, B = 60° and C = 45°. Let D divides BC internally in the ratio 1 : 3, then, (A)
2 3
(B)
1 3
(C)
1 6
(D)
sin ∠BAD = sin ∠CAD
1 3
14.
Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ∆ ABC on the sides a, b and a b c abc c respectively. If + + = λ then the value of λ is: f g h f gh (A) 1/4 (B) 1/2 (C) 1 (D) 2
15.
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of length 3, 4 and 5 units. Then area of the triangle is equal to: 18
(A)
9 3 (1 + 3 ) π
2
(B)
9 3 ( 3 − 1) π
(C)
2
9 3 (1 + 3 ) 2π
2
(D)
9 3 ( 3 − 1) 2π 2
16.
If in a triangle ABC, the line joining the circumcentre and incentre is parallel to BC, then cos B + cos C is equal to: (A) 0 (B) 1 (C) 2 (D) none of these
17.
If the incircle of the ∆ ABC touches its sides respectively at L, M and N and if x, y, z be the circumradii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz is equal to: (A) R r2
18.
19.
(B) r R2
(C)
1 R r2 2
(D)
1 r R2 2
r 1 A tan B + tan C is equal to : If in a ∆ABC, r = , then the value of tan 2 2 2 2 1 1 (B) (C) 1 (D) None of these (A) 2 2
In any ∆ABC, then minimum value of (A) 3
(B) 9
r1 r2 r3 r3
is equal to (C) 27
(D) None of these
20.
In a acute angled triangle ABC, AP is the altitude. Circle drawn with AP as its diameter cuts the sides AB and AC at D and E respectively, then length DE is equal to ∆ ∆ ∆ ∆ (B) (C) (D) (A) 3R 2R 4R R
21. 22.
AA1, BB1 and CC1 are the medians of triangle ABC whose centroid is G. If the concyclic, then points A, C1, G and B1 are (A) 2b2 = a2 + c2 (B) 2c2 = a2 + b2 (C) 2a2 = b2 + c2 (D) None of these In a ∆ABC, a, b, A are given and c1, c2 are two values of the third side c. The sum of the areas of two triangles with sides a, b, c1 and a, b, c2 is 1 1 (A) b2 sin 2A (B) a2 sin 2A (C) b2 sin 2A (D) none of these 2 2
23.
In a triangle ABC, let ∠C = is equal to (A) a + b – c
π . If r is the inradius and R is the circumradius of the triangle, then 2(r + R) 2 [IIT - 2000] (B) b + c (C) c + a (D) a + b + c
24.
Which of the following pieces of data does NOT uniquely determine an acute - angled triangle ABC (R being the radius of the circumcircle )? [IIT - 2002] (A) a , sin A, sin B (B) a, b, c (C) a, sin B, R (D) a, sin A, R
25.
If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the longest side to the perimeter is (A) 3 : (2 + 3 ) (B) 1 : 6 (C) 1 : 2 + 3 (D) 2 : 3 [IIT - 2003]
26.
The sides of a triangle are in the ratio 1 : (A) 1 : 3 : 5
27.
In an equilateral triangle, 3 coincs of radii 1 unit each are kept so that they touche each other and also the sides of the triangle. Area of the triangle is [IIT - 2005]
(A) 4 + 2 3
28.
(B) 6 + 4
3
(C) 12 +
If P is a point on C1 and Q is a point on C2, then (A) 1/2
29.
(B) 2 : 3 : 4
3 : 2, then the angle of the triangle are in the ratio [IIT - 2004] (C) 3 : 2 : 1 (D) 1 : 2 : 3
(B) 3/4
7 3 4
(D) 3 +
7 3 4
PA 2 + PB 2 + PC 2 + PD 2
equals QA 2 + QB 2 + QC 2 + QD 2 (C) 5/6 (D) 7/8
A circle C touches a line L and circle C1 externally. If C and C1 are on the same side of the line L, then locus of the centre of circle C is (A) an ellipse (B) a circle (C) a parabola (D) a hyperbola 19
30.
Let be a line through A and parallel to BD. A point S moves such that its distance from the line BD and the vertex A are equal. If the locus of S meets AC in A1, and in A2 and A3, then area of ∆A1 A2A3 is (A) 0.5 (unit)2 (B) 0.75 (unit)2 (C) 1 (unit)2 (D) (2/3) (unit)2
Part : (B) May have more than one options correct 31.
In a ∆ABC, following relations hold good. In which case(s) the triangle is a right angled triangle? (A) r2 + r3 = r1 − r (B) a2 + b2 + c2 = 8 R2 (C) r1 = s (D) 2 R = r1 − r
32.
In a triangle ABC, with usual notations the length of the bisector of angle A is : A abc cos ec 2 bc cos A 2 bc sin A 2∆ . A 2 2 2 (A) (B) (C) (D) b + c cos ec 2 b+c b+c 2R (b + c ) AD, BE and CF are the perpendiculars from the angular points of a ∆ ABC upon the opposite sides, then : Perimeter of ∆DEF r (B) Area of ∆DEF = 2 ∆ cosA cosB cosC (A) = Perimeter of ∆ABC R R (C) Area of ∆AEF = ∆ cos2A (D) Circum radius of ∆DEF = 2
33.
34.
The product of the distances of the incentre from the angular points of a ∆ ABC is: (abc ) R (abc ) r (A) 4 R2 r (B) 4 Rr 2 (C) (D) s s
35.
In a triangle ABC, points D and E are taken on side BC such that BD = DE = EC. If angle ADE = angle AED = θ, then: (A) tanθ = 3 tan B (B) 3 tanθ = tanC 6 tan θ = tan A (D) angle B = angle C (C) tan2 θ − 9
36.
With usual notation, in a ∆ ABC the value of Π (r 1 − r) can be simplified as:
A (A) abc Π tan 2
1.
(B) 4 r R
If in a triangle ABC, angled.
2
(C)
(a b c)2 2 R (a + b + c)
(D) 4 R r2
cos A + 2 cos C sin B = , prove that the triangle ABC is either isosceles or right cos A + 2 cos B sin C
A + B , prove that triangle is isosceles. 2
2.
In a triangle ABC, if a tan A + b tan B = (a + b) tan
3.
r r If 1 − 1 1 − 1 = 2 then prove that the triangle is the right triangle. r2 r3
4. 5. 6.
In a ∆ ABC, ∠ C = 60° & ∠ A = 75°. If D is a point on AC such that the area of the ∆ BAD is 3 times the area of the ∆ BCD, find the ∠ ABD. The radii r1, r 2, r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm, find the lengths of its sides. ABC is a triangle. D is the middle point of BC. If AD is perpendicular to AC, then prove that
7.
8.
9.
(
)
2 c 2 −a 2 . 3ac Two circles, of radii a and b, cut each other at an angle θ. Prove that the length of the common chord is 2ab sin θ . 2 a + b 2 + 2ab cos θ In the triangle ABC, lines OA, OB and OC are drawn so that the angles OAB, OBC and OCA are each equal to ω, prove that (i) cot ω = cot A + cot B + cot C cosec2 ω = cosec2 A + cosec2 B + cosec2 C (ii) In a plane of the given triangle ABC with sides a, b, c the points A′, B′, C′ are taken so that the ∆ A′ BC, ∆ AB′C and ∆ ABC′ are equilateral triangles with their circum radii Ra, Rb, Rc ; in−radii ra, rb, r c & ex − radii ra′, rb′ & rc ′ respectively. Prove that; cos A. cos C =
[∑ (3R +6r +2r′ )] Πtan A 3
(i) 10.
Π r a: Π Ra: Π r a′ = 1: 8: 27
(ii)
r 1 r2 r 3 =
a
a
a
2 648 3 The triangle ABC is a right angled triangle, right angle at A. The ratio of the radius of the circle 20
(
)
(
)
11.
circumscribed to the radius of the circle escribed to the hypotenuse is, 2 : 3 + 2 . Find the acute angles B & C. Also find the ratio of the two sides of the triangle other than the hypotenuse. The triangle ABC is a right angled triangle, right angle at A. The ratio of the radius of the circle
12.
circumscribed to the radius of the circle escribed to the hypotenuse is, 2 : 3 + 2 . Find the acute angles B & C. Also find the ratio of the two sides of the triangle other than the hypotenuse. If the circumcentre of the ∆ ABC lies on its incircle then prove that,
13.
cosA + cosB + cosC = 2 Three circles, whose radii area a, b and c, touch one another externally and the tangents at their points of contact meet in a point; prove that the distance of this point from either of their points of contacts 1
abc 2 is . a+b+c
EXERCISE # 2
EXERCISE # 1 1. C
2. B
3. B
4. D
5. A
6. B
7. B
8. A
9. C
10. B
11. A
12. B
13. C
14. A
15. A
16. B
17. C
18. B
19. C
20. D
21. C
22. A
23. A
24. D
25. A
26. A
27. B
28. B
29. C
30. C
31. ABCD 32. ACD
34. BD 35. ACD
4. ∠ ABD = 30°
5. 6, 8, 10 cms
10. B =
5π π b ,C= , = 2+ 3 12 12 c
11. B =
5π π b ,C= , = 2+ 3 12 12 c
33. ABCD
36. ACD
21
Some questions (Assertion–Reason type) are given below. Each question contains Statement – 1 (Assertion) and Statement – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. So select the correct choice : (A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1. (C) Statement – 1 is True, Statement – 2 is False.(D) Statement – 1 is False, Statement – 2 is True. 576.
−1 −1 Statement-1: The value of tan 2 + tan 3 =
3π 4
x+y . 1 − xy
Statement-2: If x > 0, y > 0, xy > 1, then tan–1x + tan–1y = π + tan −1 577. 578. 579. 580.
7π 7π −1 –1 is the principal value of cos cos Statement-2: cos (cos x) = x if x∈[0, π] 6 6 3 π −1 −1 Statement-1: The value of cot–1(–1) is Statement-2: cot (− x) = π − cot x, x ∈ R 4 1 π Statement-1: If x + = 2 then the principal value of sin–1x is x 2 Statement-1:
Statement-2: sin–1(sin x) = x ∀ x∈R. Statement-1: If A, B, C are the angles of a triangle such that angle A is obtuse then tan B tan C > 1. Statement-2: In any triangle tan A =
581.
582. 583. 584. 585.
tan B + tan C . tan B tan C − 1
Let f(θ) = sinθ.sin (π/3 + θ) . sin (π/3 – θ) Statement-1: f(θ) ≤ 1/4 Statement-2: f(θ) = 1/4 sin2 Statement–1 : Number of ordered pairs (θ, x) satisfying 2sinθ = ex + e–x, θ∈[0, 3π] is 2. Statement–2 : Number of values of x for which sin2x + cos4x = 2 is zero. Statement–1 : The number of values of x∈ [0, 4π] satisfying | 3 cosx – sinx| ≥ 2 is 2. Statement–2 : |cos (x + π/6)| = 1 ⇒ number of solutions of | 3 cosx – sinx| ≥ 2 is 4 Statement–1 : Number of solutions of sin–1 (sinx) = 2π – x; x∈[3π/2, 5π/2] is 1 Statement–2 : sin–1 (sinx) = x, x∈ [–π/2, π/2] Statement–1 : Number of ordered pairs (x, y) satisfying sin–1x = π – sin–1y and cos–1x + cos–1y = 0 simultaneously is 1 Statement–2 : Ordered pairs (x, y) satisfying sin–1x = π – sin–1y and cos–1x + cos–1y = 0 will lie on x2 + y2 = 2.
586.
Statement–1
: The equation k cos x – 3 sin x = k + 1 is solvable only if k belongs to the interval ( −∞, 4
587.
Statement–2 Statement–1 Statement–2
: − a + b ≤ a sin x ± b cos x ≤ a + b . : The equation 2 sec2x – 3 sec x + 1 = 0 has no solution in the interval (0, 2π) : sec x ≤ – 1 as sec x ≥ 1. 2
2
2
2
588.
Statement–1
: The number of solution of the equation sin x =| x | is only one.
589.
Statement–2 Statement–1 Statement–2
: The number of point of intersection of the two curves y = |sin x| and y = |x| is three. : The equation sin x = 1 has infinite number of solution. : The domain of f(x) = sin x is (– ∞, ∞).
590.
Statement–1
: There is no solution of the equation | sin x | + | cos x |= tan x + cot x .
591.
592.
2
Statement–1 :If sin θ = a for exactly one value of θ ∈ 0,
594. 595.
Statement–2 : – 1 ≤ sin θ ≤ 1. Statement–1 : tan 5° is an irrational number. Let θ be an acute angle Statement–1 : sin6 θ + cos6 θ ≤ 1.
596.
2
Statement–2 : 0 ≤ | sin x | + | cos x |≤ 2 and tan2 x + cot2x ≥ 2. Statement–1 : The equation sin2x + cos2 y = 2 sec2 z is only solvable cos y = 1 an sec z = 1 where x, y , z ∈ R. Statement–2 : Maximum value of sin x and cos y is 1 and minimum value of sec z is 1. Statement–1 : If cot–1x < n, n∈ R then x < cot (n) Statement–2 : cot–1 (x) is an decreasing function.
593.
S–1 : sin
π is a root of 8x3 – 6x + 1 = 0. S–2 18
]
when
sin
x
7π , then a can take infinite value in the interval [– 1, 1]. 3
Statement–2
:
tan 15° is an irrational number.
Statement–2
:
sin θ + cos θ ≤ 1
: For any θ ∈ R, sin 3θ = 3 sin θ – 4 sin3 θ. 1
=
22
1,
597. 598.
Let f be any one of the six trigonometric functions. Let A, B ∈ R satisfying f(2A) = f(2B). Statement–1 : A = nπ + B, for some n ∈ I. Statement–2 : 2π is one of the period of f. Let x ∈ [-1, 1] Statement–1
(
: 2 sin-1 x = sin -1 2x 1 − x
599.
Let f(x) = cos–1 x Statement–1 : f is a decreasing function. Statement–2 : f(– x) = π – f(x).
600.
Statement–1
601.
602.
603.
2
).
Statement–2
: - 1 ≤ 2x
1 − x 2 ≤ 1.
: The total number of 2 real roots of the equation x2 tan x = 1 lies between the interval (0, 2π).
Statement–2
: The total number of solution of equation cos x − sin x = 2 cos x in [0, 2π] is 3.
Statement–1
: The number of real solutions of equation sin ex cos ex = 2x – 2 + 2- x – 2 is 0.
Statement–2
: The number of solutions of the eqution 1 + sin x sin2
Statement–1
x = 0 n [- π, π] is 0. 2 2 2 −1 −1 4 −1 : Equation tan x + − tan − tan x − = 0 has 3 real roots. x x x
Statement–2
: the number of real solution of
Statement–1
: If
1 + cos 2x = 2 sin −1 ( sin x ) ; x ∈ [ −π, π] is 2.
1 1 1 1 n + tan −1 + tan −1 ... + tan −1 tan-1 = tan -1 θ, then θ = . 1+ 2 1 + 2.3 1 + 3.4 1 + n ( n + 1) n +1 Statement–2
: The sum of series cos-1 2 + cot-1 8 + cot-1 18 + . . . is
604.
Statement-1: If tanθ + secθ = 3 , 0 < θ < π, then θ = π/6 Statement-2: General solution of cosθ = cosα is θ = α, if 0 < α < π/2
605.
Statement-1: If x < 0, tan-1x + tan -1
π . 4
1 = π/2 x
Statement-2: tan-1x + cos-1x = π/2,∀x∈R 606.
Statement-1: sin-1 (sin10) = 10 Statement-2: For principal value sin -1 (sinx) = x
607.
Statement-1: cos
π 2π 4π 1 cos cos = − 7 7 7 8
Statement-2: cosθ cos2θ cos23θ .... cos2n-1θ = -
1 π if θ = n , n ∈ N, n ≥ 2. n 2 2 −1
TRI 608.
Statement-1: sin3 < sin1 < sin2 is true Statement-2: sinx is positive in first and second quadrants.
609.
Statement-1: The equation 2sin2x – (P + 3) sinx + (2P – 2) = 0 possesses a real solution if P∈[-1, 3] Statement-2 : -1 ≤ sinx ≤ 1
610.
Statement-1: The maximum value of 3sinθ + 4cos θ + Statement-2:: - a + b 2
2
≤ asinθ + bcos θ ≤
π is 5 here θ∈R. 4
a 2 + b2
2
23
611.
Statement-1: If A + B + C = π, cosA + cosB + cosC ≤ 3/2 Statement-2:: If A + B + C = π, sin
612.
A B C 1 sin sin ≤ 2 2 2 8
Statement-1: The maximum & minimum values of the function f(x) =
1 does not exists. 6sin x − 8cos x + 5
Statement-2: The given function is an unbounded function. 613.
1 = π/2 x
Statement-1: If x < 0 tan-1x +tan-1
Statement-2: tan-1x + cot-1x = π/2 ∀ x∈R. 614.
Statement-1: In any triangle square of the length of the bisector AD is bc 1 −
Statement-2: In any triangle length of bisector AD =
a2 (b + c) 2
bc A cos b+c 2
615.
Statement-1: If in a triangle ABC, C = 2acosB, then the triangle is isosceles. Statement-2: Triangle ABC, the two sides are equal i.e. a = b.
616.
Statement-1: If the radius of the circumcircle of an isosceles triangle pqR is equal to pq = PR then the angle p = 2π/3. Statement-2: OPQ and oPR will be equilateral i.e., ∠OPq = 60°, ∠OPR = 60°
617.
Statement-1: The minimum value of the expression sinα + sinβ + sinγ is negative, where α, β, γ are real numbers such that α + β + γ = π. Statement-2: If α, β, γ are angle of a triangle then sinα + sinβ + sinγ = 4cos
α β γ cos cos . 2 2 2
618.
Statement-1: If in a triangle sin2A + sin2B + sin2C = 2 then one of the angles must be 90°. Statement-2: In any triangle sin2A + sin2B + sin2C = 2 + 2cosA cosB cosC.
619.
Statement-1: If in a ∆ABC a → 2c and b → 3c then cosB must tend to –1. Statement-1: In a ∆ABC cosB =
c2 + a 2 − b2 . 2ac
620.
Statement-1: cos(45 − A) cos(45 − B) − sin(45 − A) × sin (45 − B) = sin(A + B). Statement-2: cos(90 − θ) = − sin θ.
621.
Statement-1: The maximum and minimum values of 7cosθ + 24sinθ are 25 and − 25 respectively. Statement-2:
622.
a 2 + b 2 ≤ a cos θ ± b sin θ ≤ a 2 + b 2 for all θ.
Statement-1: If sin
−1
x + sin −1 (1 − x) = sin −1 1 − x 2 then x = 0,
−1
Statement-2: sin sin x = x ∀x ∈ R
1 2
TE 623.
Statement-1: The numbers sin 18° and –sin54° are roots of same quadratic equation with integer coefficients. Statement-2: If x = 18°, then 5x = 90°, if y = -54°, then 5y = -270°.
Inverse Trigonometry 624.
(
) (
)
Statement-1: The number of solution of the equation cos( π x − 4 cos π x = 1 is one. Statement-2: cosx = cosα ⇒ x = 2nπ ± α n∈I 3
24
Inverse Trigonometric Function
π 3π
627.
Statement-1: The range of sin-1x + cos-1x + tan −1x is , 4 4 -1 -1 Statement-2: sin x + cos x = π/2 for every x∈R.
628.
Statement-1: sin-1 (sin10) = 10 Statement-2: sin-1 (sinx) = x for - π/2 ≤ x ≤ π/2
Statement-1: 7π/6 is the principal value of cos–1 cos
7π 6
Statement-2: cos–1 (cosx) = x, if x∈ [0, π]. 631.
Statement-1:
3 cos θ + sin θ = 5 has no solution.
Statement-2: a cos θ + b sin θ = c has solution if | c |≤
632.
a 2 + b2 3 1 , 2 2
Statement-1: The equation sin4x + cos4x + sin2x + a = 0 is valid if a ∈ −
Statement-2: If discriminant of a quadratic equation is −ve. Then its roots are real. 633.
Statement-1: In a ∆ABC cosAcosB + sinAsinBsinC = 1 then ∆ABC must be isosceles as well as right angled triangle. Statement-2: In a ∆ABC if A =
634.
π tanA tanB = k. then k must satisfy k2 − 6k + 1 ≥ 0 4
Statement-1: If r1, r2, r 3 in a ∆ABC are in H.P. then sides a, b, c are in A.P. Statement-2:: r1 =
∆ ∆ ∆ . , r2 = , r3 = s−a s−b s−c
Answer Key 576. A
577. D
578. A
579. C
580. D
581.C
582.B
583. D
584. B
585. B
586. A
587. B
588. C
589.A
590. C
591. A
592. D
593. D
594. A
595. C
596. A
597. A
598. D
599. B
600. C
601. B
602. D
603. D
604. A
605. D
606. D
607. D
608. B
609. A
610. D
611. B
612. A
613. D
614. C
615. A
616. A
617. B
618. A
619. A
620. C
621. A
622. C
623. A
624. B
627. C
628. D
629. A
630. D
631. A
4
25
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: MATHEMATICS TOPIC: 4 XI M 4. Functions Index: 1. Key Concepts 2. Exercise I to V 3. Answer Key 4. Assertion and Reasons 5. 34 Yrs. Que. from IIT-JEE 6. 10 Yrs. Que. from AIEEE
1
A.
Definition :
Functions
Function is a special case of relation, from a non empty set A to a non empty set B, that associates each member of A to a unique member of B. Symbolically, we write f: A → B. We read it as "f is a function from A to B". Set 'A' is called domain of f and set 'B' is called co-domain of f. For example, let A ≡ {–1, 0, 1} and B ≡ {0, 1, 2}. Then A × B ≡ {(–1, 0), (–1, 1), (–1, 2), (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)} Now, " f : A → B defined by f(x) = x 2 " is the function such that f ≡ {(–1, 1), (0, 0), (1, 1)} f can also be show diagramatically by following picture.
Every function say f : A → B satisfies the following conditions: (a) f ⊆ A x B, (b) ∀ a ∈ A ⇒ (a, f(a)) ∈ f and (c) (a, b) ∈ f & (a, c) ∈ f ⇒ b = c Illustration # 1: (i) Which of the following correspondences can be called a function ? (A) f(x) = x 3 ; {–1, 0, 1} → {0, 1, 2, 3} (B) f(x) = ± x ; {0, 1, 4} → {–2, –1, 0, 1, 2} ; {0, 1, 4} → {–2, –1, 0, 1, 2} (C) f(x) = x ; {0, 1, 4} → {–2, –1, 0, 1, 2} (D) f(x) = – x Solution: f(x) in (C) & (D) are functions as definition of function is satisfied. while in case of (A) the given relation is not a function, as f(–1) ∉ codomain. Hence definition of function is not satisfied. While in case of (B), the given relation is not a function, as f(1) = ± 1 and f(4) = ± 2 i.e. element 1 as well as 4 in domain is related with two elements of codomain. Hence definition of function is not satisfied. (ii) Which of the following pictorial diagrams represent the function (A)
(B)
(C) (D) Solution: B & D. In (A) one element of domain has no image, while in (C) one element of domain has two images in codomain Assignment: 1. Let g(x) be a function defined on [−1, 1]. If the area of the equilateral triangle with two of its vertices at (0,0) & (x,g(x)) is 3 / 4 sq. units, then the function g(x) may be. 2.
(B*) g(x) = (1 − x 2 ) (C*) g(x) = − (1 − x 2 ) (A) g(x)= ± (1 − x 2 ) Represent all possible functions defined from {α, β} to {1, 2} Answer (1) B
B.
Domain, Co-domain & Range of a Function :
(2)
(i)
(ii)
(iii)
(D) g(x) =
(1 + x 2 )
(iv)
Let f: A → B, then the set A is known as the domain of f & the set B is known as co−domain of f. If a member 'a' of A is associ at ed to t he member 'b' of B, t hen ' b' i s cal led the f -image of 'a' and we writ e b = f (a). Further 'a' is called a pre-image of 'b'. The set {f(a): ∀ a ∈A} is called the range of f and is denoted by f(A). Clearly f(A) ⊂ B. Sometimes if only definition of f (x) is given (domain and codomain are not mentioned), then domain is set of those values of ' x' for which f (x) is defined, while codomain is considered to be (– ∞, ∞) A function whose domain and range both are sets of real numbers is called a real function. Conventionally the word "FUNCTION” is used only as the meaning of real function. Illustration # 2 : Find the domain of following functions : (i)
Solution :(i)
(ii)
sin–1 (2x – 1)
f(x) =
x2 − 5
f(x) =
2 x 2 − 5 is real iff x – 5 ≥ 0
(ii)
⇒
|x| ≥
5
⇒
x ≤ – 5 or x ≥
5
∴ the domain of f is (–∞ , – 5 ] ∪ [ 5 , ∞) –1 ≤ 2x – 1 ≤ + 1 ∴ domain is x ∈ [0, 1] Algebraic Operations on Functions : If f & g are real valued functions of x with domain set A and B respectively, then both f & g are defined in A ∩ B. Now we define f + g, f − g, (f . g) & (f /g) as follows:
f f( x ) (x) = g g ( x ) domain is {x x ∈ A ∩ B such that g(x) ≠ 0}. Note : For domain of φ(x) = {f(x)}g(x) , conventionally, the conditions are f(x) > 0 and g(x) must be defined. For domain of φ(x) = f(x) Cg(x) or φ(x) = f(x)Pg(x) conditions of domain are f(x) ≥ g(x) and f(x) ∈ N and g(x) ∈ W Illustration # 3: Find the domain of following functions :
(iii)
2
(i) Solution:
f(x) =
sin x − 16 − x 2
(ii)
f(x) =
3
log(x 3 − x)
4−x sin x is real iff sin x ≥ 0 ⇔ x∈[2nπ, 2nπ + π], n∈I.
(i)
2
(iii)
f(x) = x cos
−1
x
2 16 − x is real iff 16 − x ≥ 0 ⇔ − 4 ≤ x ≤ 4. Thus the domain of the given function is {x : x∈[2nπ, 2nπ + π], n∈I }∩[−4, 4] = [−4, −π] ∪ [0, π]. (ii) Domain of 4 − x 2 is [−2, 2] but 4 − x 2 = 0 for x = ± 2 ⇒ x ∈ (–2, 2) log(x 3 − x) is defined for x 3 − x > 0 i.e. x(x − 1)(x + 1) > 0. ∴ domain of log(x 3 − x) is (−1, 0 ) ∪ (1, ∞). Hence the domain of the given function is {(−1, 0 ) ∪ (1, ∞)}∩ (−2, 2) = (−1, 0 ) ∪ (1, 2). (iii) x > 0 and –1 ≤ x ≤ 1 ∴ domain is (0, 1] Assignment : 3. Find the domain of following functions. 1 –1 2 x − 1 f(x) = log( 2 − x ) + x + 1 (ii) f(x) = 1 − x – sin (i) 3 Ans. (i) [–1, 1) ∪ (1, 2) (ii) [–1, 1] Methods of determining range : (i) Representing x in terms of y Definition of the function is usually represented as y (i.e. f(x) which is dependent variable) in terms of an expression of x (which is independent variable). To find range rewrite given definition so as to represent x in terms of an expression of y and thus obtain range (possible values of y). If y = f(x) ⇔ x = g(y), then domain of g(y) represents possible values of y, i.e. range of f(x).
2
Find the range of f(x) =
Illustration # 4: x + x +1
x2 + x + 1 x2 + x − 1
2
Solution
x2 + x + 1
Illustration # 5: Solution
∴
(iii)
{x 2 + x + 1 and x 2 + x – 1 have no common factor}
x2 + x − 1
⇒ yx 2 + yx – y = x 2 + x + 1 x2 + x − 1 2 ⇒ (y – 1) x + (y – 1) x – y – 1 = 0 If y = 1, then the above equation reduces to –2 = 0. Which is not true. Further if y ≠ 1, then (y – 1) x 2 + (y – 1) x – y – 1 = 0 is a quadratic and has real roots if (y – 1)2 – 4 (y – 1) (–y – 1) ≥ 0 i.e. if y ≤ –3/5 or y ≥ 1 but y ≠ 1 Thus the range is (–∞, –3/5] ∪ (1, ∞) Graphical Method : Values covered on y-axis by the graph of function is range
y=
(ii)
f(x) =
f(x) =
Find the range of f(x) = x2 − 4 x−2
x2 − 4 x−2
= x + 2; x ≠ 2
graph of f(x) would be
Thus the range of f(x) is R – {4} Using Monotonocity/Maxima-Minima (a) Continuous function: If y = f(x) is continuous in its domain then range of f(x) is y ∈ [min f(x), max. f(x)] (b) Sectionally continuous function: In case of sectionally continuous functions, range will be union of [min f(x), max. f(x)] over all those intervals where f(x) is continuous, as shown by following example. Let graph of function y = f(x) is
Then range of above sectionally continuous function is [y 2, y 3] ∪ (y4, y5] ∪ (y 6, y7] Note : In case of monotonic functions minimum and maximum values lie at end points of interval. Illustration # 6 : Find the range of following functions : (i) y = n (2x – x 2) (ii) y = sec–1 (x 2 + 3x + 1) Solution : (i) Step – 1 Using maxima-minima, we have 2x – x 2 ∈ (–∞, 1] Step – 2 For log to be defined accepted values are 2x – x 2 ∈ (0, 1] {i.e. domain (0, 1]} Now, using monotonocity n (2x – x 2) ∈ (–∞, 0] ∴ range is (– ∞, 0] Ans. 3
(ii)
y = sec –1 (x 2 + 3x + 1) Let t = x 2 + 3x + 1 for x ∈ R 5 5 but y = sec –1 (t) ⇒ t ∈ − , − 1 ∪ [1, ∞) then t ∈ − , ∞ 4 4
π −1 5 from graph range is y ∈ 0, ∪ sec − 4 , π 2
Assignment: Find domain and range of following functions. 4. x 2 − 2x + 5 (i) y = x3 (ii) y= 2 x + 2x + 5 Answer
C.
(i)
1
domain R; range R
(iii)
y=
(iv)
y = cot–1 (2x – x 2)
(v)
3 2 y = n sin–1 x + x + Answer 4
x −x 2
Answer Answer
(ii)
3 − 5 3 + 5 domain R ; range 2 , 2
domain R – [0, 1] ; range (0, ∞) π domain R ; range , π 4 − 2 − 5 − 2 + 5 π π , domain x ∈ ; range n 6 , n 2 4 4
Classification of Functions : Functions can be classified as : (i) One − One Function (Injective Mapping) and Many − One Function: One − One Function : A function f : A → B is said to be a one-one function or injective mapping if different elements of A have different f images in B. Thus for x 1, x 2 ∈ A & f(x 1), f(x 2) ∈ B, f(x 1) = f(x 2) ⇔ x 1 = x 2 or x 1 ≠ x 2 ⇔ f(x 1) ≠ f(x 2). Diagrammatically an injective mapping can be shown as
OR Many − One function : A function f : A → B is said to be a many one function if two or more elements of A have the same f image in B. Thus f : A → B is m any one iff there exi sts atleast two elem ents x 1 , x 2 ∈ A, such that f(x 1) = f(x 2) but x 1 ≠ x 2. Diagrammatically a many one mapping can be shown as
OR Note : If a function is one−one, it cannot be many−one and vice versa. Methods of determining whether function is ONE-ONE or MANY-ONE : (a) If x 1, x 2 ∈ A & f(x 1), f(x 2) ∈ B, f(x 1) = f(x 2) ⇔ x 1 = x 2 or x 1 ≠ x 2 ⇔ f(x 1) ≠ f(x 2), then function is ONE-ONE otherwise MANY-ONE. (b) If there exists a straight line parallel to x-axis, which cuts the graph of the function atleast at two points, then the function is MANY-ONE, otherwise ONE-ONE. (c) If either f′(x) ≥ 0, ∀ x ∈ complete domain or f′(x) ≤ 0 ∀ x ∈ complete domain, where equality can hold at discrete point(s) only, then function is ONE-ONE, otherwise MANY-ONE. (ii) Onto function (Surjective mapping) and Into function : Onto function : If the function f : A → B is such that each element in B (co−domain) must have atleast one pre−image in A, then we say that f is a function of A 'onto' B. Thus f : A → B is surjective iff ∀ b ∈ B, there exists some a ∈ A such that f (a) = b. Diagrammatically surjective mapping can be shown as
OR Method of determining whether function is ONTO or INTO : Find the range of given function. If range ≡ co−domain, then f(x) is onto, otherwise into Into function : If f : A → B is such that there exists atleast one element in co−domain which is not the image of any element in domain, then f(x) is into. 4
Diagrammatically into function can be shown as
OR Note : If a function is onto, it cannot be into and vice versa. Thus a function can be one of these four types:
(a)
one−one onto (injective & surjective)
(b)
one−one into (injective but not surjective)
(c)
many−one onto (surjective but not injective)
(d)
many−one into (neither surjective nor injective)
Note :
If f is both injective & surjective, then it is called a bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions. If a set A contains 'n' distinct elements then the number of different functions defined from A → A is nn and out of which n! are one one. Illustration # 7 (i) Find whether f(x) = x + cos x is one-one. Solution The domain of f(x) is R. f′ (x) = 1 − sin x. ∴ f′ (x) ≥ 0 ∀ x ∈ complete domain and equality holds at discrete points only ∴ f(x) is strictly increasing on R. Hence f(x) is 2one-one. 3 (ii) Identify whether the function f(x) = –x + 3x – 2x + 4 ; R → R is ONTO or INTO Solution As codomain ≡ range, therefore given function is ONTO (iii) f(x) = x 2 – 2x + 3; [0, 3] → A. Find whether f(x) is injective or not. Also find the set A, if f(x) is surjective. Solution f′(x) = 2(x – 1); 0 ≤ x ≤ 3 − ve ; 0 ≤ x < 1 f′(x) = + ve ; 1 < x < 3 ∴ f(x) is not monotonic. Hence it is not injective. For f(x) to be surjective, A should be equal to its range. By graph range is [2, 6] ∴ A ≡ [2, 6]
∴
Assignment: 5. For each of the following functions find whether it is one-one or many-one and also into or onto (i) f(x) = 2 tan x; (π/2, 3π/2) → R
one-one onto
Answer
(ii) (iii)
D.
1
; (–∞, 0) → R 1+ x2 Answer one-one into f(x) = x 2 + n x f(x) =
Answer
one-one onto
Various Types of Functions :
Polynomial Function : If a function f is defined by f (x) = a0 x n + a1 x n−1 + a2 x n−2 +... + an−1 x + an where n is a non negative integer and a0, a1, a2,........., an are real numbers and a0 ≠ 0, then f is called a polynomial function of degree n. Note : There are two polynomial functions, satisfying the relation; f(x).f(1/x) = f(x) + f(1/x), which are f(x) = 1 ± x n (ii) Algebraic Function : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form, P0 (x) yn + P1 (x) yn−1 +....... + Pn−1 (x) y + Pn (x) = 0 where n is a positive integer and P0 (x), P1 (x)....... are polynomials in x. e.g. y = x is an algebraic function, since it satisfies the equation y² − x² = 0. Note : All polynomial functions are algebraic but not the converse. A function that is not algebraic is called Transcendental Function . g( x ) (iii) Fractional / Rational Function : A rational function is a function of the form, y = f (x) = , where g (x) h( x ) & h (x) are polynomials and h (x) ≡/ 0. (iv) Exponential Function : A function f(x) = ax = ex In a (a > 0, a ≠ 1, x ∈ R) is called an exponential function. Graph of exponential function can be as follows : Case - Ι Case - ΙΙ For a > 1 For 0 < a < 1 (i)
5
(v)
Logarithmic Function : f(x) = logax is called logarithmic function where a > 0 and a ≠ 1 and x > 0. Its graph can be as follows Case- Ι For a > 1
(vi)
Case- ΙΙ For 0 < a < 1
Absolute Value Function / Modulus Function : x if The symbol of modulus function is f (x) = x and is defined as: y = x= − x if
(vi)
Signum Function :
x≥0 . x<0
A function f (x) = sgn (x) is defined as follows :
1 for x > 0 f (x) = sgn (x) = 0 for x = 0 − 1 for x < 0
| x | ; x≠0 It is also written as sgn x = x 0 ; x = 0 | f ( x ) | ; f ( x) ≠ 0 Note : sgn f(x) = f ( x ) 0 ; f (x) = 0
(vii)
Greatest Integer Function or Step Up Function : The function y = f (x) = [x] is called the greatest integer function where [x] equals to the greatest integer less than or equal to x. For example : for − 1 ≤ x < 0 ; [x] = − 1 ; for 0 ≤ x < 1 ; [x] = 0 for 1 ≤ x < 2 ; [x] = 1 ; for 2 ≤ x < 3 ; [x] = 2 and so on. Alternate Definition : The greatest integer occur on real number line while moving L.H.S. of x (starting from x) is [x]
(a) (c) (viii)
Properties of greatest integer function : x − 1 < [x] ≤ x (b) [x ± m] = [x] ± m iff m is an integer. 0 ; if x is an int eger [x] + [y] ≤ [x + y] ≤ [x] + [y] + 1 (d) [x] + [− x] = − 1 otherwise Fractional Part Function: It is defined as, y = {x} = x − [x]. e.g. the fractional part of the number 2.1 is 2.1 − 2 = 0.1 and the fractional part of − 3.7 is 0.3. The period of this function is 1 and graph of this function is as shown. 6
Identity function : The function f : A → A defined by, f(x) = x ∀ x ∈ A is called the identity function on A and is denoted by ΙA. It is easy to observe that identity function is a bijection. (x) Constant function : A function f : A → B is said to be a constant function, if every element of A has the same f image in B. Thus f : A → B; f(x) = c, ∀ x ∈ A, c ∈ B is a constant function. Illustration # 8 (i) Let {x} & [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x] Solution As x = [x] + {x} 2 [ x] 4{x} = [x] + {x} + [x] ⇒ {x} = ∴ Given equation ⇒ 3 As [x] is always an integer and {x} ∈ [0, 1), possible values are [x] {x} x = [x] + {x} 0 0 0 2 5 1 3 3 5 ∴ There are two solution of given equation x = 0 and x = 3 (ii) Draw graph of f(x) = sgn ( n x) (ix)
Solution
Assignment: 6. If f : R → R satisfying the conditions f(0) = 1, f(1) = 2 and f(x + 2) = 2f (x) + f(x + 1), then find f (6). Answer 64 7. Draw the graph of following functions where [.] denotes greatest integer function (i) y=[2x]+ 1 (ii) y = x [x], 1 ≤ x ≤ 3 (iii) y = sgn (x 2 – x)
Answer (i)
E.
(ii)
(iii)
Odd & Even Functions :
(i) If f (−x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. If f (x) − f (−x) = 0 ⇒ f (x) is even. e.g. f (x) = cos x; g (x) = x² + 3. (ii) If f (−x) = −f (x) for all x in the domain of ‘f’ then f is said to be an odd function. If f (x) + f (−x) = 0 ⇒ f (x) is odd. e.g. f (x) = sin x; g (x) = x 3 + x. Note : A function may neither be odd nor even. (e.g. f(x) = ex , cos–1x) If an odd function is defined at x = 0, then f(0) = 0 Properties of Even/Odd Function (a) Every even function is symmetric about the y−axis & every odd function is symmetric about the origin. For example graph of y = x 2 is symmetric about y-axis, while graph of y = x 3 is symmetric about origin
(b)
All functions (whose domain is symmetrical about origin) can be expressed as the sum of an even & an odd function, as follows f(x) =
(c) The only function which is defined on the entire number line and is even & odd at the same time is f(x) = 0. 7
(d)
If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd. (e) If f(x) is even then f′(x) is odd but converse need not be true. x + x 2 + 1 is an odd function. Illustration # 9: Show that log 2 x + x 2 + 1 . Solution Let f(x) = log Then f(–x) = log − x + ( − x ) + 1 x 2 + 1 − x x 2 + 1 + x 1 2 = log = log – log x + x + 1 = –f(x) 2 2 x + 1 + x x +1 + x Hence f(x) is an odd function. Illustration # 10 Show that ax +a–x is an even function. Let f(x) = ax + a–x Then f(–x) = a–x + a–(–x) = a–x +ax = f(x). Solution Hence f(x) is an even function Illustration # 11 Show that cos–1 x is neither odd nor even. Let f(x) = cos–1x. Then f(–x) = cos–1 (–x) = π – cos–1 x which is neither equal to f(x) nor equal to f(–x). Solution Hence cos–1 x is neither odd nor even Assignment: 8. Determine whether following functions are even or odd? e x + e −x (i) Answer Odd e x − e−x 2 (ii) log x + 1 − x Answer Odd 2 (iii) x log x + x + 1 Answer Even Answer Odd (iv) sin–1 2x 1− x 2 Even extension / Odd extension : Let the defincition of the function f(x) is given only for x ≥ 0. Even extension of this function implies to define the function for x < 0 assuming it to be even. In order to get even extension replace x by –x in the given defincition Similarly, odd extension implies to define the function for x < 0 assuming it to be odd. In order to get odd extension, multiply the definition of even extension by –1 Illustration # 12 What is even and odd extension of f(x) = x 3 – 6x 2 + 5x – 11 ; x > 0 Solution Even extension f(x) = –x 3 – 6x 2 + 5x – 11 ;x<0 Odd extension 3 2 f(x) = x + 6x + 5x + 11 ;x<0 Periodic Function : A function f(x) is called periodic with a period T if there exists a real number T > F. 0 such that for each x in the domain of f the numbers x – T and x + T are also in the domain of f and f(x) = f(x + T) for all x in the domain of 'f'. Domain of a periodic function is always unbounded. Graph of a periodic function with period T is repeated after every interval of 'T'. e.g. The function sin x & cos x both are periodic over 2π & tan x is periodic over π. The least positive period is called the principal or fundamental period of f or simply the period of f. Note : f (T) = f (0) = f (−T), where ‘T’ is the period. Inverse of a periodic function does not exist. Every constant function is always periodic, with no fundamental period. Properties of Periodic Function 1 (a) If f(x) has a period T, then and f( x ) also have a period T. f( x ) T (b) If f(x) has a period T then f (ax + b) has a period | a | . f ( x) (c) If f (x) has a period T 1 & g (x) also has a period T 2 then period of f(x) ± g(x) or f(x) . g(x) or is L.C.M. g( x ) of T1 & T2 provided their L.C.M. exists. However that L.C.M. (if exists) need not to be fundamental period. f ( x) If L.C.M. does not exists f(x) ± g(x) or f(x) . g(x) or is aperiodic. g( x ) e.g. |sinx| has the period π, | cosx | also has the period π π ∴ |sinx| + |cosx| also has a period π. But the fundamental period of |sinx| + |cosx| is . 2 Illustration # 13 Find period of following functions x x (i) f(x) = sin + cos (ii) f(x) = {x} + sin x 3 2 x 2x 3x – cos – tan (iii) f(x) = cos x . cos 3x (iv) f(x) = sin 3 3 2 x x x x Solution (i) Period of sin is 4π while period of cos is 6π . Hence period of sin + cos is 12 π 3 3 2 2 {L.C.M. of 4 & 6 is 12} (ii) Period of sin x = 2p Period of {x} = 1 ∴ it is aperiodic
but L.C.M. of 2π & 1 is not possible 8
f(x) = cos x . cos 3x 2π = 2π period of f(x) is L.C.M. of 2π, 3 (iii)
2π , where n ∈ N. Hence crossn checking for n = 1, 2, 3, ....we find π to be fundamental period f(π + x) = (– cos x) (– cos 3x) = f(x) 2π 2π π , , (iv) Period of f(x) is L.C.M. of 3 / 2 1/ 3 3 / 2 4π 2π = L.C.M. of , 6π , = 12π 3 3 = NOTE : Assignment: 9. Find the period of following function. (i) f(x) = sin x + | sin x | Answer 2π x (ii) f(x) = 3 cos x – sin Answer 6π 3 2x 3x (iii) sin – cos Answer 70 π 5 7 2 4 (iv) f(x) = sin x + cos x Answer π
but 2π may or may not be fundamental periodic, but fundamental period =
L.C.M.(a, p, ) L.C.M. of ba , pq , m H.C.F. (b, q, m)
G.
Composite Function :
(ii)
f(x) = x , g(x) = x 2 − 1. Domain of f is [0, ∞), range of f is [0, ∞). Domain of g is R, range of g is [−1, ∞). Since range of f is a subset of the domain of g, ∴ domain of gof is [0, ∞) and g{f(x)}= g(√x) = x − 1. Range of gof is [−1, ∞) Further since range of g is not a subset of the domain of f i.e. [−1, ∞) ⊄ [0, ∞) ∴ fog is not defined on whole of the domain of g. Domain of fog is {x∈R, the domain of g : g(x)∈ [0, ∞), the domain of f}. Thus the domain of fog is D = {x∈R: 0 ≤ g(x) < ∞} i.e. D = { x∈R: 0 ≤ x 2 − 1}= { x∈R: x ≤ −1 or x ≥ 1 }= (−∞, −1] ∪ [1, ∞) fog (x) = f{g(x)} = f(x 2−1) = x 2 − 1 Its range is [0, ∞).
Let f: X→Y1 and g: Y2→ Z be two functions and the set D = {x∈ X: f(x)∈ Y2}. If D ≡/ φ, then the function h defined on D by h(x) = g{f(x)} is called composite function of g and f and is denoted by gof. It is also called function of a function. Note : Domain of gof is D which is a subset of X (the domain of f ). Range of gof is a subset of the range of g. If D = X, then f(x) ⊂ Y2. Properties of Composite Functions : (a) In general gof ≠ fog (i.e. not commutative) (b) The com posit e of f unct ions are associ at iv e i .e. i f three f unctions f , g, h are such that fo (goh) & (fog) oh are defined, then fo (goh) = (fog) oh. (c) If f and g both are one-one, then gof and fog would also be one-one. (d) If f and g both are onto, then gof or fog may or may not be onto. (e) The composite of two bijections is a bijection iff f & g are two bijections such that gof is defined, then gof is also a bijection only when co-domain of f is equal to the domain of g . If g is a function such that gof is defined on the domain of f and f is periodic with T, then gof is also (f) periodic with T as one of its periods. Further if # g is one-one, then T is the period of gof # g is also periodic with T′ as the period and the range of f is a sub-set of [0, T′ ], then T is the period of gof Illustration # 14 Describe fog and gof wherever is possible for the following functions (i) f(x) = x + 3 , g(x) = 1 + x 2 (ii) f(x) = x , g(x) = x 2 − 1. (i) Domain of f is [−3, ∞), range of f is [0, ∞). Solution Domain of g is R, range of g is [1, ∞). Since range of f is a subset of domain of g, ∴ domain of gof is [−3, ∞) {equal to the domain of f } gof (x) = g{f(x)} = g ( x + 3 ) = 1 + (x+3) = x + 4. Range of gof is [1, ∞). Further since range of g is a subset of domain of f, ∴ domain of fog is R {equal to the domain of g} 2 2 fog (x) = f{g(x)}= f(1+ x ) = x + 4 Range of fog is [2, ∞).
π π Let f(x) = ex ; R+ → R and g(x) = sin–1 x; [–1, 1] → − , . Find domain and range of fog (x) 2 2 Solution π π Domain of f(x) : (0, ∞) Range of g(x) : − , 2 2
(iii)
π values in range of g(x) which are accepted by f(x) are 0, 2 π π ⇒ 0 < g(x) ≤ 0 < sin–1x ≤ 0 9< x ≤ 1 2 2
Hence domain of fog(x) is x ∈ (0, 1]
Therefore
Domain : (0, 1] Range : (1, eπ/2] Example of composite function of non-uniformly defined functions : Illustration # 15 If f(x) = | |x – 3| – 2 | 0≤x≤4 g(x) = 4 – |2 – x| –1≤x≤3 then find fog(x) and draw rough sketch of fog(x). Solution f(x) = | | x – 3| – 2| 0 ≤ x ≤ 4 | x − 1 | 0 ≤ x < 3 = | x − 5 | 3 ≤ x ≤ 4 1 − x 0 ≤ x < 1 x −1 1≤ x < 3 = 5 − x 3 ≤ x ≤ 4 g(x) = 4 – |2 – x| −1 ≤ x ≤ 3 4 − ( 2 − x ) − 1 ≤ x < 2 = 4 − ( x − 2) 2 ≤ x ≤ 3
=
∴
=
=
2 + x − 1 ≤ x < 2 6 − x 2 ≤ x ≤ 3 1 − g( x ) 0 ≤ g( x ) < 1 g( x ) − 1 1 ≤ g( x ) < 3 fog (x) = 5 − g( x ) 3 ≤ g( x ) ≤ 4
1 − ( 2 + x ) 2 + x −1 5 − (2 + x ) 1− 6 + x 6 − x −1 5 − 6 + x
0 ≤ 2 + x < 1 and − 1 ≤ x < 2 1 ≤ 2 + x < 3 and − 1 ≤ x < 2 3 ≤ 2 + x ≤ 4 and − 1 ≤ x < 2 0 ≤ 6 − x < 1 and 1 ≤ 6 − x ≤ 3 and
2≤x≤3 2≤x≤3
3 ≤ 6 − x ≤ 4 and
2≤x≤3
−2≤ x <1 − 1 − x −1≤ x < 1 1+ x 3 − x 1≤ x ≤ 2 x − 5 − 6 ≤ − x < −5 5 − x − 5 ≤ − x < −3 x − 1 − 3 ≤ − x ≤ −2
− 1 − x − 2 ≤ x < −1 −1≤ x < 1 1+ x 3 − x 1≤ x ≤ 2 = x − 5 5
and − 1 ≤ x < 2 and − 1 ≤ x < x and − 1 ≤ x < 2 and 2 ≤ x ≤ 3 and and
2≤x≤3 2≤x≤3
and − 1 ≤ x < 2 and − 1 ≤ x < 2 and − 1 ≤ x < 2 and and and
2≤x≤3 2≤x≤3 2≤x≤3
graph of g(x) is 10
1 − g( x ) 0 ≤ g( x ) < 1 g( x ) − 1 1 ≤ g( x ) < 3 ∴ fog(x) = 5 − g( x ) 3 ≤ g( x ) ≤ 4 1 − g( x ) for no value 2 + x −1 −1≤ x < 1 x +1 −1≤ x < 1 g( x ) − 1 −1≤ x < 1 5 − (2 + x ) 1 ≤ x < 2 3 − x 1≤ x < 2 = = = 5 − g( x ) 5 − (6 − x ) 2 ≤ x ≤ 3 x −1 2 ≤ x ≤ 3 ≤ 1 ≤ x 3 Assignment: 10. Define fog(x) and gof(x). Also their Domain & Range. (i) f(x) = [x], g(x) = sin x (ii) f(x) = tan x, x ∈ (–π/2, π/2); g(x) = 1− x 2 gof = sin [x] Answer (i) domain : R range { sin a : a ∈ Ι} fog = [ sin x] domain : R range : {–1, 0, 1} 1 − tan 2 x π π domain : − , 4 4
(ii)
Answer
gof =
range : [0, 1]
fog = tan 1− x 2 domain : [–1, 1] range [0, tan 1] Let f(x) = e : R → R and g(x) = x 2 – x : R → R. Find domain and range of fog (x) & gof (x) 11. Answer fog (x) gof f(x) Domain : (0, ∞) Domain : (–∞, 0) ∪ (1, ∞) 1 Range : [1, ∞) Range : − , ∞ 4 Inverse of a Function : Let f : A → B be a function. Then f is invertible iff there is a function g : B H. → A such that go f is an identity function on A and fog is an identity function on B. Then g is called inverse of f and is denoted by f −1. For a function to be invertible it must be bijective Note : The inverse of a bijection is unique. Inverse of an even function is not defined. Properties of Inverse Function : (a) The graphs of f & g are the mirror images of each other in the line y = x. For example f(x) = ax and g(x) = loga x are inverse of each other, and their graphs are mirror images of each other on the line y = x as shown below. x
+
Normally points of intersection of f and f –1 lie on the straight line y = x. However it must be noted that f(x) –1 and f (x) may intersect otherwise also. (c) In general fog(x) and gof(x) are not equal but if they are equal then in majority of cases either f and g are inverse of each other or atleast one of f and g is an identity function. (d) I f f & g are t wo bi j ect i on s f : A → B, g : B → C t hen t he i nv erse of gof ex i sts and (gof)−1 = f −1 o g−1. 1 (e) If f(x) and g are inverse function of each other then f′(g(x)) = g′( x ) 2x + 3 Illustration # 16 (i) Determine whether f(x) = ; R → R, is invertible or not? If so find it. 4 2x + 3 Solution: As given function is one-one and onto, therefore it is invertible. y = 4 4y − 3 4x − 3 –1 ⇒ x= ∴ f (x) = 2 2 –1 2 (ii) Is the function f(x) = sin 2 x 1 − x invertible? Solution: Domain of f is [–1, 1] and f is continuous (b)
−1 1 2 if 1 − x 2 2 2 −1 1 and is decreasing in each of the intervals , f(x) is increasing in 2 2
(
2 1 − 2x 2
∴
)
−1 −1 − 1, and , 1 2 2
11
∴ (iii)
f(x) is not one-one, so is not invertible. Let f(x) = x 2 + 2x; x ≥ –1. Draw graph of f –1(x) also find the number of solutions of the equation, f(x) = f –1(x)
Solution
f(x) = f –1(x) is equilavent to solving y = f(x) and y = x ⇒ x 2 + 2x = x ⇒ x(x + 1) = 0 ⇒ x = 0, –1 Hence two solution for f(x) = f –1(x) (iv) If y = f(x) = x 2 – 3x + 1, x ≥ 2. Find the value of g′(1) where g is inverse of f Solution y=1 ⇒ x 2 – 3x + 1 = 1 But x≥2 ∴ x=3 Now g(f(x)) = x Differentiating both sides w.r.t. x ⇒
⇒
g′(f(x)). f′(x) = 1
1 f ′(3) Alternate Method y = x 2 – 3x + 1 x 2 – 3x +1 – y = 0
⇒
⇒
g′(f(3)) =
x= =
x (x – 3) = 0
1 f ′( x ) 1 g′ (1) = = 6−3
x = 0, 3
g′(f(x)) =
(As f′(x) = 2x – 3)
=
1 3
2 3 ± 5 + 4y
2
3 + 5 + 4y
g(x) =
2 3 + 5 + 4y
g′(x) = 0 +
2 1
x
g′(1) =
1
x 5 + 4 x–1 5+4 Assignment: 12. Determine f (x), if given function is invertible 2 (i) f : (–∞, –1) → (–∞, –2) defined f(x) = –(x + 1) – 2 π π 7π (ii) f: , [–1, 1] defined by f(x) = sin x + → 3 6 6 2π –1 Answer (i) – 1 + − x − 2 (ii) – sin x 3
I.
⇒
3 ± 9 − 4(1 − y )
x≥2 x=
⇒
=
1 9
=
1 3
Equal or Identical Function :
Two functions f & g are said to be identical (or equal) iff : (i) The domain of f ≡ the domain of g. (ii) The range of f ≡ the range of g and x 1 (iii) f(x) = g(x), for every x belonging to their common domain. e.g. f(x) = & g(x) = 2 are identical functions. x x x2 But f(x) = x and g(x) = are not identical functions. x Illustration # 17 Examine whether following pair of functions are identical or not x2 & g(x) = x Answer No, as domain of f(x) is R – {0} while domain of g(x) is R x2 2 2 (ii) f(x) = sin x + cos x & g(x) = sec x – tan2x π Answer No, as domain are not same. Domain of f(x) is R while that of g(x) is R – (2n + 1) ; n ∈ I 2 (i)
Assignment:
(i)
f(x) =
Examine whether following pair of functions are identical or not x x≠0 f(x) = sgn (x) & g(x) = | x | 12 0 x = 0 13.
π Answer (i) Yes (ii) No 2 General : If x, y are independent variables, then: (i) f (xy) = f (x) + f (y) ⇒ f (x) = k ln x or f (x) = 0. (ii) f (xy) = f (x). f (y) ⇒ f (x) = x n, n ∈ R (iii) f (x + y) = f (x). f (y) ⇒ f (x) = akx. (iv) f (x + y) = f (x) + f (y) ⇒ f(x) = kx, where k is a constant. 1 1 ⇒ f(x) = 1 ± x n where n ∈ N (v) f(x) . f = f(x) + f x x
(ii)
J.
f(x) = sin–1x + cos–1x & g(x) =
1 1 If f(x) is a polynomial function satisfying f(x) . f = f(x) + f ∀ x ∈ R – {0} and x x f(2) = 9, then find f (3) Solution f(x) = 1 ± x n As f(2) = 9 ∴ f(x) = 1 + x 3 Hence f(3) = 1 + 33 = 28 1 1 Assignment: 14. If f(x) is a polynomial function satisfying f(x) . f = f(x) + f ∀ x ∈ R – {0} and f(3) = –8, x x then find f(4) Answer – 15
Illustration # 18
15.
If f(x + y) = f(x) . f(y) for all real x, y and f(0) ≠ 0 then prove that the function, g(x) =
13
f(x) 1 + f 2 ( x)
is an even function
SHORT REVISION (FUNCTIONS) THINGS TO REMEMBER : 1.
GENERAL DEFINITION : If to every value (Considered as real unless other−wise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only one finite value of the quantity y, then y is said to be a function (Single valued) of x or a dependent variable defined on the set E ; x is the argument or independent variable . If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word "FUNCTION” is used only as the meaning of a single valued function, if not otherwise stated. x Pictorially : →
input
f (x ) = y output
→ , y is called the image of x & x is the pre-image of y under f.
Every function from A → B satisfies the following conditions . (i) f ⊂ AxB (ii) ∀ a ∈ A ⇒ (a, f(a)) ∈ f (iii) (a, b) ∈ f & (a, c) ∈ f ⇒ b = c
and
2.
DOMAIN, CO− −DOMAIN & RANGE OF A FUNCTION : Let f : A → B, then the set A is known as the domain of f & the set B is known as co-domain of f . The set of all f images o f elements of A is known as the range of f . Thus : Domain of f = {a a ∈ A, (a, f(a)) ∈ f} Range of f = {f(a) a ∈ A, f(a) ∈ B} It should be noted that range is a subset of co−domain . If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.
3. (i)
IMPORTANT TYPES OF FUNCTIONS : POLYNOMIAL FUNCTION : If a function f is defined by f (x) = a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an where n is a non negative integer and a0, a1, a2, ..., an are real numbers and a0 ≠ 0, then f is called a polynomial function of degree n . NOTE : (a) A polynomial of degree one with no constant term is called an odd linear function . i.e. f(x) = ax , a ≠ 0 (b)
There are two polynomial functions , satisfying the relation ; f(x).f(1/x) = f(x) + f(1/x). They are : (i) f(x) = xn + 1 & (ii) f(x) = 1 − xn , where n is a positive integer .
(ii)
ALGEBRAIC FUNCTION : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form P0 (x) yn + P1 (x) yn−1 + ....... + Pn−1 (x) y + Pn (x) = 0 Where n is a positive integer and P0 (x), P1 (x) ........... are Polynomials in x. e.g. y = x is an algebraic function, since it satisfies the equation y² − x² = 0. Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called TRANSCEDENTAL FUNCTION .
(iii)
FRACTIONAL RATIONAL FUNCTION : A rational function is a function of the form. y = f (x) =
(IV)
g(x ) h (x )
, where
g (x) & h (x) are polynomials & h (x) ≠ 0. EXPONENTIAL FUNCTION : A function f(x) = ax = ex ln a (a > 0 , a ≠ 1, x ∈ R) is called an exponential function. The inverse of the exponential function is called the logarithmic function . i.e. g(x) = loga x . Note that f(x) & g(x) are inverse of each other & their graphs are as shown .
14
+∞
>1 , a a (0, 1) ) =← f(x
+∞
x
)45º
y
)= g(x
x
lo g a
x
=
=
x
→
)45º (1, 0)
y
(1, 0)
(v)
f(x) = ax , 0 < a < 1
(0, 1)
g(x) = loga x
ABSOLUTE VALUE FUNCTION : A function y = f (x) = x is called the absolute value function or Modulus function. It is defined as x
if x ≥ 0
: y = x= − x if x < 0 SIGNUM FUNCTION : A function y= f (x) = Sgn (x) is defined as follows :
y
1 for x > 0 y = f (x) = 0 for x = 0 − 1 for x < 0
(b) (c) (d)
4.
[x] ≤ x < [x] + 1 and x − 1 < [x] ≤ x , 0 ≤ x − [x] < 1 [x + m] = [x] + m if m is an integer . [x] + [y] ≤ [x + y] ≤ [x] + [y] + 1 [x] + [− x] = 0 if x is an integer = − 1 otherwise .
y = −1 if x < 0
y
graph of y = [x]
3 •
2 1 −3 3
FRACTIONAL PART FUNCTION : It is defined as : g (x) = {x} = x − [x] . e.g. the fractional part of the no. 2.1 is 2.1− 2 = 0.1 and the fractional part of − 3.7 is 0.3. The period of this function is 1 and graph of this function is as shown .
−2 •
•
• º −1 • º −1 −2 º
º
º 1
x
2
−3
y
• −1
graph of y = {x}
1− − − º º −− • • 1
º • 2
DOMAINS AND RANGES OF COMMON FUNCTION : Function (y = f (x) )
A.
y = Sgn x
GREATEST INTEGER OR STEP UP FUNCTION : The function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x . Note that for : −1 ≤ x < 0 ; [x] = − 1 0≤x< 1 ; [x] = 0 1≤x< 2 ; [x] = 1 2≤x < 3 ; [x] = 2 and so on . Properties of greatest integer function : (a)
(viii)
> x
O
It is also written as Sgn x = |x|/ x ; x ≠ 0 ; f (0) = 0 (vii)
y = 1 if x > 0
Domain (i.e. values taken by x)
Range (i.e. values taken by f (x) )
R, if n is odd + R ∪ {0} , if n is even
Algebraic Functions (i)
xn , (n ∈ N)
R = (set of real numbers)
(ii)
1 ∈ x n , (n N)
R – {0}
15
R – {0} , if n is odd
º
−−−− −−
(vi)
•
x
R+ , (iii)
(iv)
B.
1 x
1/ n
, (n ∈ N)
R – {0} , if n is odd
R – {0} , if n is odd
R+ ,
R+ ,
if n is even
R R
(iii)
tan x
R – (2k + 1)
π , k ∈I 2
R
π , k ∈I 2 (v) cosec x R – kπ , k ∈ I (vi) cot x R – kπ , k ∈ I Inverse Circular Functions (Refer after Inverse is taught )
EQUAL OR IDENTICAL FUNCTION : Two functions f & g are said to be equal if : The domain of f = the domain of g. The range of f = the range of g and f(x) = g(x) , for every x belonging to their common domain. eg. x 1 f(x) = & g(x) = 2 are identical functions . x x CLASSIFICATION OF FUNCTIONS : One − One Function (Injective mapping) : A function f : A → B is said to be a one−one function or injective mapping if different elements of A have different f images in B . Thus for x1, x2 ∈ A & f(x1) , f(x2) ∈ B , f(x1) = f(x2) ⇔ x1 = x2 or x1 ≠ x2 ⇔ f(x1) ≠ f(x2) . Diagramatically an injective mapping can be shown as
OR Note : (i)
Any function which is entirely increasing or decreasing in whole domain, then f(x) is one−one . (ii) If any line parallel to x−axis cuts the graph of the function atmost at one point, then the function is one−one . Many–one function : A function f : A → B is said to be a many one function if two or more elements of A have the same f image in B . Thus f : A → B is many one if for17; x1, x2 ∈ A , f(x1) = f(x2) but x1 ≠ x2 .
Diagramatically a many one mapping can be shown as
OR Note : (i)
(ii)
Any continuous function which has atleast one local maximum or local minimum, then f(x) is many−one . In other words, if a line parallel to x−axis cuts the graph of the function atleast at two points, then f is many−one . If a function is one−one, it cannot be many−one and vice versa .
Onto function (Surjective mapping) : If the function f : A → B is such that each element in B (co−domain) is the f image of atleast one element in A, then we say that f is a function of A 'onto' B . Thus f : A → B is surjective iff ∀ b ∈ B, ∃ some a ∈ A such that f (a) = b . Diagramatically surjective mapping can be shown as
OR Note that : if range = co−domain, then f(x) is onto. Into function : If f : A → B is such that there exists atleast one element in co−domain which is not the image of any element in domain, then f(x) is into . Diagramatically into function can be shown as
OR Note that : If a function is onto, it cannot be into and vice versa . A polynomial of degree even will always be into.
Thus a function can be one of these four types : (a)
one−one onto (injective & surjective)
(b)
one−one into (injective but not surjective)
(c)
many−one onto (surjective but not injective)
(d)
many−one into (neither surjective nor injective)
Note : (i) (ii)
If f is both injective & surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions. If a set A contains n distinct elements then the number of different functions defined from A → A is nn & out of it n ! are one one.
Identity function : The function f : A → A defined by f(x) = x ∀ x ∈ A is called the identity of A and is denoted by IA. It is easy to observe that identity function is a bijection . 18
Constant function : A function f : A → B is said to be a constant function if every element of A has the same f image in B . Thus f : A → B ; f(x) = c , ∀ x ∈ A , c ∈ B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into . 7.
ALGEBRAIC OPERATIONS ON FUNCTIONS : If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in A ∩ B. Now we define f + g , f − g , (f . g) & (f/g) as follows : (i) (f ± g) (x) = f(x) ± g(x) (f . g) (x) = f(x) . g(x) (ii) (iii)
8.
f f (x) (x) = g g (x)
domain is {x x ∈ A ∩ B s . t g(x) ≠ 0} .
COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS : Let f : A → B & g : B → C be two functions . Then the function gof : A → C defined by (gof) (x) = g (f(x)) ∀ x ∈ A is called the composite of the two functions f & g . f (x)
x Diagramatically → → g (f(x)) . → Thus the image of every x ∈ A under the function gof is the g−image of the f−image of x . Note that gof is defined only if ∀ x ∈ A, f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f & g, the range of f must be a subset of the domain of g. PROPERTIES OF COMPOSITE FUNCTIONS : (i) The composite of functions is not commutative i.e. gof ≠ fog . (ii) The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog) oh are defined, then fo (goh) = (fog) oh . (iii) The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection.
9.
10. 11.
12.
HOMOGENEOUS FUNCTIONS : A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables . For example 5 x2 + 3 y2 − xy is homogeneous in x & y . Symbolically if , f (tx , ty) = tn . f (x , y) then f (x , y) is homogeneous function of degree n . BOUNDED FUNCTION : A function is said to be bounded if f(x) ≤ M , where M is a finite quantity . IMPLICIT & EXPLICIT FUNCTION : A function defined by an equation not solved for the dependent variable is called an IMPLICIT FUNCTION . For eg. the equation x3 + y3 = 1 defines y as an implicit function. If y has been expressed in terms of x alone then it is called an EXPLICIT FUNCTION. INVERSE OF A FUNCTION : Let f : A → B be a one−one & onto function, then their exists a unique function g : B → A such that f(x) = y ⇔ g(y) = x, ∀ x ∈ A & y ∈ B . Then g is said to be inverse of f . Thus g = f−1 : B → A = {(f(x), x) (x, f(x)) ∈ f} . PROPERTIES OF INVERSE FUNCTION : (i) The inverse of a bijection is unique . (ii) If f : A → B is a bijection & g : B → A is the inverse of f, then fog = IB and gof = IA , where IA & IB are identity functions on the sets A & B respectively. Note that the graphs of f & g are the mirror images of each other in the line y = x . As shown in the figure given below a point (x ',y ' ) corresponding to y = x2 (x >0)
changes to (y ',x ' ) corresponding to y = + x , the changed form of x = y .
19
(iii) (iv) 13.
The inverse of a bijection is also a bijection . If f & g are two bijections f : A → B , g : B → C then the inverse of gof exists and (gof)−1 = f−1 o g−1 .
ODD & EVEN FUNCTIONS : If f (−x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x ; g (x) = x² + 3 . If f (−x) = −f (x) for all x in the domain of ‘f’ then f is said to be an odd function. e.g. f (x) = sin x ; g (x) = x3 + x .
NOTE : (a) f (x) − f (−x) = 0 => f (x) is even & f (x) + f (−x) = 0 => f (x) is odd . (b) A function may neither be odd nor even . (c) Inverse of an even function is not defined . (d) Every even function is symmetric about the y−axis & every odd function is symmetric about the origin . (e) Every function can be expressed as the sum of an even & an odd function.
e.g. f ( x) =
(f) (g)
f ( x) + f ( − x ) f ( x) − f ( − x) + 2 2
The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0. If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd .
14.
PERIODIC FUNCTION : A function f(x) is called periodic if there exists a positive number T (T > 0) called the period of the function such that f (x + T) = f(x), for all values of x within the domain of x. e.g. The function sin x & cos x both are periodic over 2π & tan x is periodic over π . NOTE : (a) f (T) = f (0) = f (−T) , where ‘T’ is the period . (b) Inverse of a periodic function does not exist . (c) Every constant function is always periodic, with no fundamental period . (d) If f (x) has a period T & g (x) also has a period T then it does not mean that f (x) + g (x) must have a period T . e.g. f (x) = sinx + cosx.
15.
1 and f (x )
(e)
If f(x) has a period p, then
(f)
if f(x) has a period T then f(ax + b) has a period T/a (a > 0) .
f (x) also has a period p .
GENERAL : If x, y are independent variables, then : (i) f(xy) = f(x) + f(y) ⇒ f(x) = k ln x or f(x) = 0 . (ii) f(xy) = f(x) . f(y) ⇒ f(x) = xn , n ∈ R (iii) f(x + y) = f(x) . f(y) ⇒ f(x) = akx . (iv) f(x + y) = f(x) + f(y) ⇒ f(x) = kx, where k is a constant .
EXERCISE–1 Q.1
Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.)
(i) f (x) = cos2x + 16 − x 2
(ii) f (x) = log7 log5 log3 log2 (2x3 + 5x2 − 14x)
(iii) f (x) = ln x 2 − 5x − 24 − x − 2
(iv) f (x) =
(v) y = log10 sin (x − 3) + 16 − x 2
2 log10 x + 1 log100x (vi) f (x) = 20 −x
log10 (log10 x ) − log10 ( 4 − log10 x ) − log10 3
1 2−| x|
1
+
sec(sin x)
(7 x − 5 − 2x ) + ln 2
7 − x 2
(xix) If f(x) = x 2 − 5 x + 4 & g(x) = x + 3 , then find the domain of
5
(
)
2 (sin x − cos x) + 3
(ii) y =
x (iv) f (x) = 1+ | x |
Q.5
f (x) . g
2x 1+ x2
(iii) f(x) =
x 2 − 3x + 2 x2 + x − 6
(v) y = 2 − x + 1+ x
x +4 −3 x −5 Draw graphs of the following function , where [ ] denotes the greatest integer function. (i) f(x) = x + [x] (ii) y = (x)[x] where x = [x] + (x) & x > 0 & x ≤ 3 (iii) y = sgn [x] (iv) sgn (x −x) Classify the following functions f(x) definzed in R → R as injective, surjective, both or none .
(vi) f (x) = log(cosec x - 1) (2 − [sin x] − [sin x]2)
Q.4
−1
Find the domain & range of the following functions . ( Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) y = log
1 . Let f2(x) denote f [f (x)] and f3(x) denote f [f {f(x)}]. Find f3n(x) where n is a natural 1− x
(b) f(x) = x3 − 6 x2 + 11x − 6
(c) f(x) = (x2 + x + 5) (x2 + x − 3)
number. Also state the domain of this composite function. Q.6 Q.7
π π 5 If f(x) = sin²x + sin² x + + cos x cos x + and g = 1 , then find (gof) (x). 3 3 4
The function f(x) is defined on the interval [0,1]. Find the domain of definition of the functions. (a) f (sin x) (b) f (2x+3)
Q.8(i) Find whether the following functions are even or odd or none (a) f(x) = log x + 1 + x 2 (d) f(x) = x sin2 x − x3
(b) f(x) =
(
)
x ax +1 a −1 x
(e) f(x)= sin x − cos21 x
(c) f(x) = sin x + cos x
(1 + 2 ) (f) f(x) = x
2x
2
(g) f(x)=
x x + +1 e −1 2
(h) f(x) = [(x+1)²]1/3 + [(x −1)²]1/3
x
(ii) If f is an even function defined on the interval (−5, 5), then find the 4 real values of x satisfying the x +1 equation f (x) = f .. x+2
Q.9
Write explicitly, functions of y defined by the following equations and also find the domains of definition of the given implicit functions : (a) 10x + 10y = 10 (b) x + y= 2y
Q.10 Show if f(x) = n a − x n , x > 0 n ≥ 2 , n ∈ N , then (fof) (x) = x . Find also the inverse of f(x). Q.11
(a)
Represent the function f(x) = 3x as the sum of an even & an odd function.
(b)
For what values of p ∈ z , the function f(x) = n x p , n ∈ N is even.
Q.12 A function f defined for all real numbers is defined as follows for x ≥ 0 : f ( x) = [1x,,x0>≤1x≤1 How is f defined for x ≤ 0 if : (a) f is even
Q.13 If f (x) = max x ,
(b) f is odd?
1 for x > 0 where max (a, b) denotes the greater of the two real numbers a and b. x
Define the function g(x) = f(x) . f 1 and plot its graph. x
Q.14 The function f (x) has the property that for each real number x in its domain, 1/x is also in its domain and 1 f (x) + f = x. Find the largest set of real numbers that can be in the domain of f (x)? x Q.15 Compute the inverse of the functions: x
(a) f(x) = ln x + x 2 + 1 1
x −1 (b) f(x) = 2
3
(c) y =
10 x − 10 − x 10 x + 10 − x
Q.16 A function f : , ∞ → , ∞ defined as, f(x) = x2 − x + 1. Then solve the equation f (x) = f −1 (x). 2 4 Q.17 Function f & g are defined by f(x) = sin x, x∈R ; g(x) = tan x , x∈R − K + 1 π
where K ∈ I . Find
(i) periods of fog & gof.
Q.18 Find the period for each of the following functions : (a) f(x)= sin4x + cos4x (b) f(x) = cosx (d) f(x)= cos
2
(ii) range of the function fog & gof .
(c) f(x)= sinx+cosx
3 2 x − sin x . 5 7
Q.19 Prove that the functions ; (c) f(x) = x + sin x
(a) f(x) = cos x (d) f(x) = cos x2
(b) f(x) = sin x are not periodic .
Q.20 Find out for what integral values of n the number 3π is a period of the function : f(x) = cos nx . sin (5/n) x.
EXERCISE–2 Q.1
Let f be a one−one function with domain {x,y,z} and range {1,2,3}. It is given that exactly one of the following statements is true and the remaining two are false . f(x) = 1 ; f(y) ≠ 1 ; f(z) ≠ 2 . Determine f−1(1)
Q.2 (a)
Solve the following problems from (a) to (e) on functional equation. The function f (x) defined on the real numbers has the property that f ( f ( x ) )· (1 + f ( x ) ) = – f (x) for all x in the domain of f. If the number 3 is in the domain22and range of f, compute the value of f (3).
(b)
Suppose f is a real function satisfying f (x + f (x)) = 4 f (x) and f (1) = 4. Find the value of f (21).
(c)
Let 'f' be a function defined from R+ → R+ . If [ f (xy)]2 = x ( f ( y) )2 for all positive numbers x and y and
f (2) = 6, find the value of f (50). (d)
Let f (x) be a function with two properties (i) for any two real number x and y, f (x + y) = x + f (y) and (ii) f (0) = 2. Find the value of f (100).
(e)
Let f be a function such that f (3) = 1 and f (3x) = x + f (3x – 3) for all x. Then find the value of f (300).
Q.3(a) A function f is defined for all positive integers and satisfies f(1) = 2005 and f(1)+ f(2)+ ... + f(n) = n2f(n) for all n > 1. Find the value of f(2004). (b) If a, b are positive real numbers such that a – b = 2, then find the smallest value of the constant L for which
x 2 + ax − x 2 + bx < L for all x > 0.
(c) Let f (x) = x2 + kx ; k is a real number. The set of values of k for which the equation f (x) = 0 and f ( f ( x ) ) = 0 have same real solution set. (d) If f (2x + 1) = 4x2 + 14x, then find the sum of the roots of the equation f (x) = 0. ax + b 5 for real a, b and c with a ≠ 0. If the vertical asymptote of y = f (x) is x = – and the Q.4 Let f (x) = 4x + c 4 3 vertical asymptote of y = f –1 (x) is x = , find the value(s) that b can take on. 4 Q.5
A function f : R → R satisfies the condition, x2 f (x) + f (1 – x) = 2x – x4 . Find f (x) and its domain and range.
Q.6
Suppose p(x) is a polynomial with integer coefficients. The remainder when p(x) is divided by x – 1 is 1 and the remainder when p(x) is divided by x – 4 is 10. If r (x) is the remainder when p(x) is divided by (x – 1)(x – 4), find the value of r (2006). − | ln{ x }|
Q.7
e Prove that the function defined as , f (x) = {x}
− {x}
1 | ln{ x }|
where ever it exists otherwise , then
f (x) is odd as well as even. ( where {x} denotes the fractional part function ) Q.8
In a function Prove that
Q.9
1
1
πx
π
+ x cos 2 f(x) + xf − 2f 2 sin π x + = 4 cos2 x 2 x 4 (i) f(2) + f(1/2) = 1 and (ii) f(2) + f(1) = 0
A function f , defined for all x , y ∈ R is such that f (1) = 2 ; f (2) = 8 & f (x + y) − k xy = f (x) + 2 y2 , where k is some constant . Find f (x) & show that : 1 = k for x + y ≠ 0. x + y
f (x + y) f
Q.10 Let ‘f’ be a real valued function defined for all real numbers x such that for some positive constant ‘a’ the 2 equation f (x + a ) = + f (x) − ( f (x)) holds for all x . Prove that the function f is periodic .
1 2
Q.11
f (x) = −1 + x − 2 , 0 ≤ x ≤ 4 g (x) = 2 − x , − 1 ≤ x ≤ 3 Then find fog (x) & gof (x) . Draw rough sketch of the graphs of fog (x) & gof (x) .
If
Q.12 Find the domain of definition of the implicit function defined by the implicit equation , 4
3y + 2x = 24 x
2 − 1
.
23
Q.13 Let {x} & [x] denote the fractional and integral part of a real number x respectively. Solve 4{x}= x + [x] 9x Q.14 Let f (x) = x then find the value of the sum 9 +3
Q.15 Let f (x) = (x + 1)(x + 2)(x + 3)(x + 4) + 5 where x ∈ [–6, 6]. If the range of the function is [a, b] where a, b ∈ N then find the value of (a + b). Q.16 Find a formula for a function g (x) satisfying the following conditions (a) domain of g is (– ∞, ∞) (b) range of g is [–2, 8] (c) g has a period π and (d) g (2) = 3 3 4 Q.17 The set of real values of 'x' satisfying the equality + = 5 (where [ ] denotes the greatest integer x x b b function) belongs to the interval a , where a, b, c ∈ N and is in its lowest form. Find the value of c c a + b + c + abc.
Q.18 Find the set of real x for which the function f(x) =
[
1 is not defined, where [x] x − 1 + 12 − x − 11
] [
]
denotes the greatest integer function. Q.19 A is a point on the circumference of a circle. Chords AB and AC divide the area of the circle into three equal parts . If the angle BAC is the root of the equation, f (x) = 0 then find f (x) . Q.20 If for all real values of u & v, 2 f(u) cos v = f(u + v) + f(u − v), prove that, for all real values of x (i) f(x) + f(− x) = 2a cos x (ii) f(π − x) + f(− x) = 0 (iii) f(π − x) + f(x) = − 2b sin x . Deduce that f(x) = a cos x − b sin x, a, b are arbitrary constants.
EXERCISE–3 Q.1
If the functions f , g , h are defined from the set of real numbers R to R such that ; 0, if x ≤ 0 f (x)= x2 − 1, g (x) = x 2 + 1 , h(x) = ; then find the composite function ho(fog) & determine x , if x ≥ 0
whether the function (fog) is invertible & the function h is the identity function.
(
)
[REE '97, 6]
2
Q.2(a) If g (f(x)) = sin x & f (g(x)) = sin x , then : (A) f(x) = sin2 x , g(x) = x
(B) f(x) = sin x , g(x) = x
(C) f(x) = x2 , g(x) = sin x
(D) f & g cannot be determined
(b) If f(x) = 3x − 5, then f−1(x) (A) is given by
1 3x − 5
(B) is given by
x +5 3
(C) does not exist because f is not one−one (D) does not exist because f is not onto [JEE'98, 2 + 2] Q.3
If the functions f & g are defined from the set of real numbers R to R such that f(x) = ex, g(x) = 3x − 2, then find functions fog & gof. Also find the domains of functions (fog)−1 & (gof)−1. [ REE '98, 6 ]
Q.4
If the function f : [1, ∞) → [1, ∞) is defined by f(x) = 2x (x − 1), then f−1(x) is : 1 2
(A)
x (x − 1)
(B)
(
1 1 + 1 + 4 log2 x 2
)
(C)
(
1 1 − 1 + 4 log2 x 2 24
)
(D) not defined
[ JEE '99, 2 ]
Q.5 Q.6
The domain of definition of the function, y (x) given by the equation, 2x + 2y = 2 is : (A) 0 < x ≤ 1 (B) 0 ≤ x ≤ 1 (C) − ∞ < x ≤ 0 (D) − ∞ < x < 1 Given x = {1, 2, 3, 4}, find all one−one, onto mappings, f : X → X such that, f (1) = 1 , f (2) ≠ 2 and f (4) ≠ 4 . [ REE 2000, 3 out of 100 ]
− 1 , x < 0 Q.7(a) Let g (x) = 1 + x − [ x ] & f (x) = 0 , x = 0 . Then for all x , f (g (x)) is equal to 1 , x>0
(A) x
(B) 1
(C) f (x)
(D) g (x)
1 (b) If f : [1 , ∞) → [2 , ∞) is given by , f (x) = x + , then f −1 (x) equals x 2 x x+ x −4 x − x2 − 4 (B) (C) (D) 1 − (A) 2 1+ x 2 2 log (x + 3) (c) The domain of definition of f (x) = 2 2 is : x + 3x + 2
x2 − 4
(d)
(A) R \ {− 1, − 2} (B) (− 2, ∞) (C) R\{− 1, − 2, − 3} (D) (− 3, ∞) \ {− 1, − 2} Let E = {1, 2, 3, 4 } & F = {1, 2}. Then the number of onto functions from E to F is (A) 14 (B) 16 (C) 12 (D) 8
(e)
Let f (x) = (A)
αx , x ≠ − 1 . Then for what value of α is f (f (x)) = x ? x+1
(B) − 2
2
(D) − 1.
(C) 1
Q.8(a) Suppose f(x) = (x + 1)2 for x > –1. If g(x) is the function whose graph is the reflection of the graph of f(x) with respect to the line y = x, then g(x) equals 1 , x > –1 (C) x + 1 , x > –1 (D) x – 1, x > 0 (x + 1) 2 (b) Let function f : R → R be defined by f (x) = 2x + sinx for x ∈ R. Then f is (A) one to one and onto (B) one to one but NOT onto (C) onto but NOT one to one 2 (D) neither one to one nor onto x +x+2 is Q.9(a) Range of the function f (x) = 2 x + x +1 7 7 (A) [1, 2] (B) [1, ∞ ) (C) 2 , (D) 1, 3 3 x (b) Let f (x) = defined from (0, ∞ ) → [ 0, ∞ ) then by f (x) is 1+ x (A) one- one but not onto (B) one- one and onto (C) Many one but not onto (D) Many one and onto [JEE 2003 (Scr),3+3] 2 Q.10 Let f (x) = sin x + cos x, g (x) = x – 1. Thus g ( f (x) ) is invertible for x ∈
(A) – x – 1, x > 0
(B)
π π π π (A) − , 0 (B) − , π (C) − , (D) 2 2 4 4 Q.11(a) If the functions f (x) and g (x) are defined on R → R such that 0, f (x) = x,
x ∈ rational
x ∈ irrational
0, , g (x) = x, x ∈ irrational
then (f – g)(x) is (A) one-one and onto (B) neither one-one nor onto
π 0, 2 [JEE 2004 (Screening)]
x ∈ rational
(C) one-one but not onto (D) onto but not one-one
(b) X and Y are two sets and f : X → Y. If {f (c) = y; c ⊂ X, y ⊂ Y} and {f –1(d) = x; d ⊂ Y, x ⊂ X}, then the true statement is
( (C) f (f
) (b) ) = b , b ⊂ y
(A) f f −1 ( b) = b −1
(B) f −1 (f (a ) ) = a (D) f −1 (f (a ) ) = a , a ⊂ x 25
(xiv) φ (xiii) [– 3,– 2) ∪ [ 3,4) (xv) 2Kπ < x < (2K + 1)π but x ≠ 1 where K is non−negative integer (xvi) {x 1000 ≤ x < 10000} (xvii) (–2, –1) U (–1, 0) U (1, 2)
(c) neither injective nor surjective Q.5 f3n(x) = x ; Domain = R − {0 , 1} Q.6 1 Q.7 (a) 2Kπ ≤ x ≤ 2Kπ + π where K ∈ I (b) [−3/2 , −1] Q.8 (i) (a) odd, (b) even, (c) neither odd nor even, (d) odd, (e) neither odd nor even, (f) even,
(g) even,
(h) even;
(ii)
−1 + 5 −1 − 5 −3 + 5 −3 − 5 , , , 2 2 2 2
(a) y = log (10 − 10x) , − ∞ < x < 1 (b) y = x/3 when − ∞ < x < 0 & y = x when 0 ≤ x < + ∞ Q.10 f−1(x) = (a − xn)1/n Q.12 (a) f(x) = 1 for x < −1 & −x for −1 ≤ x ≤ 0; (b) f(x) = −1 for x < −1 and x for −1 ≤ x ≤ 0 Q.9
Q.13
1 if 01
x −x Q.15 (a) e − e ;
2
(b)
Q.14
{–1, 1}
1 log2 x 1+ x ; (c) log 2 log2 x − 1 1− x
Q.16 x = 1
Q.17 (i) period of fog is π , period of gof is 2π ; (ii) range of fog is [−1 , 1] , range of gof is [−tan1, tan1] 26
15 Q5. f (x) = 1 – x2, D = x ∈ R ; range =(– ∞, 1] 4 Q 9. f (x) = 2 x2
− (1 + x) , − 1 ≤ x ≤ 0 ; x−1 , 0
x
,
0≤x≤1
fof (x) = 4 − x , 3 ≤ x ≤ 4 ;
Q 12. −
Q.14
x +1 3− x gof (x) = x −1 5−x −x
Q.17 20
Q.15
0≤x<1 1≤ x ≤ 2 2
4−x ,
3 + 1 2
2< x ≤3
Q.13 Q.16
5049
;
, −1 ≤ x ≤ 0 , 0
gog (x) = x
3 −1 3 + 1 1 − 3 ∪ , , 2 2 2
1002.5
, , , ,
x = 0 or 5/3
g (x) = 3 + 5 sin(nπ + 2x – 4), n ∈ I
Q 18. (0 , 1) ∪ {1, 2, ....., 12} ∪ (12, 13)
Q 19. f (x) = sin x + x −
π 3
EXERCISE–3 Q.1 (hofog)(x) = h(x2) = x2 for x ∈ R , Hence h is not an identity function , fog is not invertible Q.2 (a) A, (b) B Q.3 (fog) (x) = e3x − 2 ; (gof) (x) = 3 ex − 2 ; Domain of (fog)–1 = range of fog = (0, ∞); Domain of (gof)–1 = range of gof = (− 2, ∞) Q.4 B Q.5 D Q.6 {(1, 1), (2, 3), (3, 4), (4, 2)} ; {(1, 1), (2, 4), (3, 2), (4, 3)} and {(1, 1), (2, 4), (3, 3), (4, 2)} Q.7 (a) B, (b) A, (c) D, (d) A, (e) D Q.8 (a) D ; (b) A Q.9 (a) D , (b) A Q.10 C Q.11 (a) A ; (b) D
Exercise-4 Part : (A) Only one correct option 1.
The domain of the function f(x) = (A) (1, 4)
− log0.3 ( x − 1) x 2 + 2x + 8
(B) (– 2, 4) ( x +3 )x + (B) (0, 3)
cot−1
is (C) (2, 4)
cos−1
(D) [2, ∞)
3.
x +3 x +1 is defined on the set S, where S is equal to: (C) {0, − 3} (D) [− 3, 0] 1 2 1 2 The range of the function f (x) = sin−1 x + + cos−1 x − , where [ ] is the greatest integer 2 2 function, is: π π π (A) , π (B) 0, (C) { π } (D) 0, 2 2 2
4.
Range of f(x) = log
2.
The function f(x) = (A) {0, 3}
(A) [0, 1]
5
2
{ 2 (sinx – cosx) + 3} is (B) [0, 2]
3 0, (C) 27 2
(D) none of these
5. 6. 7. 8.
9. 10.
11.
12. 13.
14. 15.
16.
Range of f(x) = 4x + 2x + 1 is (A) (0, ∞) (B) (1, ∞) (C) (2, ∞) (D) (3, ∞) If x and y satisfy the equation y = 2 [x] + 3 and y = 3 [x – 2] simultaneously, the [x + y] is (A) 21 (B) 9 (C) 30 (D) 12 The function f : [2, ∞) → Y defined by f(x) = x 2 − 4x + 5 is both one−one & onto if (A) Y = R (B) Y = [1, ∞ ) (C) Y = [4, ∞) (D) Y = [5, ∞) Let S be the set of all triangles and R + be the set of positive real numbers. Then the function, f : S → R+, f (∆ ) = area of the ∆ , where ∆ ∈ S is : (A) injective but not surjective (B) surjective but not injective (C) injective as well as surjective (D) neither injective nor surjective Let f(x) be a function whose domain is [– 5, 7]. Let g(x) = |2x + 5|. Then domain of (fog) (x) is (A) [– 4, 1] (B) [– 5, 1] (C) [– 6, 1] (D) none of these e x − e −x is The inverse of the function y = x e + e −x 1 1+ x 1 2+x 1 1− x log (B) log (C) log (D) 2 log (1 + x) (A) 2 1− x 2 2−x 2 1+ x The fundamental period of the function, f(x) = x + a − [x + b] + sin πx + cos 2πx + sin 3πx + cos 4πx +...... + sin (2n − 1) πx + cos 2 nπx for every a, b ∈ R is: (where [ ] denotes the greatest integer function) (A) 2 (B) 4 (C) 1 (D) 0 The period of e cos (A) 1
18.
is ______(where [ ] denotes the greatest integer function) (C) 3 (D) 4
(B) 2
(
)
1
a x + a−x (a > 0). If f(x + y) + f(x − y) = k f(x). f(y) then k has the value equal to: 2 (A) 1 (B) 2 (C) 4 (D) 1/2 A function f : R → R satisfies the condition, x 2 f(x) + f(1 − x) = 2x − x 4. Then f(x) is: (A) – x 2 – 1 (B) – x 2 + 1 (C) x 2 − 1 (D) – x 4 + 1
Given the function f(x) =
The domain of the function, f (x) =
(x
1 − 1 cos −1 (2 x + 1) tan 3 x is:
)
π (B) (− 1, 0) − − 6
π π (C) (− 1, 0] − − , − 2 6
π (D) − , 0 6
If f (x) = 2 [x] + cos x, then f: R → R is: (where [. ] denotes greatest integer function) (A) one−one and onto (B) one−one and into (C) many −one and into (D) many−one and onto If q2 − 4 p r = 0, p > 0, then the domain of the function, f (x) = log (p x 3 + (p + q) x 2 + (q + r) x + r) is:
q 2p
(A) R − − 19.
πx + x − [ x ] + cos πx
If y = f(x) satisfies the condition f x + x1 = x 2 + 2 (x ≠ 0) then f(x) = x (A) − x 2 + 2 (B) − x 2 − 2 (C) x 2 + 2 (D) x 2 − 2
If [ 2 cos x ] + [ sin x ] = − 3, then the range of the function, f (x) = sin x + 3 cos x in [0, 2 π] is: (where [. ] denotes greatest integer function) (A) [ − 2, − 1)
(B) (− 2, − 1]
(C) (− 2, − 1)
20.
The domain of the function f (x) = log1/2 − log2
21.
The range of the functions f (x) = log
(A) 0 < x < 1 (A) (− ∞, 1)
(B) 0 < x ≤ 1 (B) (− ∞, 2)
(D) [–2, – 3 )
1 1 + − 1 is: 4x (C) x ≥ 1
(D) null set
(C) (− ∞, 1]
(D) ( − ∞, 2]
(2 − log2 (16sin2 x + 1)) is 2
22.
1 + x3 + sin (sin x) + log The domain of the function, f (x) = sin−1 (x 2 + 1), (3{x} + 1) 2 x3/ 2
23.
where {x} represents fractional part function is: (A) x ∈ {1} (B) x ∈ R − {1, − 1} (C) x > 3, x ≠ I (D) none of these The minimum value of f(x) = a tan2 x + b cot2 x equals the maximum value of g(x) = a sin2x + b cos2x where a > b > 0, when (A) 4a = b (B) 3a = b (C) a = 3b (D) a = 4b
24.
x
Let f (2, 4) → (1, 3) be a function defined by f (x) = x − (where [. ] denotes the greatest integer function), then 2 f −1 (x) is equal to :
25.
x
(B) x + (C) x + 1 (D) x − 1 2 The image of the interval R when the mapping f: R → R given by f(x) = cot–1 (x2 – 4x + 3) is π 3π π 3π (A) , (B) , π (C) (0, π) (D) 0, 4 4 4 4 (A) 2x
28
26.
If the graph of the function f (x) =
ax − 1 x (a x + 1) n
is symmetric about y-axis, then n is equal to:
(A) 2
(B) 2 / 3 (C) 1 / 4 (D) – 1 / 3 π 27. If f(x) = cot–1x : R+ → 0, 2 and g(x) = 2x – x 2 : R → R. Then the range of the function f(g(x)) wherever define is π π π π π (D) (A) 0, (B) 0, (C) , 2 4 4 2 4 28. Let f: (e2, ∞) → R be defined by f(x) =n (n(n x)), then (A) f is one one but not onto (B) f is on to but not one - one (C) f is one-one and onto (D) f is neither one-one nor onto 29. Let f: (e, ∞) → R be defined by f(x) =n (n(n x)), then (A) f is one one but not onto (B) f is on to but not one - one (C) f is one-one and onto (D) f is neither one-one nor onto 30. Let f(x) = sin x and g(x) = | n x| if composite functions fog(x) and gof (x) are defined and have ranges R1 & R2 respectively then. (A) R1 = {u: – 1 < u < 1} R2 = {v: 0 < v < ∞} (B) R1 = {u: – ∞ < u < 0} R2 = {v: –1< v < 1} (C) R1 = {u: 0 < u < ∞} R2 = {v:– 1 < v < 1; v ≠ 0} (D) R1 = {u: –1 < u < 1 } R2 = {v:0 < v < ∞} −( x 2 −3 x + 2 ) 31. Function f : (– ∞, 1) → (0, e5] defined by f(x) = e is (A) many one and onto (B) many one and into (C) one one and onto (D) one one and into 32. The number of solutions of the equation [sin–1 x] = x – [x], where [ . ] denotes the greatest integer function is (A) 0 (B) 1 (C) 2 (D) infinitely many x x 33. The function f(x) = x + + 1 is e −1 2 (B) an even function (A) an odd function (C) neither an odd nor an even function (D) a periodic function Part : (B) May have more than one options correct (sin −1 og2 x),
34.
For the function f(x) = n
35.
π (C) Domain is (1, 2] (B) Range is − ∞ , n 2 A function ' f ' from the set of natural numbers to integers defined by,
1
(A) Domain is , 2 2
n − 1 , when n is odd f (n) = 2n is: , when n is even − 2
periodic with fundamental period 1 range is singleton
(B)
even
{x } − 1, where {x} denotes fractional part function and [ . ] denotes greatest {x} integer function and sgn (x) is a signum function. (D)
38.
identical to sgn sgn
D ≡ [− 1, 1] is the domain of the following functions, state which of them are injective. (B) g(x) = x 3 (C) h(x) = sin 2x (D) k(x) = sin (π x/2) (A) f(x) = x 2
Exercise-5 1 + log10 (1 − x )
x+2
1.
Find the domain of the function f(x) =
2.
Find the domain of the function f(x) =
3.
Find the inverse of the following functions. f(x) = n (x +
4.
π π Let f : − , → B defined by f (x) = 2 cos2x + 3 6 f –1 (x).
5.
29 Find for what values of x, the following functions would be identical.
3x − 1 –1 1− 2 x + 3 sin 2
1+ x2 )
3 sin2x + 1. Find the B such that f
–1
exists. Also find
x −1 f (x) = log (x - 1) - log (x - 2) and g (x) = log x − 2 . 4x
6.
If f(x) =
7.
1 1 Let f(x) be a polynomial function satisfying the relation f(x). f = f(x) + f ∀ x ∈ R – {0} and x x f(3) = –26. Determine f ′(1).
8.
Find the domain of definitions of the following functions.
9.
4x + 2
, then show that f(x) + f(1 – x) = 1
3 − 2 x − 21 − x
(i)
f (x) =
(iii)
f (x) = og10 (1 – og10(x 2 – 5x + 16))
x−2 + x+2
1− x 1+ x
(ii)
f (x) =
(ii)
4 − x2 f (x) = sin og 1 − x f (x) = sin2 x + cos4x
Find the range of the following functions. x 2 − 2x + 4
(i)
f (x) =
(iii)
f (x)= x 4 − 2 x 2 + 5
x 2 + 2x + 4
(iv)
10.
Solve the following equation for x (where [x] & {x} denotes integral and fractional part of x) 2x + 3 [x] – 4 {–x} = 4
11.
Draw the graph of following functions where [.] denotes greatest integer function and { .} denotes fractional part function. (i) y = {sin x } (ii) y = [x] + { x}
12. 13.
2 Draw the graph of the function f(x) = x – 4 | x | + 3 a has exactly four distinct real roots.
Examine whether the following functions are even or odd or none. x | x |, x ≤ −1 (1 +2 x )7 [1 + x ] + [1 − x ], − 1 < x < 1 (i) f (x) = (ii) f (x) = − x | x |, 2x x ≥1 2x (sinx + tanx ) , where [ ] denotes greatest integer function. x + 2π 2 π −3 Find the period of the following functions.
(iii)
14.
and also find the set of values of ‘a’ for which the equation f(x) =
(i) (ii) (iii)
f (x) =
sin2 x cos 2 x − 1 + cot x 1 + tanx π f (x) = tan [ x ] , where [.] denotes greatest integer function. 2 sin x + sin 3x 1 |sinx | + sinx f (x) = (iv) f (x) = cosx | cosx | cos x + cos 3 x 2
f (x) = 1 −
15.
1 + x 2 x ≤1 If f(x) = and g(x) = 1 – x ; – 2 < x < 1 then define the function fog(x). x + 1 1 < x ≤ 2
16.
Find the set of real x for which the function, f (x) = greatest integer not greater than x.
17.
18.
19.
( (
1 is not defined, where [x] denotes the [| x − 1 |] + [| 12 − x | ] −11
))
4 − 2cosx & the function , g(x) = cosec−1 3 h(x) = f(x) defined only for those values of x, which are common to the domains of the functions f(x) and g(x). Calculate the range of the function h(x). Let ‘f’ be a real valued function defined for all real numbers x such that for some positive constant ‘a’ the 1 equation f ( x + a) = + f ( x ) − (f ( x ))2 holds for all x. Prove that the function f is periodic. 2 If f (x) = −1 + x − 2, 0 ≤ x ≤ 4 g (x) = 2 − x , − 1 ≤ x ≤ 3
Given the functions f(x) = e
cos −1 sin x + π 3
30
Then find fog (x), gof (x), fof(x) & gog(x). Draw rough sketch of the graphs of fog (x) & gof (x). Find the integral solutions to the equation [x] [y] = x + y. Show that all the non-integral solutions lie on exactly two lines. Determine these lines. Here [ .] denotes greatest integer function.
20.
Exercise-4 1. 8. 15. 22. 29. 35.
D B B D C AC
2. 9. 16. 23. 30. 36.
C C D D D B
3. 10. 17. 24. 31. 37.
C 4. A 11. C 18. C 25. D 32. ABCD
B A B D B
5. 12. 19. 26. 33. 38.
B B D D B BD
6. 13. 20. 27. 34.
C D D C BC
7. 14. 21. 28.
B B D A
13. (i) neither even nor odd (ii) even (iii) odd 14. (i) π (ii) 2 (iii) 2 π (iv) π
Exercise-5 1. [–2, 0) ∪ (0, 1)
1 − 3 ,
1 2
2 − 2 x + x 2 15. f(g(x)) = 2 − x
e x − e−x 2
3. f –1 =
4. B = [0, 4] ; f 5. (2, ∞) 9. (i)
2.
(x) =
1 2
−1 x − 2 π sin − 2 6
(ii) φ
(iii) (2, 3)
(ii) [ − 1, 1] (iii) [4, ∞)
3 (iv) , 1 4
7. – 3 8. (i) [0, 1]
1 3 , 3
3 10. 2
–1
a ∈ (1, 3) ∪ {0}
12.
0≤ x ≤1 −1≤ x < 0
π 16. (0, 1) U {1, 2,......., 12} U (12, 13) 17. e 6 , e π 18. Period 2 a − (1 + x ) , − 1 ≤ x ≤ 0 19. fog (x) = ; x −1 , 0 < x ≤ 2
x +1 3 − x gof(x) = x −1 5 − x
, , , ,
0 ≤ x <1 1≤ x ≤ 2 2
, 0≤x≤2 x fof (x) = ; 4 − x , 2 < x ≤ 2
11. (i)
− x , −1≤ x ≤ 0 , 0
20. Integral solution (0, 0); (2, 2). x + y = 6, x + y = 0
(ii)
31
FUNCTIONS (ASSERTION AND REASON) Some questions (Assertion–Reason type) are given below. Each question contains Statement – 1 (Assertion) and Statement – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. So select the correct choice : Choices are : (A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1. (C) Statement – 1 is True, Statement – 2 is False. (D) Statement – 1 is False, Statement – 2 is True. 1.
2.
3.
Let f(x) = cos3πx + sin
3πx .
Statement – 1 : f(x) is not a periodic function. Statement – 2 : L.C.M. of rational and irrational does not exist Statement – 1: If f(x) = ax + b and the equation f(x) = f –1(x) is satisfied by every real value of x, then a∈R and b = –1. Statement – 2: If f(x) = ax + b and the equation f(x) = f –1(x) is satisfied by every real value of x, then a = –1 and b∈R.
x2 , then F(x) = f(x) always x Statements-2: At x = 0, F(x) is not defined. Statements-1: If f(x) = x and F(x) =
1 , x ≠ 0, 1, then the graph of the function y = f (f(f(x)), x > 1 is a straight 1− x Statement–2 : f(f(x)))) = x
4.
Statement–1 : If f(x) =
5.
line Let f(1 + x) = f(1 – x) and f(4 + x) = f(4– x) Statement–1 : f(x) is periodic with period 6
Statement–2 : 6 is not necessarily fundamental period of f(x) 6.
Statement–1 : Period of the function f(x) = 1 + sin 2x + e{x} does not exist Statement–2 : LCM of rational and irrational does not exist
7.
Statement–1 : Domain of f(x) =
8.
Statement–1 : Range of f(x) =
9.
Let a, b ∈ R, a ≠ b and let f(x) =
10.
1 is (–∞, 0) Statement–2:| x | – x > 0 for x ∈ R– | x | −x
4 − x 2 is [0, 2] Statement–2 : f(x) is increasing for 0 ≤ x ≤ 2 and decreasing for – 2 ≤ x ≤ 0. a+x . b+x Statement–1 : f is a one–one function.
Statement–2 : Range of f is R – {1}
Statement–1 : sin x + cos (πx) is a non–periodic function. Statement–2 : Least common multiple of the periods of sin x and cos (πx) is an irrational number.
11.
Statement–1: The graph of f(x) is symmetrical about the line x = 1, then, f(1 + x) = f(1 – x). 32 32
Statement–2 : even functions are symmetric about the y-axis.
πx πx + cos is 2(n)! n! ( n − 1)!
12.
Statement–1 : Period of f(x) = sin
13.
Statement–2 : Period of |cos x| + |sin x| + 3 is π. Statement–1 : Number of solutions of tan(|tan–1x|) = cos|x| equals 2
Statement–2 : ?
14.
Statement–1 : Graph of an even function is symmetrical about y–axis Statement–2 : If f(x) = cosx has x (+)ve solution then total number of solution of the above equation is 2n. (when f(x) is continuous even function).
15.
If f is a polynomial function satisfying 2 + f(x).f(y) = f(x) + f(y) + f(xy) ∀ x, y∈R
Statement-1: f(2) = 5 which implies f(5) = 26 Statement-2: If f(x) is a polynomial of degree 'n' satisfying f(x) + f(1/x) = f(x). f(1/x), then f(x) = 1 xn + 1
16.
Statement-1: The range of the function sin-1 + cos-1x + tan-1x is [π/4, 3π/4] Statement-2: sin-1x, cos-1x are defined for |x| ≤ 1 and tan-1x is defined for all 'x'.
17.
0 where x is rational A function f(x) is defined as f(x) = 1 where x is irrational Statement-1 : f(x) is discontinuous at xll x∈R Statement-2 : In the neighbourhood of any rational number there are irrational numbers and in the vincity of any irrational number there are rational numbers.
(
)
(
)
18.
Let f(x) = sin 2 3 π x + cos 3 3 πx
19.
Statement-1 : f(x) is a periodic function Statement-2: LCM of two irrational numbers of two similar kind exists. Statements-1: The domain of the function f(x) = cos-1x + tan-1x + sin-1x is [-1, 1]
20.
Statements-2: sin-1x, cos-1x are defined for |x| ≤ 1and tan-1x is defined for all x. Statement-1 : The period of f(x) = = sin2x cos [2x] – cos2x sin [2x] is 1/2 Statements-2: The period of x – [x] is 1, where [⋅] denotes greatest integer function.
21.
Statements-1: If the function f : R → R be such that f(x) = x – [x], where [⋅] denotes the greatest integer less than or equal to x, then f-1(x) is equals to [x] + x Statements-2: Function ‘f ’ is invertible iff is one-one and onto.
22.
Statements-1 : Period of f(x) = sin 4π {x} + tan π [x] were, [⋅] & {⋅} denote we G.I.F. & fractional part respectively is 1. Statements-2: A function f(x) is said to be periodic if there exist a positive number T independent of x such that f(T + x) = f(x). The smallest such positive value of T is called the period or fundamental period.
23.
Statements-1: f(x) = Statements-2:
24.
x +1 is one-one function x −1
x +1 is monotonically decreasing function and every decreasing function is one-one. x −1
Statements-1: f(x) = sin2x (|sinx| - |cosx|) is periodic with fundamental period π/2 33 33
25. 26.
Statements-2: When two or more than two functions are given in subtraction or multiplication form we take the L.C.M. of fundamental periods of all the functions to find the period. Statements-1: e x = lnx has one solution. Statements-2: If f(x) = x ⇒ f(x) = f−1(x) have a solution on y = x. Statements-1: F(x) = x + sinx. G(x) = -x H(x) = F(X) + G(x), is a periodic function. Statements-2: If F(x) is a non-periodic function & g(x) is a non-periodic function then h(x) = f(x) ± g(x) will be a periodic function.
27.
28.
29.
x + 1, x ≥ 0 Statements-1: f (x) = is an odd function. x − 1, x < 0 Statements-2: If y = f(x) is an odd function and x = 0 lies in the domain of f(x) then f(0) = 0 x; x ∈ Q is one to one and non-monotonic function. Statements-1: f (x) = C − x; x ∈ Q Statements-2: Every one to one function is monotonic. x + 4, x ∈ [1, 2] Statement–1 : Let f : [1, 2] ∪ [5, 6] → [1, 2] ∪ [5, 6] defined as f (x) = then the − x + 7, x ∈ [5, 6] equation f(x) = f−1(x) has two solutions.
Statements-2: f(x) = f−1(x) has solutions only on y = x line. 30.
Statements-1: The function
px + q (ps − qr ≠ 0) cannot attain the value p/r. rx + s
Statements-2: The domain of the function g(y) =
q − sy is all real except a/c. ry − p
31.
Statements-1: The period of f(x) = sin [2] xcos [2x] – cos2x sin [2x] is 1/2 Statements-2: The period of x – [x] is 1.
32.
Statements-1: If f is even function, g is odd function then
b (g ≠ 0) is an odd function. g
Statements-2: If f(–x) = –f(x) for every x of its domain, then f(x) is called an odd function and if f(–x) = f(x) for every x of its domain, then f(x) is called an even function. 33.
Statements-1: f : A → B and g : B → C are two function then (gof)–1 = f–1 og–1. Statements-2: f : A → B and g : B → C are bijections then f–1 & g–1 are also bijections.
34.
Statements-1: The domain of the function f (x) = log 2 sin x is (4n + 1)
π , n ∈ N. 2
Statements-2: Expression under even root should be ≥ 0 35.
Statements-1: The function f : R → R given f (x) = log a (x + x 2 + 1) a > 0, a ≠ 1 is invertible. Statements-2: f is many one into.
36.
π Statements-1: φ(x) = sin (cos x) x ∈ 0, is a one-one function. 2
34 34
π Statements-2: φ '(x) ≤ ∀x ∈ 0, 2 37.
38.
Statements-1: For the equation kx2 + (2 − k)x + 1 = 0 k ∈ R − {0} exactly one root lie in (0, 1). Statements-2: If f(k1) f(k2) < 0 (f(x) is a polynomial) then exactly one root of f(x) = 0 lie in (k1, k2). 1+ x2 Statements-1: Domain of f (x) = sin −1 is {−1, 1} 2x
39.
1 1 ≥ 2 when x > 0 and x + ≤ −2 when x < 0. x x Statements-1: Range of f(x) = |x|(|x| + 2) + 3 is [3, ∞)
40.
Statements-2: If a function f(x) is defined ∀ x ∈ R and for x ≥ 0 if a ≤ f(x) ≤ b and f(x) is even function than range of f(x) f(x) is [a, b]. Statements-1: Period of {x} = 1. Statements-2: Period of [x] = 1
41.
Statements-1: Domain of f = φ. If f(x) =
42.
Statements-2: [x] ≤ x ∀ x∈ R Statements-1: The domain of the function sin–1x + cos–1x + tan–1x is [–1, 1]
Statements-2: x +
1 [x] − x
Statements-2: sin–1x, cos–1x are defined for |x| ≤ 1 and tan–1x is defined for all ‘x’
ANSWER KEY 1. A 8. C 15. A 22. A 29. C 36. A
2. D 9. B 16. A 23. A 30. A 37. C
3. A 10. C 17. A 24. A 31. A 38. A
4. C 11. A 18. A 25. D 32. A 39. A
5. A 12. C 19. A 26. C 33. D 40. A
6. A 13. B 20. A 27. D 34. A 41. A
SOLUTIONS 4.
f(f(x)) =
1 1 x −1 = = 1 1 − f (x) 1 − x 1− x
1 1 = =x 1 − f (f (x)) 1 − x − 1 x f(1 + x) = f(1 – x) f(4 + x) = f(4 – x)
... (1) ... (2)
x → 1– x in (1) ⇒ f(1 – x) = f(x)
... (3)
x → 4 – x in (2) ⇒ f(2 – x) f(8 – x) = f(x)
... (4)
∴ f(f(f(x))) =
5.
Ans. C
35 35
7. A 14. A 21. D 28. C 35. C 42. A
(1) and (4) ⇒ f(2 – x) = f(8 – x)
.... (5)
Use x → x – x in (5), we get f(x) = f(6 + x)
⇒ f(x) is periodic with period 6 Obviously 6 is not necessary the fundamental period.
Ans. A
∴ (A) is the correct option.
6.
L.C.M. of {π, 1} does not exist
7.
(a) Clearly both are true and statement – II is correct explantion of Statement – I .
8.
(c) f ′(x) =
9.
Suppose a > b. Statement – II is true as f ′(x) =
−x 4 − x2
∴ f(x) is increasing for – 2 ≤ x ≤ 0 and decreasing for 0 ≤ x ≤ 2.
b−a
( b + x )2
, which is always negative and hence monotonic
in its continuous part. Also lim + f (x) = ∞ and lim − f (x) = −∞ . Moreover x →− b
x →− b
lim f (x) = 1 + and lim f (x) = −1 − . Hence range of f is R – {1}.
x →∞
x →−∞
F is obviously one–one as f(x1) = f(x2) ⇒ x1 = x2. However statement – II is not a correct reasoning for statement – I Hence (b) is the correct answer.
10. 11.
Statement – I is true, as period of sin x and cos πx are 2π and 2 respectively whose L.C.M does not exist. Obviously statement – II is false Hence (c) is the correct answer. Graph of f(x) is symmetric about the line x = 0 if f(- x) = f(x) i.e. if f(0 – x) = f(0 + x) ∴ Graph of y = f(x) is symmetric about x = 1, if f(1 + x) = f(1 – x). Hence (a) is the correct answer.
12.
Period of sin
πx = 2 ( n − 1) ! ( n − 1)!
Period of cos
πx = 2 ( n )! n!
⇒ Period of f(x) = L.C.M of 2(n – 1)! And 2(n)! = 2(n!) Now,
f(x) = | copsx | + | sin x | +3 = 1+ | sin 2x | + 3
∴ f(x) is periodic function with period =
13.
14. 19.
π . 2
Hence (c) is the correct answer.
tan(|tan–1x|) = |x|, since |tan–1x| = tan–1|x| Obviously cos|x| and |x| meets at exactly two points ∴ (B) is the correct option. (A)Since cos n is also even function. Therefore solution of cosx = f(x) is always sym. also out y–axis. (a) Both A and R are obviously correct. 20. (a) f(x) = x [x] f(x + 1) = x + 1 − ([x] + 1) = x – [x] So, period of x – [x] is 1. Let f(x) = sin (2x – [2x])
36 36
1 1 1 f x + = sin 2 x + − 2 x + 2 2 2 21.
Clearly tan π[x] = 0 ∀ x∈R and period of sin 4 π {x} = 1. Ans. (A)
23.
f(x) =
x +1 x −1
f′(x) =
(x − 1) − (x + 1) −2 = <0 2 (x − 1) (x − 1) 2
So f(x) is monotonically decreasing & every monotonic function is one-one. So ‘a’ is correct.
24.
f(x) = sin2x (|sinx| -|cosx|) is periodic with period π/2 because f(π/2 + x) = sin 2 (π/2 + x) (|sin (π/2 + x)| -|cos (π/2 + x)|) = sin (π + 2x) (|cosx| - |sinx|) = -sin2x (|cosx| - |sinx|) = sin2x (|sinx| - |cosx|) Sometimes f(x + r) = f(x) where r is less than the L.C.M. of periods of all the function, but according to definition of periodicity, period must be least and positive, so ‘r’ is the fundamental period. So ‘f’ is correct.
27.
(D) If f(x) is an odd function, then f(x) + f(−x) = 0 ∀ x ∈ Df
28.
(C) For one to one function if x1 ≠ x2
⇒ f(x1) ≠ f(x2) for all x1, x2 ∈ Df
3 >1
but f ( 3) < f (1)
and 3 > 1
f(5) > f(1)
29.
30.
f(x) is one-to-one
3 11 11 3 (C) , and , both lie on y = f(x) then they will also lie on y = f−1(x) ⇒ there are two 2 2 2 2 solutions and they do not lie on y = x. If we take y =
q − sx px + q then x = ⇒ x does not exist if y = p/r rx + s rx − p
Thus statement-1 is correct and follows from statement-2
31.
f(x) = sin(2x – [2x] = sin (2x + 1 – [2x] – 1] f(x) = x – [x] f(x + 1) = x + 1 – ([x] + 1) = x – [x]
32.
but non-monotonic
(A) h(–x) =
Let h(x) =
(A)
1 f(x + 1/2) = sin 2x + 1 − 2 x + 2 = sin (2x – [2x].) i.e., period is 1/2. i.e., period is 1.
f (x) g(x)
f (− x) f (x) f (x) = = = − h(x) g(− x) g(− x) −g(x)
37 37
(A)
∴ h(x) =
f is an odd function. g
33.
(D) Assertion : f : A → B, g : B → C are two functions then (gof)–1 ≠ f–1 og–1 (since functions need not posses inverses. Reason : Bijective functions are invertibles.
34.
(A) for f(x) to be real log2(sin x) ≥ 0
⇒ sin x ≥ 2º
⇒ sin x = 1
(C) f is injective since x ≠ y (x, y ∈ R)
35.
{
}
{
⇒
x = (4n + 1)
π , n ∈ N. 2
}
⇒ log a x + x 2 + 1 ≠ log a y + y 2 + 1 ⇒ f(x) ≠ f(y)
)
(
⇒ x=
f is onto because log a x + x 2 + 1 = y 40.
a y − a−y . 2
Since {x} = x – [x] ∴ {x + 1} = x + 1 – [x + 1] = x + 1 – [x] – 1 Period of [x] = 1
41.
f(x) =
= x – [x] = [x] Ans (A)
1 [x] – x ≠ 0 [x] − x
[x] ≠ x → [x] > x It is imposible or [x] ≤ x So the domain of f is φ
because reason [x] ≤ x
Ans. (A)
Imp. Que. From Competitive exams 1.
If f (x ) =
cos 2 x + sin 4 x for x ∈ R , then f (2002 ) = sin 2 x + cos 4 x
(a) 1
(b) 2
[EAMCET 2002]
(c)
3
(d)
4
n
2.
If f : R → R satisfies f (x + y ) = f (x ) + f (y ) , for all x , y ∈ R and f (1) = 7 , then
∑ f (r) is
[AIEEE 2003]
r =1
(a)
3.
7n 2
7(n + 1) 2
(b) {0}
If f (x ) = sgn( x 3 ) , then
5.
for 0 ≤ x ≤ 2
, then { x ∈ (−2, 2) : x ≤ 0 and f (| x |) = x } = {−1 / 2} (d)
(c)
7 n(n + 1) 2
φ
[EAMCET 2003]
f ' (0 − ) = 1
f ' (0 + ) = 2
(b)
f is not derivable at x = 0
(d)
If f : R → R and g : R → R are given by f (x ) = | x | and g(x ) = | x | for each x ∈ R , then {x ∈ R : g( f (x )) ≤ f (g(x ))} = (a)
6.
for − 2 ≤ x ≤ 0
(d)
[DCE 2001]
(a) f is continuous but not derivable at x = 0
(c)
7 n(n + 1)
(c)
−1 Suppose f : [2, 2] → R is defined by f (x ) = x − 1
(a) {−1}
4.
(b)
Z ∪ (−∞, 0 )
(b) (−∞,0)
Z
(c)
For a real number x , [x ] denotes the integral part of x. The value of
If x 1 = 3 and x n +1 = 2 + x n , n ≥ 1, then lim x n is equal to n →∞
(b) 2
The value of lim x→
π
2
∫
x
π /2
t dt
sin(2 x − π )
(a) ∞
is
(b) cot x
The lim (cos x ) x →0
28.
Greater then 1
If [.] denotes the greatest integer less than or equal to x, then the value of lim (1 − x + [ x − 1] + [1 − x ]) is
(a) –1 27.
(d)
lim sin[π n + 1 ] =
(a) a = 1 26.
1 and 1 2
n →∞
(a) 25.
Lies between
2
(a) 0 24.
1 (c) 2
[IIT 1992]
(a) ∞ 23.
0
[BIT Ranchi 1982]
(b) Lies between 0 and
(a) No value of n 22.
(d)
π
is
2 + 3x − 2 − 3x
(a) Does not exist
2
(c)
[MP PET 1998]
π
π
(c)
2
4
(d)
π 8
is [RPET 1999](a)–1(b)0c)1(d) None of these
The integer n for which lim
x →0
(a) 1
(cos x − 1) (cos x − e x ) is a finite non-zero number is xn
(b) 2
(c)
3
[IIT Screening 2002]
(d)
f (x ) − f (x ) is equal to f ( x ) − f (0 )
4
2
29.
If f is strictly increasing function, then lim
x →0
(a) 0
(b) 1
[IIT Screening 2004]
(c)
–1
(d)
2
x − 3, 2 < x < 3 If f ( x ) = , the equation whose roots are lim− f ( x ) and lim+ f (x ) is x →3 x →3 2 x + 5, 3 < x < 4 2
30.
(a) 31.
x2 −7x + 3 = 0
x 2 − 20 x + 66 = 0
(c)
x 2 − 17 x + 66 = 0 (d) x 2 − 18 x + 60 = 0
2x − 1 The function f ( x ) = [ x ] cos π , where [.] denotes the greatest integer function, is discontinuous at 2
(a) All x 32.
(b)
[Orissa JEE 2004]
(b) No x
(c)
All integer points (d)
[IIT 1995]
x which is not an integer
x Let f (x ) be defined for all x > 0 and be continuous. Let f (x ) satisfy f = f ( x ) − f (y ) for all x, y and f (e ) = 1, then [IIT 1995] y
40 40
(a)
33.
f (x ) = ln x
(b)
35.
36.
(b) 2
38.
(d)
(c) 3
(d)
x f ( x ) → 1 as x → 0
[Orissa JEE 2004]
None of these
The function f (x ) = [ x ] − [ x ] , (where [y] is the greatest integer less than or equal to y),is discontinuous at
[IIT 1999]
(a) All integers
All integers except 1
2
2
(b) All integers except 0 and 1 (c)
− 1 + 1 | x| x , x ≠ 0 , then f (x ) is If f ( x ) = xe 0 x =0 ,
All integers except 0
(d)
[AIEEE 2003]
(a) Continuous as well as differentiable for all x
(b)
Continuous for all x but not differentiable at x = 0
(c) Neither differentiable nor continuous at x = 0
(d)
Discontinuous every where
1 − tan x π π π π Let f ( x ) = , x ≠ , x ∈ 0, , If f (x ) is continuous in 0, , then f is 4x −π 4 2 2 4
(a) –1
37.
1 f → 0 as x → 0 x
(c)
(4 x − 1)3 ,x ≠ 0 x x2 The value of p for which the function f ( x ) = sin log 1 + may be continuous at x = 0 , is p 3 12(log 4 )3 , x = 0
(a) 1 34.
f (x ) is bounded
(b)
1 2
−
(c)
1 x sin , x ≠ 0 Let g( x ) = x . f ( x ), where f ( x ) = at x = 0 x 0, x = 0
1 2
[AIEEE 2004]
(d)
1
[IIT Screening 1994; UPSEAT 2004]
(a) g is differentiable but g ' is not continuous
(b)
g is differentiable while f is not
(c) Both f and g are differentiable
(d)
g is differentiable and g ' is continuous
The function f (x ) = max[( 1 − x ), (1 + x ), 2], x ∈ (−∞, ∞), is
[IIT 1995]
(a) Continuous at all points
Differentiable at all points except at x = 1 and x = −1
(b)Differentiable at all points(c)
(d) Continuous at all points except at x = 1 and x = −1 where it is discontinuous 39.
The function f ( x ) =| x | + | x − 1 | is
[RPET 1996; Kurukshetra CEE 2002]
(a) Continuous at x = 1, but not differentiable at x = 1
(b)
Both continuous and differentiable at x = 1
(c) Not continuous at x = 1
(d)
Not differentiable at x = 1
ANSWER: Imp. Que. From Competitive exams 1
a
2
d
3
c
4
d
5
6
b
7
a
8
b
9
d
10
d
11
d
12
d
13
d
14
a
15
d
16
a
17
a
18
c
19
c
20
b
21
b
22
b
23
c
24
c
25
a
26
b
27
c
28
c
29
c
30
c
31
c
32
c
33
a
34
d
35
d
36
b
37
c
38
a,b
39
a,c
40
a
41 41
d
STUDY PACKAGE Target: IIT-JEE (Advanced) SUBJECT: MATHEMATICS TOPIC: 5 XI M 5. Complex Numbers Index: 1. Key Concepts 2. Exercise I to V 3. Answer Key 4. Assertion and Reasons 5. 34 Yrs. Que. from IIT-JEE 6. 10 Yrs. Que. from AIEEE
1
1.
The complex number system
2 of 38
Complex Numbers
There is no real number x which satisfies the polynomial equation x 2 + 1 = 0. To permit solutions of this and similar equations, the set of complex numbers is introduced. We can consider a complex number as having the form a + bi where a and b are real number and i, which is called the imaginary unit, has the property that i 2 = – 1. It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z which is denoted by (Re z) and ‘b’ is called as imaginary part of z which is denoted by (Im z). Any complex number is : (i) Purely real, if b = 0 ; (ii) Purely imaginary, if a = 0 (iii) Imaginary, if b ≠ 0. NOTE : (a) The set R of real numbers is a proper subset of the Complex Numbers. Hence the complete number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C. (b) Zero is purely real as well as purely imaginary but not imaginary. (c) (d)
i = −1 is called the imaginary unit. Also i² = − 1; i 3 = − i ; i 4 = 1 etc.
a
b = a b only if atleast one of a or b is non - negative.
(e) is z = a + ib, then a – ib is called complex conjugate of z and written as z = a – ib Self Practice Problems 1. Write the following as complex number (i) (ii) − 16 x , (x > 0)
(iii) 2.
2.
–b +
− 4ac , (a, c> 0)
Ans. (i) 0 + i 16 (ii) x + 0i (iii) –b + i 4ac Write the following as complex number (i) (ii) roots of x 2 – (2 cosθ)x + 1 = 0 x (x < 0)
Algebraic Operations:
Fundamental operations with complex numbers In performing operations with complex numbers we can proceed as in the algebra of real numbers, replacing i 2 by – 1 when it occurs. 1. Addition (a + bi) + (c + di) = a + bi + c + di = (a + c) + (b + d) i 2. Subtraction (a + bi) – c + di) = a + bi – c – di = (a – c) + (b – d) i 3. Multiplication (a + bi) (c + di) = ac + adi + bci + bdi 2 = (ac – bd) + (ad+ bc)i a + bi c − bi ac − adi + bci − bdi 2 . = c + di c − di c 2 − d 2i 2 ac + bd + (bc − ad)i ac + bd bc − ad i = 2 + 2 = c 2 − d2 c + d2 c + d2 Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative. e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless.
4.
a + bi c + di
Division
=
In real numbers if a2 + b2 = 0 then a = 0 = b however in complex numbers, z 12 + z22 = 0 does not imply z 1 = z2 = 0. Example : Solution
Find multiplicative inverse of 3 + 2i. Let z be the multiplicative inverse of 3 + 2i. then ⇒ z . (3 + 2i) = 1 3 − 2i 1 ⇒ z= = (3 + 2i) (3 − 2i) 3 + 2i
3 2 – i 13 13 2 3 − i Ans. 13 13 Self Practice Problem 1. Simplify i n+100 + i n+50 + i n+48 + i n+46 , n ∈ Ι . Ans. 0
⇒
3.
z=
Equality In Complex Number: Two complex numbers z1 = a1 + ib1 & z 2 = a2 + ib2 are equal if and only if their real and imaginary parts are equal respectively i.e. z1 = z 2 ⇒ Re(z 1) = Re(z2) and Ιm (z1) = Ιm (z2). 2
Example: Solution.
Example: Solution.
Find the value of x and y for which (2 + 3i) x 2 – (3 – 2i) y = 2x – 3y + 5i where x, y ∈ R. (z + 3i)x 2 – (3 – 2i)y = 2x – 3y + 5i ⇒ 2x 2 – 3y = 2x – 3y ⇒ x2 – x = 0 ⇒ x = 0, 1 and 3x 2 + 2y = 5 5 ⇒ if x = 0,y = and if x = 1, y = 1 2 5 and x = 1, y = 1 ∴ x = 0, y = 2 5 are two solutions of the given equation which can also be represented as 0, & (1, 1) 2
3 of 38
Example: Solution
5 0, , (1, 1) Ans. 2 Find the value of expression x 4 – 4x 3 + 3x 2 – 2x + 1 when x = 1 + i is a factor of expression. x=1+i ⇒ x–1=i ⇒ (x – 1) 2 = –1 ⇒ x 2 – 2x + 2 = 0 Now x 4 – 4x 3 + 3x 2 – 2x + 1 = (x 2 – 2x + 2) (x 2 – 3x – 3) – 4x + 7 ∴ when x = 1 + i i.e. x 2 – 2x + 2 = 0 x 4 – 4x 3 + 3x 2 – 2x + 1 = 0 – 4 (1 + i) + 7 = –4 + 7 – 4i = 3 – 4i Ans.
Solve for z if z2 + |z| = 0 Let z= x + iy ⇒
(x + iy) 2 +
⇒ ⇒ when x ⇒ ⇒ when y ⇒
x 2 – y2 + x 2 + y 2 = 0 and 2xy = 0 x = 0 or y = 0 =0 – y2 + | y | = 0 y = 0, 1, –1 z = 0, i, –i =0 x2 + | x | = 0 x=0 ⇒ z = 0 Ans. z = 0, z = i, z = – i
x2 + y2 = 0
Find square root of 9 + 40i Let (x + iy)2 = 9 + 40i ∴ x 2 – y2 = 9 ...............(i) and xy = 20 ...............(ii) squing (i) and adding with 4 times the square of (ii) we get x 4 + y4 – 2x 2 y2 + 4x 2 y2 = 81 + 1600 ⇒ (x 2 + y 2)2 = 168 ⇒ x 2 + y2 = 4 ...............(iii) from (i) + (iii) we get x 2 = 25 ⇒ x=±5 and y = 16 ⇒ y=±4 from equation (ii) we can see that x & y are of same sign ∴ x + iy = +(5 + 4i) or = (5 + 4i) ∴ sq. roots of a + 40i = ± (5 + 4i) Ans. Self Practice Problem
Example: Solution.
1 3 – i, 0, i 2 2
1.
Solve for z : z = i z 2
4.
Representation Of A Complex Number: (a)
Ans.
±
± (5 + 4i)
Cartesian Form (Geometric Representation) : Every complex number z = x + i y can be represented by a point on the Cartesian plane known as complex plane (Argand diagram) by the ordered pair (x, y).
Length OP is called modulus of the complex number which is denoted by z & θ is called the argument or amplitude. y z = x 2 + y 2 & θ = tan−1 (angle made by OP with positive x −axis) x 3
(ii) (iii) (b)
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NOTE : (i) Argument of a complex number is a many valued function. If θ is the argument of a complex number then 2 nπ + θ; n ∈ I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nπ.
The unique value of θ such that − π < θ ≤ π is called the principal value of the argument. Unless otherwise stated, amp z implies principal value of the argument. By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is only given by its modulus. Trignometric/Polar Representation : z = r (cos θ + i sin θ) where z = r; arg z = θ ; z = r (cos θ − i sin θ)
NOTE : cos θ + i sin θ is also written as CiS θ or ei θ. e ix +e −ix eix −e −ix & sin x = are known as Euler's identities. 2 2 Euler's Representation : z = rei θ; z = r; arg z = θ; z = re− i θ
Also cos x = (c) (d)
Vectorial Representation : Every complex number can be considered as if it is the position vector of a point. If the point →
→
Example:
P represents the complex number z then, OP = z & OP = z. Express the complex number z = – 1 + 2 i in polar form.
Solution.
z = –1 + i 2 |z|=
( −1)2 +
( 2)
2
=
1+ 2 =
3
2 Arg z = π – tan–1 1 = π – tan–1
∴ z= Self Practice Problems 1.
3 (cos θ + i sin θ )
2
−1 (9 + i) 2−i
17 82 , 11 5 Find the |z| and principal argument of the complex number z = 6(cos 310º – i sin 310°) Ans. 6, 50°
Ans.
5.
where θ = π – tan–1
Find the principal argument and |z| z=
2.
2 = θ (say)
– tan–1
Modulus of a Complex Number : If z = a + ib, then it's modulus is denoted and defined by |z| = a 2 + b 2 . Infact |z| is the distance of z from origin. Hence |z 1 – z 2| is the distance between the points represented by z 1 and z2. Properties of modulus
(i)
|z 1z2| = |z1| . |z 2|
(ii)
(iii)
|z 1 + z2| ≤ |z1| + |z2|
(iv)
z1 z1 z2 = z2 |z 1 – z2| ≥ ||z1| – |z2||
(provided z2 ≠ 0)
(Equality in (iii) and (iv) holds if and only if origin, z1 and z2 are collinear with z1 and z2 on the same side of origin). Example: If |z – 5 – 7i| = 9, then find the greatest and least values of |z – 2 – 3i|. Solution. We have 9 = |z – (5 + 7i)| = distance between z and 5 + 7i. Thus locus of z is the circle of radius 9 and centre at 5 + 7i. For such a z (on the circle), we have to find its greatest and least distance as from 2 + 3i, which obviously 14 and 4. Example: Find the minimum value of |1 + z| + |1 – z|. Solution |1 + z| + |1 – z| ≥ |1 + z + 1 – z| (triangle inequality) ⇒ |1 + z | + |1 – z| ≥ 2 ∴ minimum value of (|1 + z| + |1 – z|) = 2 Geometrically |z + 1| + |1 – 2| = |z + 1| + |z – 1| which represents sum of distances of z from 1 and – 1 it can be seen easily that minimu (PA + PB) = AB = 2 Ans.
1/ 4
2
π 1 + nπ 8
e
4
Solution.
z−
2 z
= 1 then find the maximum and minimum value of |z|
2 =1 z Let | z | = r
|z|−
z−
⇒
r−
2 r
2 z
≤1 ≤ r+
≤ z−
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Example:
2 2 ≤|z|+ − 2 z
2 r
2 ≥1 ⇒ r ∈ R+ ..............(i) r 2 2 ≤ 1⇒ –1 ≤ r – ≤1 and r − r r ⇒ r ∈ (1, 2) ..............(ii) ∴ from (i) and (ii) r ∈ (1, 2) Ans. r ∈ (1, 2)
r+
Self Practice Problem 1.
|z – 3| < 1 and |z – 4i| > M then find the positive real value of M for which these exist at least one complex number z satisfy both the equation. Ans. M ∈ (0, 6)
6.
Agrument of a Complex Number : Argument of a non-zero complex number P(z) is denoted and defined by arg(z) = angle which OP makes with the positive direction of real axis. If OP = |z| = r and arg(z) = θ, then obviously z = r(cosθ + isinθ), called the polar form of z. In what follows, 'argument of z' would mean principal argument of z(i.e. argument lying in (– π, π] unless the context requires otherwise. Thus argument of a complex number z = a + ib = r(cosθ + isinθ) is the value of θ satisfying rcosθ = a and rsinθ = b. Thus the argument of z = θ , π – θ , – π + θ, – θ, θ = tan–1 or ΙVth quadrant.
b , according as z = a + ib lies in Ι, ΙΙ, ΙΙΙ a
Properties of arguments (i) arg(z1z2) = arg(z1) + arg(z2) + 2m π for some integer m. arg(z1/z2) = arg (z1) – arg(z2) + 2m π for some integer m. (ii) arg (z2) = 2arg(z) + 2m π for some integer m. (iii) (iv) arg(z) = 0 ⇔ z is real, for any complex number z ≠ 0 (v) arg(z) = ± π/2 ⇔ z is purely imaginary, for any complex number z ≠ 0 (vi) arg(z2 – z1) = angle of the line segment P′Q′ || PQ, where P′ lies on real axis, with the real axis.
Example: Solution
Example:
π 2π and Arg (z – 3 – 4i) = . 6 3 From the figure, it is clear that there is no z, which satisfy both ray
Solve for z, which satisfy Arg (z – 3 – 2i) =
Sketch the region given by (i) Arg (z – 1 – i) ≥ π/3 (ii) |z| = ≤ 5 & Arg (z – i – 1) > π/3
5
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(i)
Solution
(ii)
Self Practice Problems 1.
Sketch the region given by |Arg (z – i – 2)| < π/4 (i)
Arg (z + 1 – i) ≤ π/6
(ii)
2.
Consider the region |z – 15i| ≤ 10. Find the point in the region which has (i) max |z| (ii) min |z| (iii) max arg (z) (iv) min arg (z)
7.
Conjugate of a complex Number Conjugate of a complex number z = a + b is denoted and defined by z = a – ib. In a complex number if we replace i by – i, we get conjugate of the complex number. z is the mirror image of z about real axis on Argand's Plane.
(ix) If w = f(z), then w = f( z ) ( z1 ) = z arg(z) + arg( z ) = 0 z −1 is purely imaginary, then prove that | z | = 1 If z +1 z − 1 =0 Re z + 1
Example: Solution.
⇒ ⇒ ⇒ ⇒
z − 1 z −1 z −1 z −1 =0 ⇒ + + =0 z + 1 z +1 z + 1 z +1 zz – z + z – 1 + zz – z + z – 1 = 0 zz = 1 ⇒ | z |2 = 1 |z|=1 Hence proved
Self Practice Problem z 1 − 2z 2 1. If is unmodulus and z2 is not unimodulus then find |z1|. 2 − z1z 2 Ans. |z1| = 2
8. (i) (ii)
Rotation theorem
and Q(zz) are two complex numbers such that |z1| = |z2|, then z2 = z1 ei θ where θ = ∠POQ If P(z1), Q(z2) and R(z3) are three complex numbers and ∠PQR = θ, then
I f
P
( z
1)
z3 − z2 = z1 − z 2
z3 − z2 iθ z1 − z 2 e 6
If P(z1), Q(z2), R(z3) and S(z4) are four complex numbers and ∠STQ = θ , then z3 − z2 z1 − z 2 =
Example: Solution
7 of 38
(iii)
z3 − z 4 iθ z1 − z 2 e
z − 1 π = then interrupter the locus. If arg z + i 3 z − 1 π = arg 3 z+i 1− z π = arg 3 − 1− z 1− z represents the angle between lines joining –1 and z and 1 + z. As this angle Here arg − 1− z is constant, the locus of z will be a of a circle segment. (angle in a segment is count). It can be 1− z 2π will be equal to – seen that locus is not the complete side as in the major are arg . − 1 − z 3 Now try to geometrically find out radius and centre of this circle. 1 2 centre ≡ 0, Radius ≡ Ans. 3 3 If A(z + 3i) and B(3 + 4i) are two vertices of a square ABCD (take in anticlock wise order) then find C and D. Let affix of C and D are z3 + z4 respectively Considering ∠DAB = 90º + AD = AB z − (2 + 3 i) (3 + 4 i) − (2 + 3 i) iπ we get 4 = e AD AB 2 ⇒ z4 – (2 + 3i) = (1 + i) i ⇒ Z4 = 2 + 3i+ i – 1 = 1 + zi z 3 − (3 + 4i) ( z + 3i) − (3 − 4i) iπ e– and = CB AB 2 ⇒ z3 = 3 + 4i – (1 + i) (–i) z3 = 3 + 4i + i – 1 = z + 5i
⇒
Example: Solution.
Self Practice Problems 1.
z1, z2, z3, z4 are the vertices of a square taken in anticlockwise order then prove that 2z2 = (1 + i) z1 + (1 – i) z3 Ans. (1 + i) z1 + (1 – i)z3
2.
Check that z1z2 and z3z4 are parallel or, not where, z1 = 1 + i z3 = 4 + 2i z2 = 2 – i z4 = 1 – i Ans. Hence, z1z2 and z3z4 are not parallel.
3.
P is a point on the argand diagram on the circle with OP as diameter “two point Q and R are taken such that ∠POQ = ∠QOR If O is the origin and P, Q, R are represented by complex z1, z2, z3 respectively then show that z22 cos 2θ = z1z3cos2θ Ans. z1z3 cos2θ
9.
Demoivre’s Theorem: Case Ι Statement : If n is any integer then (i) (cos θ + i sin θ )n = cos nθ + i sin nθ (ii) (cos θ1 + i sin θ 1) (cos θ 2) + i sin θ 2) (cosθ3 + i sin θ 2) (cos θ 3 + i sin θ3) .....(cos θ n + i sin θ n) = cos (θ 1 + θ2 + θ 3 + ......... θn) + i sin (θ1 + θ 2 + θ3 + ....... + θn) Case ΙΙ Statement : If p, q ∈ Z and q ≠ 0 then 2kπ + pθ 2k π + pθ + i sin (cos θ + i sin θ)p/q = cos q q where k = 0, 1, 2, 3, ......, q – 1 7
NOTE : Continued product of the roots of a complex quantity should be determined using theory of equations. 8 of 38
1 0 . Cube Root Of Unity :
The cube roots of unity are 1, − 1 + i 3 , − 1 − i 3 . 2 2 ω is one of the imaginary cube roots of unity then 1 + ω + ω² = 0. In general 1 + ωr + ω2r = 0; where r ∈ I but is not the multiple of 3. In polar form the cube roots of unity are : 2π 2π 4π 4π cos 0 + i sin 0; cos + i sin , cos + i sin 3 3 3 3 The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral triangle. The following factorisation should be remembered : (a, b, c ∈ R & ω is the cube root of unity) a3 − b3 = (a − b) (a − ωb) (a − ω²b) ; x 2 + x + 1 = (x − ω) (x − ω2) ; 3 3 2 a + b = (a + b) (a + ωb) (a + ω b) ; a2 + ab + b2 = (a – bw) (a – bw2) a3 + b3 + c3 − 3abc = (a + b + c) (a + ωb + ω²c) (a + ω²b + ωc)
If 1, α1, α2, α3..... αn − 1 are the n, nth root of unity then :
(i)
They are in G.P. with common ratio ei(2π/n)
(ii)
1p + α 1 + α 2 +.... + α n − 1 = 0 if p is not an integral multiple of n = n if p is an integral multiple of n (1 − α1) (1 − α2)...... (1 − αn − 1) = n & (1 + α1) (1 + α2)....... (1 + αn − 1) = 0 if n is even and 1 if n is odd.
(iii) (iv) Example: Solution.
p
&
p
p
1. α1. α2. α3......... αn − 1 = 1 or −1 according as n is odd or even. Find the roots of the equation z6 + 64 = 0 where real part is positive. z6 = – 64 z6 = z6 . e+ i(2n + 1) π x∈z ⇒
z=z e
i( 2n+1)
i
π 6
π 6 i
π 2
i
π 2
i
5π 6
i
7π 6
∴
z=2 e
∴
roots with +ve real part are = e 6 + e
, 2e
, ze
, ze
= e iπ
2e
π i − 6
Ans. 8
i
, ze
11π 6
i
3π 2
, ze
i
11π 2
k =1
6
Solution.
6
2πk sin – 7 k =1
∑
6
=
∑
sin
k =0
∑ sin
Find the value
2πk 2πk − cos 7 7
∑ cos k =1
2πk 7 –
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6
Example:
2πk 7
6
∑ cos k =0
2πk 7 +1
6
=
∑
(Sum of imaginary part of seven seventh roots of unity)
k =0
6
–
∑
(Sum of real part of seven seventh roots of unity) + 1
k =0
0–0+1=1 i Ans. Self Practice Problems =
1.
Resolve z7 – 1 into linear and quadratic factor with real coefficient. 2π 4π 6π 2 z + 1 . z 2 − 2 cos z + 1 . z 2 − 2 cos z + 1 Ans. (z – 1) z − 2 cos 7 7 7
2.
Find the value of cos Ans. –
2π 4π 6π + cos + cos . 7 7 7
1 2
1 2 . The Sum Of The Following Series Should Be Remembered : (i)
cos θ + cos 2 θ + cos 3 θ +..... + cos n θ =
(ii)
sin θ + sin 2 θ + sin 3 θ +..... + sin n θ =
sin ( nθ / 2)
n + 1
sin (θ / 2) cos 2 θ.
sin ( nθ / 2)
n + 1
sin (θ / 2) sin 2 θ.
NOTE : If θ = (2π /n) then the sum of the above series vanishes.
1 3 . Logarithm Of A Complex Quantity : 1 −1 β Loge ( α² + β ²) + i 2 n π + tan where n ∈ Ι. α 2
(i)
Loge ( α + i β ) =
(ii)
ii represents a set of positive real numbers given by e
Example:
, n ∈ Ι.
Find the value of π ) 3
Ans.
log2 + i(2nπ +
(ii) (iii)
log (1 + 3 i) log(–1) zi
Ans. Ans.
iπ cos(ln2) + i sin(ln2) = ei(ln2)
(iv)
ii
Ans.
e
(v)
|(1 + i)i |
Ans.
(vi)
arg ((1 + i)i)
Ans.
4 e 1 n(2). 2
(i)
log (1 + 3 i)
(iii)
2i = ein 2
(i)
Solution.
π − 2 n π + 2
−( 4n +1). −( 8n+1).
π i + 2nπ = log 2 e 3 π = log 2 + i + 2nπ 3 = cos (n 2) cos (n 2) + i sin (n 2) ]
9
π 2 π
Self Practice Problem
Find the real part of cos (1 + i) Ans.
10 of 38
1.
1− e2 2ei
1 4 . Geometrical Properties : Distance formula :
If z1 and z2 are affixies of the two points ↓ P and Q respectively then distance between P + Q is given by |z1 – z2|. Section formula If z1 and z2 are affixes of the two points P and Q respectively and point C devides the line joining P and Q internally in the ratio m : n then affix z of C is given by mz 2 + nz1 z= m+n If C devides PQ in the ratio m : n externally then mz 2 − nz1 z= m−n (b) If a, b, c are three real numbers such that az1 + bz2 + cz3 = 0 ; where a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers z1, z2 & z3 are collinear. (1)
If the vertices A, B, C of a ∆ represent the complex nos. z1, z2, z3 respectively and a, b, c are the length of sides then,
z1 + z 2 + z 3
(i)
Centroid of the ∆ ABC =
(ii)
Orthocentre of the ∆ ABC = (asec A )z1 + (b sec B)z 2 + (c secC)z 3 z1tan A + z 2 tanB + z 3 tan C or asec A + bsec B + c secC tanA + tan B + tanC Incentre of the ∆ ABC = (az1 + bz2 + cz3) ÷ (a + b + c).
(iii) (iv)
3
:
Circumcentre of the ∆ ABC = : (Z1 sin 2A + Z2 sin 2B + Z3 sin 2C) ÷ (sin 2A + sin 2B + sin 2C).
(2)
amp(z) = θ is a ray emanating from the origin inclined at an angle θ to the x − axis.
(3)
z − a = z − b is the perpendicular bisector of the line joining a to b.
(4)
The equation of a line joining z1 & z2 is given by, z = z1 + t (z1 − z2) where t is a real parameter.
(5)
z = z1 (1 + it) where t is a real parameter is a line through the point z1 & perpendicular to the line joining z1 to the origin.
(6)
The equation of a line passing through z1 & z2 can be expressed in the determinant form as
z z1
z 1 z1 1 = 0. This is also the condition for three complex numbers to be collinear.. The above z 2 z2 1 equation on manipulating, takes the form α z + α z + r = 0 where r is real and α is a non zero complex constant. NOTE : If we replace z by zeiθ and z by ze – iθ then we get equation of a straight line which. Passes through the foot of the perpendicular from origin to given straight line and makes an angle θ with the given straightl line.
(7)
The equation of circle having centre z0 & radius ρ is : z − z0 = ρ or z z − z0 z − z 0 z + z 0 z0 − ρ² = 0 which is of the form
z z + α z + α z + k = 0, k is real. Centre is − α & radius = α α − k . Circle will be real if α α − k ≥ 0.. (8)
(9)
The equation of the circle described on the line segment joining z1 & z2 as diameter is z − z2 π = ± or (z − z1) ( z − z 2) + (z − z2) ( z − z 1) = 0. arg z − z1 2 Condition for four given points z1, z2, z3 & z4 to be concyclic is the number z 3 − z1 z 4 − z 2 . should be real. Hence the equation of a circle through 3 non collinear z 3 − z 2 z 4 − z1
Successful People Replace the words like; "wish", "try" 10 & "should" with "I Will". Ineffective People don't.
⇒ (10)
z − z1 Arg z − z = θ represent (i) a line segment if θ = π 2
(ii) (11)
is real 11 of 38
( z − z 2 ) ( z 3 − z1 ) ( z − z1 ) ( z 3 − z 2 ) ( z − z 2 ) ( z 3 − z 1 ) ( z − z 2 ) ( z 3 − z1 ) = . ( z − z 1 ) ( z 3 − z 2 ) ( z − z1 ) ( z 3 − z 2 )
points z1, z2 & z3 can be taken as
Pair of ray if θ = 0 (iii) a part of circle, if 0 < θ < π.
z1 1 z2 Area of triangle formed by the points z1, z2 & z3 is 4i z3
z1 1 z2 1 z3 1
| α z 0 + αz 0 + r | 2|α|
(12)
Perpendicular distance of a point z0 from the line α z + αz + r = 0 is
(13)
(i)
Complex slope of a line αz + αz + r = 0 is ω = –
(ii)
z1 − z 2 Complex slope of a line joining by the points z1 & z2 is ω = z − z 1 2 Complex slope of a line making θ angle with real axis = e2iθ
(iii)
α . α
(14)
ω1 & ω2 are the compelx slopes of two lines. (i) If lines are parallel then ω1 = ω2 (ii) If lines are perpendicular then ω1 + ω2 = 0
(15)
If |z – z1| + |z – z2| = K > |z1 – z2| then locus of z is an ellipse whose focii are z1 & z2
(16)
If |z – z0| =
α z + αz + r 2|α|
then locus of z is parabola whose focus is z0 and directrix is the
line α z0 + α z0 + r = 0 (17)
z − z1 If z − z 2
(18)
If z – z1 – z – z2 = K < z1 – z2 then locus of z is a hyperbola, whose focii are z1 & z2 .
= k ≠ 1, 0, then locus of z is circle.
Match the following columns : Column - Ι (i) If | z – 3+2i | – | z + i | = 0, then locus of z represents ..........
(ii) (iii)
(iv)
(v) (vi) (vii) (viii) Ans.
Column - ΙΙ (i) circle
z − 1 π = , If arg z + 1 4 then locus of z represents... if | z – 8 – 2i | + | z – 5 – 6i | = 5 then locus of z represents .......
z − 3 + 4i 5π If arg z + 2 − 5i = , 6 then locus of z represents .......
If | z – 1 | + | z + i | = 10 then locus of z represents ........ |z–3+i|–|z+2–i|=1 then locus of z represents ..... | z – 3i | = 25 z − 3 + 5i arg z + i = π Ι (i) (ii) ΙΙ (vii) (v)
(ii)
Straight line
(iii)
Ellipse
(iv)
Hyperbola
(v) Major Arc (vi) Minor arc (vii) Perpendicular bisector of a line segment (viii) Line segment
(iii) (viii)
(iv) (vi)
(v) (iii)
(vi) (iv) 11
(vii) (i)
(viii) (ii)
(a)
Reflection points for a straight line : Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z1 & z2 will be the reflection points for the straight line α z + α z + r = 0 if and only if;
12 of 38
15.
α z1 + α z2 + r = 0 , where r is real and α is non zero complex constant.
(b)
Inverse points w.r.t. a circle : Two points P & Q are said to be inverse w.r.t. a circle with centre 'O' and radius ρ, if: (i) the point O, P, Q are collinear and P, Q are on the same side of O. (ii) OP. OQ = ρ2.
Note : that the two points z1 & z2 will be the inverse points w.r.t. the circle z z + α z + α z + r = 0 if and only
if z1 z2 + α z1 + α z2 + r = 0 .
1 6 . Ptolemy’s Theorem: It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the products of lengths of the two pairs of its opposite sides. i.e. z1 − z3 z2 − z4 = z1 − z2 z3 − z4 + z1 − z4 z2 − z3. Example:
Solution.
If cos α + cos β + cos γ = 0 and also sin α + sin β + sin γ = 0, then prove that (i) cos 2α + cos2β + cos2γ = sin 2α + sin 2β + sin 2γ = 0 (ii) sin 3α + sin 3β + sin 3γ = 3 sin ( α + β + γ) (iii) cos 3α + cos 3β + cos 3γ = 3 cos (α + β + γ) Let z1 = cos α + i sin α, z2 = cos β + i sin β , z3 = cosγ + i sin γ. ∴ z 1 + z2 + z3 = (cos α + cos β + cos γ) + i (sin α + sin β + sin γ) =0+i.0=0 1 –1 (i) Also z1 = (cos α + i sin α) = cos α – i sin α
(1)
1 1 = cos β – i sin β , z2 z 3 – cos γ – sin γ
∴
(ii)
1 1 1 + + z1 z2 z 3 = (cos α + cos β + cos γ) – i (sin α + sin β + sin γ) (2)
= 0–i.0=0 Now z12 + z22 + z33 = (z1 + z2 + z3)2 – 2 (z1z2 + z2z3 + z3z1 ) 1 1 1 = 0 – 2z1z2z3 z + z + z 1 2 3 = 0 – 2z1 z2 z3. 0 = 0, using (1) and (2) or (cos α + i sin α)2 + (cos β + i sin β )2 + (cos γ + i sin γ)2 = 0 or cos 2α + i sin 2α)2 + cos 2β + i sin 2β + cos 2γ + i sin 2γ = 0 + i.0 Equation real and imaginary parts on both sides, cos 2α + cos 2β + cos 2γ = 0 and sin 2α + sin 2β + sin 2γ = 0 z 1 3 + z 23 + z 33 = (z1 + z2)3 – 3z1z2(z1 + z2) + z33 = (–z3)3 – 3z1z2 (– z3) + z33, using (1) = 3z1z2z3 ∴ (cos α + i sin α)3 + (cos β + i sin β )3 + (cos γ + i sin γ)3 = 3 (cos α + i sin α) (cos β + i sin β ) (cos γ + i sin γ) or cos 3α + i sin 3α + cos 3β + i sin 3β + cos 3γ + i sin 3γ = 3{cos(α + β + γ) + i sin (α + β + γ) Equation imaginary parts on both sides, sin 3α + sin 3β + sin 3γ = 3 sin (α + β + γ)
Alternative method Let
C ≡ cos α + cos β + cos γ = 0 S ≡ sin α + sin β + sin γ = 0 C + iS = eiα + eiβ + eiγ = 0 (1) C – iS = e–iα + e–iβ + e–iγ = 0 (2) From (1) ⇒ (e–iα )2 + (e–iβ )2 + (e–iγ )2 = (eiα) (eiβ ) + (eiβ ) (eiγ ) + (eiγ) (eiα) ⇒ ei2α + ei2β + ei2γ = eiα eiβ eiγ (e–2γ + e–iα + eiβ ) ⇒ ei(2α) + ei2β + ei2γ = 0 (from 2) Comparing the real and imaginary parts we cos 2α + cos 2β + cos 2γ – sin 2α + sin 2β + sin 2γ = 0 Also from (1) (eiα) 3 + (eiβ ) 3 + (eiγ )3 = 3eiα eiβ eiγ ⇒ ei3α + ei3β + ei3γ = 3ei(α+β+γ) Comparing the real and imaginary parts we obtain the results.
Example:
If z1 and z2 are two complex numbers and c > 0, then prove that 12
or z1 z 2 + z 2z2 ≤ c|z1|2 + c–1|z2|2
or c|z1|2 +
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Solution.
|z1 + z2|2 ≤ (I + C) |z1|2 + (I +C–1) |z2|2 We have to prove : |z1 + z2|2 ≤ (1 + c) |z1|2 + (1 + c–1) |z2|2 i.e. |z1|3 + |z2|2 + z1 z 2 + z 2z2 ≤ (1 + c) |z1|2 + (1 +c–1) |z2|3
1 |z |2 – z1 z 2 – z 2 z2 ≥ 0 c 2
(using Re (z1 z 2) ≤ |z1 z 2|) or Example:
Solution.
1 c z1 − | z2 c
2
| ≥ 0
which is always true.
If θ , ∈ [ π/6, π/3], i = 1, 2, 3, 4, 5, and z4 cos θ1 + z3 cos θ2 + z3 cos θ3. + z cos θ 4 + cosθ5 = 2 3 , 3 then show that |z| > 4 Given that 4 cosθ1 . z + cosθ2 . z3 + cosθ3 . z2 + cosθ 4 . z + cosθ5 = 2√ 3 or |cosθ1 . z4 + cosθ2 . z3 + cosθ3 . z2 + cosθ4 . z + cosθ5| = 2√ 3 2√ 3 ≤ |cosθ 1 . z4 | + |cosθ2 . z3 | + |cosθ3 . z2 | + cosθ 4 . z| + |cosθ 5 | ∵ θi ∈ [ π/6, π/3] 1 3 ≤ cosθ i ≤ 2 2 3 3 3 3 2 3 3 |z|4 + |z| + |z| + |z| + 2 3 ≤ 2 2 2 2 2 3 ≤ |z|4 + |z|3 + |z|2 + |z| 3 < |z| + |z|2 + |z|3 + |z|4 +|z|5 + .........∞ |z| 3 < 1− | z | 3 – e |z| < |z|
∴
4|z| > 3 Example:
∴
|z| >
3 4
Two different non parallel lines cut the circle |z| = r in point a, b, c, d respectively. Prove that these lines meet in the point z given by z =
Solution.
a −1 + b −1 − c −1 − d −1 a −1b −1 − c −1d −1
Since point P, A, B are collinear
z
z 1
(
)
(
)
a a 1 =0 ⇒ z a − b – z (a – b) + a b − a b = 0 b b 1 Similarlym, since points P, C, D are collinear ∴ z a − b (c – d) – z c − d (a – b) = c d − cd (a – b) – a b − a b (c – d)
∴
(
)
(
)
(
)
(
)
k k k 2 ∴ a = a , b = b , c = c etc. zz = r = k (say) From equation (iii) we get k k k k ck kd ak bk − − (a – b) – (c – d) z − (c – d) – z − (a – b) = a b c d d c a b
∵
∴
z=
a −1 + b −1 − c −1 − d −1 a −1b −1 − c −1d −1
13
(i)
(iii)
1.
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Short Revision
DEFINITION : Complex numbers are definited as expressions of the form a + ib where a, b ∈ R & i = − 1 . It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z). EVERY COMPLEX NUMBER CAN BE REGARDED AS Purely real if b = 0
Purely imaginary if a = 0
Imaginary if b ≠ 0
Note : (a) The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complete Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C. (b) Zero is both purely real as well as purely imaginary but not imaginary. (c) i = − 1 is called the imaginary unit. Also i² = − l ; i3 = −i ; i4 = 1 etc. (d)
a
b = a b only if atleast one of either a or b is non-negative.
2.
CONJUGATE COMPLEX : If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z . i.e. z = a − ib. Note that : (i) z + z = 2 Re(z) (ii) z − z = 2i Im(z) (iii) z z = a² + b² which is real (iv) If z lies in the 1st quadrant then z lies in the 4th quadrant and − z lies in the 2nd quadrant. 3. ALGEBRAIC OPERATIONS : The algebraic operations on complex numbers are similiar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative. e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless . However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers, z12 + z22 = 0 does not imply z1 = z2 = 0. 4. EQUALITY IN COMPLEX NUMBER : Two complex numbers z1 = a1 + ib1 & z2 = a2 + ib2 are equal if and only if their real & imaginary parts coincide. 5. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS : (a) Cartesian Form (Geometric Representation) : Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x, y). length OP is called modulus of the complex number denoted by z & θ is called the argument or amplitude . eg. z = x 2 + y 2 &
θ = tan−1
y (angle made by OP with positive x−axis) x
z
(ii) (iii) (iv) (v) (vi)
if z > 0
z is always non negative . Unlike real numbers z = is not correct − z if z < 0 Argument of a complex number is a many valued function . If θ is the argument of a complex number then 2 nπ + θ ; n ∈ I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nπ. The unique value of θ such that – π < θ ≤ π is called the principal value of the argument. Unless otherwise stated, amp z implies principal value of the argument. By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus. There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.
NOTE :(i)
14
(c) 6. (a)
Trignometric / Polar Representation : z = r (cos θ + i sin θ) where | z | = r ; arg z = θ ; z = r (cos θ − i sin θ) Note: cos θ + i sin θ is also written as CiS θ. eix + e −ix eix − e −ix & sin x = are known as Euler's identities. Also cos x = 2 2 Exponential Representation : z = reiθ ; | z | = r ; arg z = θ ; z = re− iθ
IMPORTANT PROPERTIES OF CONJUGATE / MODULI / AMPLITUDE : If z , z1 , z2 ∈ C then ; z + z = 2 Re (z) ; z − z = 2 i Im (z) ; z1 − z 2 = z1 − z 2
(b)
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(b)
(z) = z
;
z1 + z 2 = z1 + z 2 ;
z1 = z1 z z2 2
; z1 z 2 = z1 . z 2
; z2 ≠ 0
2 | z | ≥ 0 ; | z | ≥ Re (z) ; | z | ≥ Im (z) ; | z | = | z | = | – z | ; z z = | z | ; |z | z1 | z1 z2 | = | z1 | . | z2 | ; = 1 , z2 ≠ 0 , | zn | = | z |n ; | z2 | z2
amp 1 = amp z1 − amp z2 + 2 kπ ; k ∈ I z2 amp(zn) = n amp(z) + 2kπ . where proper value of k must be chosen so that RHS lies in (− π , π ].
VECTORIAL REPRESENTATION OF A COMPLEX : Every complex number can be considered as if it is the position vector of that point. If the point P →
→
represents the complex number z then, OP = z & OP = z. NOTE : →
(i) (ii)
(iii) 8.
9. (ii) (iii) (iv) (v)
→
→
→
If OP = z = r ei θ then OQ = z1 = r ei (θ + φ) = z . e iφ. If OP and OQ are Λ
Λ
of unequal magnitude then OQ = OP e iφ If A, B, C & D are four points representing the complex numbers z1, z2 , z3 & z4 then z 4 − z3 z −z AB CD if 4 3 is purely real ; AB ⊥ CD if z − z is purely imaginary ] z − z1 2 1 If z1, z2, z3 are the2vertices of an equilateral triangle where z0 is its circumcentre then (a) z 12 + z 22 + z 23 − z1 z2 − z2 z3 − z3 z1 = 0 (b) z 12 + z 22 + z 23 = 3 z 20 DEMOIVRE’S THEOREM : Statement : cos n θ + i sin n θ is the value or one of the values of (cos θ + i sin θ)n ¥ n ∈ Q. The theorem is very useful in determining the roots of any complex quantity Note : Continued product of the roots of a complex quantity should be determined using theory of equations. − 1 + i 3 − 1− i 3 CUBE ROOT OF UNITY : (i) The cube roots of unity are 1 , , . 2 2 If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general 1 + wr + w2r = 0 ; where r ∈ I but is not the multiple of 3. In polar form the cube roots of unity are : 2π 4π 2π 4π + i sin , cos + i sin cos 0 + i sin 0 ; cos 3 3 3 3 The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral triangle. The following factorisation should be remembered : (a, b, c ∈ R & ω is the cube root of unity) 15
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10.
a3 − b3 = (a − b) (a − ωb) (a − ω²b) ; x2 + x + 1 = (x − ω) (x − ω2) ; a3 + b3 = (a + b) (a + ωb) (a + ω2b) ; a3 + b3 + c3 − 3abc = (a + b + c) (a + ωb + ω²c) (a + ω²b + ωc) nth ROOTS OF UNITY : If 1 , α1 , α2 , α3 ..... αn − 1 are the n , nth root of unity then : (i) They are in G.P. with common ratio ei(2π/n) &
1p + α 1p + α 2p + .... +α pn − 1 = 0 if p is not an integral multiple of n = n if p is an integral multiple of n & (iii) (1 − α1) (1 − α2) ...... (1 − αn − 1) = n (1 + α1) (1 + α2) ....... (1 + αn − 1) = 0 if n is even and 1 if n is odd. (iv) 1 . α1 . α2 . α3 ......... αn − 1 = 1 or −1 according as n is odd or even. 11. THE SUM OF THE FOLLOWING SERIES SHOULD BE REMEMBERED : sin (nθ 2) n +1 cos (i) cos θ + cos 2 θ + cos 3 θ + ..... + cos n θ = θ. sin (θ 2) 2 sin (nθ 2) n + 1 (ii) sin θ + sin 2 θ + sin 3 θ + ..... + sin n θ = sin θ. sin (θ 2) 2 Note : If θ = (2π/n) then the sum of the above series vanishes. 12. STRAIGHT LINES & CIRCLES IN TERMS OF COMPLEX NUMBERS : nz + mz 2 (A) divides the joins of z1 If z1 & z2 are two complex numbers then the complex number z = 1 m+n & z2 in the ratio m : n. Note:(i) If a , b , c are three real numbers such that az1 + bz2 + cz3 = 0 ; where a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers z1 , z2 & z3 are collinear. (ii) If the vertices A, B, C of a ∆ represent the complex nos. z1, z2, z3 respectively, then : z1 + z 2 + z 3 : (a) Centroid of the ∆ ABC = 3 (b) Orthocentre of the ∆ ABC = (a sec A )z1 + (b sec B)z 2 + (c sec C)z 3 z tan A + z 2 tan B + z 3 tan C OR 1 a sec A + b sec B + c sec C tan A + tan B + tan C (c) Incentre of the ∆ ABC = (az1 + bz2 + cz3) ÷ (a + b + c) . (d) Circumcentre of the ∆ ABC = : (Z1 sin 2A + Z2 sin 2B + Z3 sin 2C) ÷ (sin 2A + sin 2B + sin 2C) . amp(z) = θ is a ray emanating from the origin inclined at an angle θ to the x− axis. (B) z − a = z − b is the perpendicular bisector of the line joining a to b. (C) (D) The equation of a line joining z1 & z2 is given by ; z = z1 + t (z1 − z2) where t is a perameter. (E) z = z1 (1 + it) where t is a real parameter is a line through the point z1 & perpendicular to oz1. (F) The equation of a line passing through z1 & z2 can be expressed in the determinant form as (ii)
(G)
(H)
(I)
(J)
z z 1 z1 z1 1 = 0. This is also the condition for three complex numbers to be collinear.. z 2 z2 1 Complex equation of a straight line through two given points z1 & z2 can be written as z (z1 − z 2 ) − z (z1 − z 2 )+ (z1z 2 − z1z 2 ) = 0, which on manipulating takes the form as α z + α z + r = 0 where r is real and α is a non zero complex constant. The equation of circle having centre z0 & radius ρ is : z − z0 = ρ or z z − z0 z − z 0 z + z 0 z0 − ρ² = 0 which is of the form
zz + αz+αz +r = 0 , r is real centre − α & radius αα −r . Circle will be real if α α − r ≥ 0 . The equation of the circle described on the line segment joining z1 & z2 as diameter is : z − z2 π (i) arg = ± or (z − z1) ( z − z 2) + (z − z2) ( z − z 1) = 0 z − z1 2 Condition for four given points z1 , z2 , z3 & z4 to be concyclic is, the number 16
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z 3 − z1 z 4 − z 2 is real. Hence the equation of a circle through 3 non collinear points z1, z2 & z3 can be . z 3 − z 2 z 4 − z1 (z − z 2 ) (z3 − z1 ) (z − z 2 )(z3 − z1 ) (z − z2 )(z3 − z1 ) taken as is real ⇒ (z − z1 ) (z3 − z 2 ) (z − z1 )(z3 − z 2 ) = (z − z1 )(z3 − z2 )
13.(a) Reflection points for a straight line : Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z1 & z2 will be the reflection points for the straight line α z + α z + r = 0 if and only if ; α z + α z + r = 0 , where r is 1 2 real and α is non zero complex constant. (b) Inverse points w.r.t. a circle : Two points P & Q are said to be inverse w.r.t. a circle with centre 'O' and radius ρ, if : (i) the point O, P, Q are collinear and on the same side of O. (ii) OP . OQ = ρ2. Note that the two points z1 & z2 will be the inverse points w.r.t. the circle zz + α z + α z + r = 0 if and only if z1 z 2 + α z1 + α z 2 + r =0 . 14. PTOLEMY’S THEOREM : It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides. i.e. z1 − z3 z2 − z4 = z1 − z2 z3 − z4 + z1 − z4 z2 − z3. 15. LOGARITHM OF A COMPLEX QUANTITY : 1 β Loge (α + i β) = Loge (α² + β²) + i 2nπ + tan −1 where n ∈ I. (i) 2 α (ii)
ii
represents a set of positive real numbers given by
π − 2 nπ+ 2 e ,
n ∈ I.
VERY ELEMENTARY EXERCISE Q.1
Simplify and express the result in the form of a + bi 2 2 4i 3 − i (2 + i )2 − (2 − i )2 3 + 2i 3 − 2i 1 + 2i − 1 (a) (d) (e) + (b) −i (9 + 6 i) (2 − i) (c) 2−i 2+i 2 − 5i 2 + 5i 2+i 2i + 1 Q.2 Given that x , y ∈ R, solve : (a) (x + 2y) + i (2x − 3y) = 5 − 4i (b) (x + iy) + (7 − 5i) = 9 + 4i (c) x² − y² − i (2x + y) = 2i (d) (2 + 3i) x² − (3 − 2i) y = 2x − 3y + 5i (e) 4x² + 3xy + (2xy − 3x²)i = 4y² − (x2/2) + (3xy − 2y²)i Q.3 Find the square root of : (a) 9 + 40 i (b) −11 − 60 i (c) 50 i Q.4 (a) If f (x) = x4 + 9x3 + 35x2 − x + 4, find f ( – 5 + 4i) (b) If g (x) = x4 − x3 + x2 + 3x − 5, find g(2 + 3i) Q.5 Among the complex numbers z satisfying the condition z + 3 − 3 i = 3 , find the number having the least positive argument. Q.6 Solve the following equations over C and express the result in the form a + ib, a, b ∈ R. (a) ix2 − 3x − 2i = 0 (b) 2 (1 + i) x2 − 4 (2 − i) x − 5 − 3 i = 0 Q.7 Locate the points representing the complex number z on the Argand plane: 2 2 z−3 (a) z + 1 − 2i = 7 ; (b) z − 1 + z + 1 = 4 ; (c) = 3 ; (d) z − 3 = z − 6
z+3
Q.8
If a & b are real numbers between 0 & 1 such that the points z1 = a + i, z2 = 1 + bi & z3 = 0 form an equilateral triangle, then find the values of 'a' and 'b'. Q.9 For what real values of x & y are the numbers − 3 + ix2 y & x2 + y + 4i conjugate complex? Q.10 Find the modulus, argument and the principal argument of the complex numbers. (i) 6 (cos 310° − i sin 310°) Q.11
(ii) −2 (cos 30° + i sin 30°)
If (x + iy)1/3 = a + bi ; prove that 4 (a2 − b2) =
Q.12(a) If
(iii)
x y + . a b
a + ib a 2 + b2 = p + qi , prove that p2 + q2 = 2 2 . c + id c +d
2+i 4 i + (1 + i) 2
(b) Let z1, z2, z3 be the complex numbers such that z1 + z2 + z3 = z1z2 + z2z3 + z3z1 = 0. Prove that | z1 | = | z2 | = | z3 |. 1+ z + z2 Q.13 Let z be a complex number such that z ∈ c\R and ∈ R, then prove that | z | =1. 1 − z + z2 Q.14 Prove the identity, | 1 − z1z 2 |2 − | z1 − z 2 |2 = 1− | z1 |2 1− | z 2 |2
(
)(
17
)
2
[
2
]
Q.15 For any two complex numbers, prove that z1 + z 2 + z1 − z 2 = 2 z1 + z 2 . Also give the geometrical interpretation of this identity. Q.16 (a) Find all non−zero complex numbers Z satisfying Z = i Z². (b) If the complex numbers z1, z2, .................zn lie on the unit circle |z| = 1 then show that |z1 + z2 + ..............+zn| = |z1–1+ z2–1+................+zn–1| . Q.17 Find the Cartesian equation of the locus of 'z' in the complex plane satisfying, | z – 4 | + | z + 4 | = 16. Q.18 If ω is an imaginary cube root of unity then prove that : (b) (1 − ω + ω²)5 + (1+ ω − ω²)5 = 32 (a) (1 + ω − ω²)3 − (1− ω + ω²)3 = 0 (c) If ω is the cube root of unity, Find the value of, (1 + 5ω2 + ω4) (1 + 5ω4 + ω2) (5ω3 + ω + ω2). Q.19 If ω is a cube root of unity, prove that ; (i) (1 + ω − ω2)3 − (1 − ω + ω2)3 2
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2
(ii)
a + bω + c ω 2 = ω2 c + aω + b ω 2
(iii) (1 − ω) (1 − ω2) (1 − ω4) (1 − ω8) = 9
Q.20 If x = a + b ; y = aω + bω2 ; z = aω2 + bω, show that (i) xyz = a3 + b3 (ii) x2 + y2 + z2 = 6ab (iii) x3 + y3 + z3 = 3 (a3 + b3) 1 1+ i + w2 w2 −1 w 2 −1 = Q.21 If (w ≠ 1) is a cube root of unity then 1 − i − i − i + w −1 −1 (A) 0 (B) 1 (C) i (D) w 7 Q.22(a) (1 + w) = A + Bw where w is the imaginary cube root of a unity and A, B ∈ R, find the ordered pair (A, B). (b) The value of the expression ; 1. (2 − w) (2 − w²) + 2. (3 − w) (3 − w²) + ............. + (n − 1) . (n − w) (n − w²), where w is an imaginary cube root of unity is ________. n +1 nπ Q.23 If n ∈ N, prove that (1 + i)n + (1 − i)n = 2 2 . cos . 4 2n 2πk 2πk Q.24 Show that the sum ∑ sin − i cos simplifies to a pure imaginary number.. 2n + 1 2n + 1 k =1
Q.25 If x = cos θ + i sin θ & 1 + 1 − a 2 = na, prove that 1 + a cos θ =
a n (1 + nx) 1 + . 2n x
Q.26 The number t is real and not an integral multiple of π/2. The complex number x1 and x2 are the roots of the equation, tan2(t) · x2 + tan (t) · x + 1 = 0 2 nπ Show that (x1)n + (x2)n = 2 cos cotn(t). 3
EXERCISE-1
Q.1
Simplify and express the result in the form of a + bi : (a) −i (9 + 6 i) (2
(2 + i )2
−
− i)−1
(2 − i )2
4i 3 − i (b) 2i + 1
2
(c)
3 + 2i 3 − 2i + 2 − 5i 2 + 5i
(e) i + − i 2−i 2+i Q.2 Find the modulus , argument and the principal argument of the complex numbers. 10π 10π (i) z = 1 + cos (ii) (tan1 – i)2 + i sin 9 9 i −1 (iii) z = 5 + 12i + 5 − 12i (iv) 2π 2π 5 + 12i − 5 − 12i i 1 − cos + sin 5 5 Q.3 Given that x, y ∈ R, solve : x y 5 + 6i + = (a) (x + 2y) + i (2x − 3y) = 5 − 4i (b) 1 + 2i 3 + 2i 8i − 1 (c) x² − y² − i (2x + y) = 2i (d) (2 + 3i) x² − (3 − 2i) y = 2x − 3y + 5i (e) 4x² + 3xy + (2xy − 3x²)i = 4y² − (x2/2) + (3xy − 2y²)i Q.4(a) Let Z is complex satisfying the equation, z2 – (3 + i)z + m + 2i = 0, where m ∈ R. (d)
18
Q.6 Solve the following for z : (a) z2 – (3 – 2 i)z = (5i – 5) Q.7(a) If i Z3 + Z2 − Z + i = 0, then show that | Z | = 1.
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Suppose the equation has a real root, then find the value of m. (b) a, b, c are real numbers in the polynomial, P(Z) = 2Z4 + aZ3 + bZ2 + cZ + 3 If two roots of the equation P(Z) = 0 are 2 and i, then find the value of 'a'. Q.5(a) Find the real values of x & y for which z1 = 9y2 − 4 − 10 i x and z2 = 8y2 − 20 i are conjugate complex of each other. (b) Find the value of x4 − x3 + x2 + 3x − 5 if x = 2 + 3i (b) z+ z = 2 + i
z1 − 2z 2 = 1 and | z2 | ≠ 1, find | z1 |. 2 − z1z 2 z − z1 π is , then (c) Let z1 = 10 + 6i & z2 = 4 + 6i. If z is any complex number such that the argument of, z − z2 4 prove that z − 7 − 9i= 3 2 . Q.8 Show that the product, 2 22 2n 1+i 1+i 1+i 1+i 1+ 2 1+ 2 1+ 2 ......1+ 2 is equal to 1 − 1n (1+ i) where n ≥ 2 . 22 Q.9 Let a & b be complex numbers (which may be real) and let, Z = z3 + (a + b + 3i) z2 + (ab + 3 ia + 2 ib − 2) z + 2 abi − 2a. (i) Show that Z is divisible by, z + b + i. (ii) Find all complex numbers z for which Z = 0. (iii) Find all purely imaginary numbers a & b when z = 1 + i and Z is a real number. Q.10 Interpret the following locii in z ∈ C. z + 2i ≤ 4 (z ≠ 2i) (a) 1 < z − 2i < 3 (b) Re iz+2 (c) Arg (z + i) − Arg (z − i) = π/2 (d) Arg (z − a) = π/3 where a = 3 + 4i. Q.11 Prove that the complex numbers z1 and z2 and the origin form an isosceles triangle with vertical angle 2π/3 if z12 + z 22 + z1 z 2 = 0 . Q.12 P is a point on the Aragand diagram. On the circle with OP as diameter two points Q & R are taken such that ∠ POQ = ∠ QOR = θ. If ‘O’ is the origin & P, Q & R are represented by the complex numbers 2 1 , Z2 & Z3 respectively, show that : Z2 . cos 2 θ = Z1 . Z3 cos² θ.
(b) Let z1 and z2 be two complex numbers such that
Z
Q.13 Let z1, z2, z3 are three pair wise distinct complex numbers and t1, t2, t3 are non-negative real numbers such that t1 + t2 + t3 = 1. Prove that the complex number z = t1z1 + t2z2 + t3z3 lies inside a triangle with vertices z1, z2, z3 or on its boundry. Q.14 If a CiS α , b CiS β , c CiS γ represent three distinct collinear points in an Argand's plane, then prove the following : (i) Σ ab sin (α − β) = 0.
b 2 + c2 − 2bc cos(β − γ) ± (b CiS β) a 2 + c 2 − 2ac cos( α − γ ) ∓ (c CiS γ) a 2 + b 2 − 2ab cos( α − β) = 0. Q.15 Find all real values of the parameter a for which the equation (a − 1)z4 − 4z2 + a + 2 = 0 has only pure imaginary roots. (ii)
(a CiS α)
Q.16 Let A ≡ z1 ; B ≡ z2; C ≡ z3 are three complex numbers denoting the vertices of an acute angled triangle. If the origin ‘O’ is the orthocentre of the triangle, then prove that z1 z 2 + z1 z2 = z2 z 3 + z 2 z3 = z3 z1 + z 3 z1 hence show that the ∆ ABC is a right angled triangle ⇔ z1 z 2 + z1 z2 = z2 z 3 + z 2 z3 = z3 z1 + z 3 z1 = 0 Q.17 If the complex number P(w) lies on the standard unit circle in an Argand's plane and z = (aw+ b)(w – c)–1 then, find the locus of z and interpret it. Given a, b, c are real. Q.18(a) Without expanding the determinant at any stage , find K ∈ R such that 4i 8 + i 4 + 3i − 8 + i 16i i has purely imaginary value. − 4 + Ki i 8i (b) If A, B and C are the angles of a triangle
19
Q.20
20 of 38
Q.19
e −2iA eiC eiB iC − 2iB e eiA D= e where i = −1 then find the value of D. iB iA e e e − 2iC If w is an imaginary cube root of unity then prove that : (a) (1 − w + w2) (1 − w2 + w4) (1 − w4 + w8) ..... to 2n factors = 22n . (b) If w is a complex cube root of unity, find the value of (1 + w) (1 + w2) (1 + w4) (1 + w8) ..... to n factors . n nπ nπ 1 + sin θ + i cos θ Prove that = cos 2 − nθ + i sin 2 − nθ . Hence deduce that 1 + sin θ − i cos θ 5
5 π π π π 1 + sin + i cos + i 1 + sin − i cos = 0 5 5 5 5 Q.21 If cos (α − β) + cos (β − γ) + cos (γ − α) = − 3/2 then prove that : (a) Σ cos 2α = 0 = Σ sin 2α (b) Σ sin (α + β) = 0 = Σ cos (α + β) (c) Σ sin2 α = Σ cos2 α = 3/2 (d) Σ sin 3α = 3 sin (α + β + γ) (e) Σ cos 3α = 3 cos (α + β + γ) (f) cos3 (θ + α) + cos3 (θ + β) + cos3 (θ + γ) = 3 cos (θ + α) . cos (θ + β) . cos (θ + γ) where θ ∈ R. π Q.22 Resolve Z5 + 1 into linear & quadratic factors with real coefficients. Deduce that : 4·sin π ·cos = 1.
10
5
Q.23 If x = 1+ i 3 ; y = 1 − i 3 & z = 2 , then prove that for every prime p > 3. Q.24 If the expression z5 – 32 can be factorised into linear and quadratic factors over real coefficients as (z5 – 32) = (z – 2)(z2 – pz + 4)(z2 – qz + 4) then find the value of (p2 + 2p). Q.25(a) Let z = x + iy be a complex number, where x and y are real numbers. Let A and B be the sets defined by A = {z | | z | ≤ 2} and B = {z | (1 – i)z + (1 + i) z ≥ 4}. Find the area of the region A ∩ B. 1 (b) For all real numbers x, let the mapping f (x) = , where i = − 1 . If there exist real number x −i a, b, c and d for which f (a), f (b), f (c) and f (d) form a square on the complex plane. Find the area of the square. xp + yp = zp
EXERCISE-2
Q.1
p q r If q r p = 0 ; where p , q , r are the moduli of non−zero complex numbers u, v, w respectively,, r p q 2
prove that, arg Q.2 Q.3
w w − u = arg . v v−u
The equation x3 = 9 + 46i where i = − 1 has a solution of the form a + bi where a and b are integers. Find the value of (a3 + b3). Show that the locus formed by z in the equation z3 + iz = 1 never crosses the co-ordinate axes in the − Im( z) 2 Re( z) Im( z) + 1 2 If ω is the fifth root of 2 and x = ω + ω , prove that x5 = 10x2 + 10x + 6. Prove that , with regard to the quadratic equation z2 + (p + ip′) z + q + iq′ = 0 where p , p′, q , q′ are all real. (i) if the equation has one real root then q ′2 − pp ′ q ′ + qp ′2 = 0 . (ii) if the equation has two equal roots then p2 − p′2 = 4q & pp ′ = 2q ′. State whether these equal roots are real or complex. If the equation (z + 1)7 + z7 = 0 has roots z1, z2, .... z7, find the value of
Argand’s plane. Further show that |z| =
Q.4 Q.5
Q.6
7
(a)
∑ Re(Zr )
7
and
(b)
r =1
Q.7
r =1
Find the roots of the equation Zn = (Z + 1)n and show that the points which represent them are collinear on the complex plane. Hence show that these roots are also the roots of the equation 2
Q.8
∑ Im(Zr )
2
mπ mπ 2 Z + 1 = 0. 2 sin Z + 2 sin n n Dividing f(z) by z − i, we get the remainder i and dividing it by z + i, we get the remainder 20
1 + i. Find the remainder upon the division of f(z) by z² + 1. Let z1 & z2 be any two arbitrary complex numbers then prove that :
Q.10
1 ( | z1 | + | z 2 | ) z1 + z 2 . 2 | z1 | | z 2 | If Zr, r = 1, 2, 3, ......... 2m, m ε N are the roots of the equation
21 of 38
Q.9
z1 + z2 ≥
2m
Q.11
1
+ ............. + Z + 1 = 0 then prove that r∑=1 Z − 1 = − m r If (1 + x)n = C0 + C1x + C2x² + .... + Cn xn (n ∈ N), prove that :
Z2m
+
Z2m-1
+
Z2m-2
(a) C0 + C4 + C8 + .... =
1 2
nπ n −1 + 2 n / 2 cos 2 4
(b) C1 + C5 + C9 + .... =
(c) C2 + C6 + C10 + ..... = 1 2 n − 1 − 2 n / 2 cos n π 2 4 (e) C0 + C3 + C6 + C9 + ........ =
1 3
1 2
nπ n −1 + 2 n / 2 sin 2 4
(d) C3 + C7 + C11 + .... = 1 2 n − 1 − 2 n / 2 sin n π 2 4
nπ n 2 + 2 cos 3
Q.12 Let z1 , z2 , z3 , z4 be the vertices A , B , C , D respectively of a square on the Argand diagram taken in anticlockwise direction then prove that : (i) 2z2 = (1 + i) z1 + (1− i)z3 & (ii) 2z4 = (1− i) z1 + (1 + i) z3 n
Q.13 Show that all the roots of the equation 1 + i x = 1 + i a a ∈ R are real and distinct. 1 − ix
1 − ia
Q.14 Prove that: (a) cos x + nC1 cos 2x + nC2 cos 3x + ..... + nCn cos (n + 1) x = 2n . cosn (b) sin x + nC1 sin 2x + nC2 sin 3x + ..... + nCn sin (n + 1) x = 2n . cosn
x 2
n + 2 x 2
. cos
n + 2 x . sin x 2 2
2nπ 4π 6π 2π 1 = − When n ∈ N. + cos + ..... + cos + cos 2 n + 1 2 2 n + 1 2 n + 1 2 n + 1
(c) cos
Q.15 Show that all roots of the equation a0zn + a1zn – 1 + ...... + an – 1z + an = n,
n −1 . where | ai | ≤ 1, i = 0, 1, 2, .... , n lie outside the circle with centre at the origin and radius n Q.16 The points A, B, C depict the complex numbers z1 , z2 , z3 respectively on a complex plane & the angle 1 B & C of the triangle ABC are each equal to ( π − α ) . Show that 2 α (z2 − z3)² = 4 (z3 − z1) (z1 − z2) sin2 . 2
Q.17 Show that the equation
2 A1
x − a1
+
2
2
A2 An + ...... + = k has no imaginary root, given that: x − a2 x − an
a1 , a2 , a3 .... an & A1, A2, A3 ..... An, k are all real numbers.
a b c = = = k. Find the value of k. 1− b 1− c 1− a Let α, β be fixed complex numbers and z is a variable complex number such that,
Q.18 Let a, b, c be distinct complex numbers such that Q.19
2 z − α + z − β = k. Find out the limits for 'k' such that the locus of z is a circle. Find also the centre and radius of the circle. 2
Q.20 C is the complex number. f : C → R is defined by f (z) = | z3 – z + 2|. What is the maximum value of f on the unit circle | z | = 1? Q.21 Let f (x) = logcos 3x (cos 2 i x ) if x ≠ 0 and f (0) = K (where i = − 1 ) is continuous at x = 0 then find the value of K. Use of L Hospital’s rule or series expansion not allowed. Q.22 If z1 , z2 are the roots of the equation az2 + bz + c = 0, with a, b, c > 0 ; 2b2 > 4ac > b2 ; z1 ∈ third quadrant ; z2 ∈ second quadrant in the argand's plane then, show that 21
Q.23 Find the set of points on the argand plane for which the real part of the complex number (1 + i) z2 is positive where z = x + iy , x, y ∈ R and i = −1 .
Q.24 If a and b are positive integer such that N = (a + ib)3 – 107i is a positive integer. Find N. Q.25 If the biquadratic x4 + ax3 + bx2 + cx + d = 0 (a, b, c, d ∈ R) has 4 non real roots, two with sum 3 + 4i and the other two with product 13 + i. Find the value of 'b'.
EXERCISE-3 p
10 2qπ 2qπ ∑ (3 p + 2) ∑ sin 11 − i cos 11 . q =1 p =1 32
Q.1
Evaluate:
[REE '97, 6]
Q.2(a) Let z1 and z2 be roots of the equation z2 + pz + q = 0 , where the co−efficients p and q may be complex numbers. Let A and B represent z1 and z2 in the complex plane. If ∠AOB = α ≠ 0 and α 2
OA = OB, where O is the origin . Prove that p2 = 4 q cos2 . n −1
(b) Prove that
∑
k =1
(n − k) cos
2kπ n =− where n ≥ 3 is an integer . n 2
Q.3(a) If ω is an imaginary cube root of unity, then (1 + ω − ω2)7 equals (B) − 128ω (C) 128ω2 (A) 128ω
∑ (i n + i n+1 ) 13
(b) The value of the sum
n =1
−1 , equals (C) − i
[JEE '97, 5] (D) − 128ω2
, where i =
(B) i − 1
(A) i
[JEE '97 , 5]
(D) 0
[JEE' 98, 2 + 2 ]
Find all the roots of the equation (3z − 1)4 + (z − 2)4 = 0 in the simplified form of a + ib. [REE ’98, 6 ]
Q.4
334
1 i 3 +3 Q.5(a) If i = −1 , then 4 + 5 − + 2 2 (A) 1 − i 3 (B) − 1 + i 3
1 i 3 − + 2 2
365
is equal to : (D) − i 3
(C) i 3
(b) For complex numbers z & ω, prove that, z ω − ω z = z − ω if and only if, z = ω or z ω = 1 [JEE '99, 2 + 10 (out of 200)] 2
2πi
Q.6
2
20
If α = e 7 and f(x) = A0 + ∑ Ak xk, then find the value of, k =1 f(x) + f(α x) + ...... + f(α6x) independent of α .
[REE '99, 6] 1
1
1
+ Q.7(a) If z1 , z2 , z3 are complex numbers such that z1 = z2 = z3 = + = 1, then z1 z 2 z 3 z1 + z2 + z3 is : (A) equal to 1 (B) less than 1 (C) greater than 3 (D) equal to 3
, 'n' a positive integer, find the equation whose roots are, Given , z = cos 2 n + 1 + i sin 2n + 1 α = z + z3 + ...... + z2n − 1 & β = z2 + z4 + ...... + z2n . [ REE 2000 (Mains) 3 out of 100 ] z1 − z 3 1 − i 3 = Q.9(a) The complex numbers z1, z2 and z3 satisfying are the vertices of a triangle which is z2 − z3 2 (A) of area zero (B) right-angled isosceles (C) equilateral (D) obtuse – angled isosceles
Q.8
22
23 of 38
(b) Let z1 and z2 be nth roots of unity which subtend a right angle at the origin. Then n must be of the form (A) 4k + 1 (B) 4k + 2 (C) 4k + 3 (D) 4k [ JEE 2001 (Scr) 1 + 1 out of 35 ] Q.10 Find all those roots of the equation z12 – 56z6 – 512 = 0 whose imaginary part is positive. [ REE 2000, 3 out of 100 ] 1 1 1 1 3 2 Q.11(a) Let ω = − + i . Then the value of the determinant 1 −1 − ω ω 2 is 2 2 1 ω2 ω4 (A) 3ω (B) 3ω (ω – 1) (C) 3ω2 (D) 3ω(1 – ω) (b) For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 – 3 – 4i| = 5, the minimum value of |z1 – z2| is (A) 0 (B) 2 (C) 7 (D) 17 [JEE 2002 (Scr) 3+3] (c) Let a complex number α , α ≠ 1, be a root of the equation zp+q – zp – zq + 1 = 0 where p, q are distinct primes. Show that either 1 + α + α2 + ...... + αp–1 = 0 or 1 + α + α2 + ...... + αq–1 = 0 , but not both together. [JEE 2002, (5) ] Q.12(a) If z1 and z2 are two complex numbers such that | z1 | < 1 < | z2 | then prove that (b) Prove that there exists no complex number z such that | z | <
1 and 3
1 − z1 z 2 < 1. z1 − z 2
n
∑ a r zr r =1
= 1 where | ar | < 2.
[JEE-03, 2 + 2 out of 60] Q.13(a) ω is an imaginary cube root of unity. If (1 + ω2)m = (1 + ω4)m , then least positive integral value of m is (B) 5 (C) 4 (D) 3 (A) 6 [JEE 2004 (Scr)] (z − α) (b) Find centre and radius of the circle determined by all complex numbers z = x + i y satisfying = k, (z − β) [JEE 2004, 2 out of 60 ] where α = α1 + iα 2 , β = β1 + iβ2 are fixed complex and k ≠ 1. Q.14(a) The locus of z which lies in shaded region is best represented by (A) z : |z + 1| > 2, |arg(z + 1)| < π/4 (B) z : |z - 1| > 2, |arg(z – 1)| < π/4 (C) z : |z + 1| < 2, |arg(z + 1)| < π/2 (D) z : |z - 1| < 2, |arg(z - 1)| < π/2 (b) If a, b, c are integers not all equal and w is a cube root of unity (w ≠ 1), then the minimum value of |a + bw + cw2| is 1 3 (A) 0 (B) 1 (C) (D) 2 2 [JEE 2005 (Scr), 3 + 3] (c) If one of the vertices of the square circumscribing the circle |z – 1| = 2 is 2 + 3 i . Find the other vertices of square. [JEE 2005 (Mains), 4] w − wz Q.15 If w = α + iβ where β ≠ 0 and z ≠ 1, satisfies the condition that is purely real, then the set of 1− z values of z is (A) {z : | z | = 1} (B) {z : z = z ) (C) {z : z ≠ 1} (D) {z : | z | = 1, z ≠ 1} [JEE 2006, 3]
(a) x =1, y = 2; (b) (2, 9); (c) (−2 , 2) or − 3 , − 3 ; (d) (1 ,1)
2
2
23
5 0 , 2
(e) x = K, y =
3K 2
, K∈R
(a) ± (5 + 4i) ; (b) ± (5 − 6i) (c) ± 5(1 + i)
Q.5
–
3 3 3 i + 2 2
Q.4
(a) −160 ; (b) − (77 +108 i)
Q.6
(a) − i , − 2i (b)
3 − 5i 1+ i or − 2 2
Q.7
(a) on a circle of radius 7 with centre (−1, 2) ; (b) on a unit circle with centre at origin (c) on a circle with centre (−15/4, 0) & radius 9/4 ; (d) a straight line Q.9 x = 1, y = − 4 or x = − 1, y = − 4 Q.8 a = b = 2 − 3 ; 5π 5π Q.10 (i) Modulus = 6 , Arg = 2 k π + (K ∈ I) , Principal Arg = (K ∈ I) (ii) Modulus = 2 , Arg = 2 k π + (iii) Modulus = Q.16 (a)
18 7π 5π , Principal Arg = − 6 6
24 of 38
Q.3
18
5 , Arg = 2 k π − tan−1 2 (K ∈ I) , Principal Arg = − tan−12 6
3 i 3 i − ,i ; − , − 2 2 2 2
Q.17
n ( n + 1) −n 2
x 2 y2 + = 1 ; Q.18 64 48
(c) 64 ;
Q.21
A
2
Q.22 (a) (1, 1) ; (b)
EXERCISE-1 Q.1 (a)
21 12 − 5 5
i (b) 3 + 4 i (c) −
8 +0i 29
(d)
22 i (e) + 2 + 0 i or 0 ± 2 i 5
4π 4π 4π Q.2 (i) Principal Arg z = − ; z = 2 cos ; Arg z = 2 k π − k∈I 9 9 9 2 (ii) Modulus = sec 1 , Arg = 2 n π + (2 – π ) , Principal Arg = (2 – π )
(iii) Principal value of Agr z = − (iv) Modulus =
3 2 π π & z = ; Principal value of Arg z = & z = 2 2 2 3
1 π 11π 11π cos ec , Arg z = 2 nπ + , Principal Arg = 5 20 20 2
2 ti 5 , ti where t ∈ R − − − 3 3t + 5 (a) The region between the co encentric circles with centre at (0 , 2) & radii 1 & 3 units 1 1 (b) region outside or on the circle with centre + 2i and radius . 2 2 (c) semi circle (in the 1st & 4th quadrant) x² + y² = 1 (d) a ray emanating from the point (3 + 4i) directed away from the origin & having equation 3 x − y + 4 − 3 3 = 0 [−3 , −2] Q.17 (1 – c2) | z |2 – 2(a + bc) (Re z) + a2 – b2 = 0 (a) K = 3 , (b) – 4 Q.19 (b) one if n is even ; − w² if n is odd (Z + 1) (Z² − 2Z cos 36° + 1) (Z² − 2Z cos 108° + 1) Q.24 4 (a) π – 2 ; (b) 1/2
(ii) z = − (b + i) ; − 2 i , − a
(iii)
EXERCISE-2 Q.2
Q.6
35
Q.19 k >
(a) –
7 , (b) zero 2
Q.8
iz 1 + +i 2 2
Q.18 – ω or – ω2
1 2 α − β Q.20 | f (z) | is maximum when z = ω, where ω is the cube root unity and | f (z) | = 13 2
Q.21 K = –
4 9
24
25 of 38
Q.23 required set is constituted by the angles without their boundaries, whose sides are the straight lines y = ( 2 − 1) x and y + ( 2 + 1) x = 0 containing the x − axis Q.24 198 Q.25 51
EXERCISE-3 Q.1 48(1 − i) Q.4
Z=
Q.3 (a) D
(b) B
(29 + 20 2 ) + i(±15 + 25 2 ) , 82
Q.6 7 A0 + 7 A7
x7
+ 7 A14
x14
Q.9 (a) C, (b) D
(29 − 20 2 ) + i(±15 − 25 2 ) 82
Q.7 (a) A (b) A Q.10
+1 + i 3 ,
(±
Q.8
z2
3+i
),
2
Q.5 (a) C
sin 2 n θ 2π +z+ = 0, where θ = sin 2 θ 2n + 1
Q.11
2i
(
(a) B ; (b) B
)(
)
k β−α 1 | α − k 2β |2 − k 2 . | β |2 − | α |2 . k 2 − 1 , Radius = 2 2 k −1 ( k − 1) Q.15 D Q.14 (a) A, (b) B, (c) z2 = – 3 i ; z3 = 1 − 3 + i ; z4 = 1 + 3 − i
Q.13 (a) D ; (b) Centre ≡
2
(
)
(
)
EXERCISE-4 Part : (A) Only one correct option z −1 1. If |z| = 1 and ω = (where z ≠ –1), the Re(ω) is [IIT – 2003, 3] z +1 1 z 1 2 (A) 0 (B) − (D) (C) z + 1 . 2 2 | z + 1| | z + 1| | z + 1 |2 2. The locus of z which lies in shaded region (excluding the boundaries) is best represented by
[IIT – 2005, 3]
(A) z : |z + 1| > 2 and |arg (z + 1)| < π/4 (C) z : |z + 1| < 2 and |arg (z + 1)| < π/2 3.
4.
(B) z : |z – 1| > 2 and |arg (z – 1)| < π/4 (D) z : |z – 1| < 2 and |arg (z + 1)| < π/2 w − wz is purely real, then the set of If w = α, + iβ , where β ≠ 0 and z ≠ 1, satisfies the condition that 1− z values of z is [IIT – 2006, (3, –1)] (A) {z : |z| = 1} (B) {z : z = z } (C) {z : z ≠ 1} (D) {z : |z| = 1, z ≠1} If ( 3 + i)100 = 299 (a + ib), then b is equal to
(A) 5. 6. 7.
3
(B)
2
z − 8i = 0, then z lies on the curve If Re z+6 (A) x2 + y2 + 6x – 8y = 0 (B) 4x – 3y + 24 = 0
(D) none of these
(C) 4ab
(D) none of these
If n1, n2 are positive integers then : (1 + i)n1 + (1 + i3 )n1 + (1 − i5 )n2 + (1 − i7 )n2 is a real number if and only if (A) n1 = n2 + 1 (B) n1 + 1 = n2 (C) n1 = n2 (D) n1, n2 are any two positive integers The three vertices of a triangle are represented by the complex numbers, 0, z1 and z2. If the triangle is equilateral, then (A) z12 – z22 = z1z2 (B) z22 – z12 = z1 z2 (C) z12 + z22 = z1z2 (D) z12 + z22 + z1z2 = 0 5
8.
(C) 1
If x2 – x + 1 = 0 then the value of (A) 8
(B) 10
∑
n =1
2
n 1 x + n is x (C) 12
(D) none of these
Successful People Replace the words like; "wish", "try" 25 & "should" with "I Will". Ineffective People don't.
9.
If α is nonreal and α = 5 1 then the value of 2|1 + α + α 2 + α −2 − α −1| is equal to (A) 4 (B) 2 (C) 1 (D) none of these
10.
If z = x + iy and z1/3 = a − ib then (B) 2 6
11. 12.
(C) 3
6
5
(B) − 1 (C) 2 (A) 1 Expressed in the form r (cos θ + i sin θ), − 2 + 2i becomes :
−1 + i 3 −1 − i 3 −1 + i 3 −1 − i 3 + + + is equal to : 2 2 2 2 (D) none
3π 3π + i sin 4 4
(A) 2 2 cos − + i sin −
13.
)
26 of 38
(A) 1
(
x y − = k a 2 − b 2 where k = a b
(B) 2 2 cos
(D)
π π 2 cos − + i sin − 4 4
The number of solutions of the equation in z, z z - (3 + i) z - (3 - i) z - 6 = 0 is : (A) 0 (B) 1 (C) 2 (D) infinite If |z| = max {|z – 1|, |z + 1|} then 1 (A) |z + z | = (B) z + z = 1 (C) |z + z | = 1 (D) none of these 2 If P, P′ represent the complex number z1 and its additive inverse respectively then the complex equation of the circle with PP′ as a diameter is z z1 (A) z = (B) z z + z1 z1 = 0 (D) none of these (C) z z1 + z z1 = 0 1 z The points z1 = 3 + 3 i and z2 = 2 3 + 6 i are given on a complex plane. The complex number lying on the bisector of the angle formed by the vectors z1 and z2 is : (3 + 2 3 ) 3 +2 + i 2 2 (C) z = − 1 − i
(A) z =
(B) z = 5 + 5 i (D) none n
17.
1 + i tan α 1 + i tan n α when simplified reduces to : − 1 − i tan n α 1 − i tan α
The expression
18. 19.
z)6
1 2 1
(C) z2 +
2
1
(1 ± i) (z1 − z2)
(1 ± i) (z1 + z2)
(B) z2 +
(1 ± i) (z2 − z 1)
(D) none of these
2
If z = x + i y then the equation of a straight line Ax + By + C = 0 where A, B, C ∈ R, can be written on the complex plane in the form a z + a z + 2 C = 0 where 'a' is equal to : (A)
21.
(D) none
z6
All roots of the equation, (1 + + =0: lie on a unit circle with centre at the origin (B)lie on a unit circle with centre at (− 1, 0) (A) (C) lie on the vertices of a regular polygon with centre at the origin (D) are collinear Points z 1 & z2 are adjacent vertices of a regular octagon. The vertex z3 adjacent to z2 (z3 ≠ z1) is represented by : (A) z 2 +
20.
(C) 2 cos n α
(B) 2 sin n α
(A) zero
(A + i B)
(B)
2
A − iB 2
(C) A + i B
(D) none
The points of intersection of the two curves z − 3 = 2 and z = 2 in an argand plane are: (A)
(
1 7±i 3 2
)
(B)
(
1 3±i 7 2
)
(C)
3 ±i 2
7 2
(D)
7 ±i 2
3 2
22.
The equation of the radical axis of the two circles represented by the equations, z − 2 = 3 and z − 2 − 3 i = 4 on the complex plane is : (A) 3iz – 3i z – 2 = 0 (B) 3iz – 3i z + 2 = 0 (C) iz – i z + 1 = 0 (D) 2iz – 2i z + 3 = 0
23.
If Π eipθ = 1 where Π denotes the continued product, then the most general value of θ is :
r
p=1
(A) 24.
2n π r (r − 1)
(B)
2n π r (r + 1)
(C)
4n π r (r − 1)
(D)
4n π r (r + 1)
The set of values of a ∈ R for which x 2 + i(a – 1) x + 5 = 0 will have a pair of conjugate imaginary roots is (A) R (B) {1} (C) |a| a2 – 2a + 21 > 0} (D) none of these Successful People Replace the words like; "wish", "try" 26 & "should" with "I Will". Ineffective People don't.
26.
If |z1 – 1| < 1, |z2 – 2| < 2, |z3 – 3| < 3 then |z1 + z2 + z3| (A) is less than 6 (B) is more than 3 (C) is less than 12 (D) lies between 6 and 12 If z1, z 2, z3, ........., zn lie on the circle |z| = 2, then the value of
1 1 1 E = |z 1 + z2 + ..... + zn| – 4 z + z + ....... + z is 1 2 n (A) 0 (B) n (C) –n (D) none of these Part : (B) May have more than one options correct 27. If z1 lies on |z| = 1 and z 2 lies on |z| = 2, then (B) 1 ≤ |z1 + z2| ≤ 3 (A) 3 ≤ |z 1 – 2z 2| ≤ 5 (C) |z1 – 3z2| ≥ 5 (D) |z 1 – z 2| ≥ 1 28. If z1, z2, z3, z4 are root of the equation a0z4 + z1z3 + z2z2 + z3z + z4 = 0, where a0, a1, a2, a3 and a4 are real, then z1 , z 2 , z3 , z 4 are also roots of the equation (B) z1 is equal to at least one of z1 , z 2 , z3 , z 4 (A) (C) – z1 ,– z2 , – z3 , – z 4 are also roots of the equation (D) none of these
29. 30.
31.
27 of 38
25.
If a3 + b3 + 6 abc = 8 c3 & ω is a cube root of unity then : (B) a, c, b are in H.P. (A) a, c, b are in A.P. (C) a + bω − 2 cω2 = 0 (D) a + bω2 − 2 c ω = 0 The points z 1, z 2, z3 on the complex plane are the vertices of an equilateral triangle if and only if : (A) Σ (z 1 − z 2) (z2 − z3) = 0 (B) z12 + z 22 + z 32 = 2 (z1 z 2 + z2 z3 + z 3 z1) (C) z12 + z22 + z 32 = z1 z 2 + z2 z 3 + z 3 z1 (D) 2 (z12 + z22 + z 32) = z1 z 2 + z 2 z3 + z3 z 1 If |z1 + z2| = |z1 – z2| then π (A) |amp z1 – amp z2| = 2 z1 (C) z is purely real 2
(B) | amp z1 – amp2| = π z1 (D) z is purely imaginary 2
EXERCISE-5 1.
Given that x, y ∈ R, solve : 4x² + 3xy + (2xy − 3x²)i = 4y² − (x 2/2) + (3xy − 2y²)i
2.
If α & β are any two complex numbers, prove that : α−
3. 4. 5. 6.
α 2 − β2 + α +
If (1 + x)n = p0 + p1 x + p2 x 2 + p3 x 3 +......., then prove that : p0 − p2 + p4 −....... = 2n/2 cos
If α, β are the numbers between 0 and 1, such that the points z1 = α + i, z2 = 1 + βi and z3 = 0 form an equilateral triangle, then find α and β. ABCD is a rhombus. Its diagonals AC and BD intersect at the point M and satisfy BD = 2AC. If the points D and M represent the complex numbers 1 + i and 2 - i respectively, then find the complex number corresponding to A. Show that the sum of the pth powers of nth roots of unity : (a) is zero, when p is not a multiple of n. (b) is equal to n, when p is a multiple of n.
(a)
9.
α 2 − β2 = α + β + α − β
i ....... ∞
1
= A + i B, principal values only being considered, prove that
tan
B 1 πA = A 2
(b)
A 2 + B2 = e − π B
Prove that the roots of the equation, (x - 1)n = x n are
1 2
r π 1 + i cot , where r
r = 0, 1, 2,....... (n − 1) & n ∈ N. If cos (α − β ) + cos ( β − γ ) + cos (γ − α) = − 3/2 then prove that : Σ cos 2α = 0 = Σ sin 2α (b) Σ sin (α + β) = 0 = Σ cos (α + β) (a) (c) Σ sin 3α = 3 sin (α + β + γ) (d) Σ cos 3 α = 3 cos (α + β + γ) (e) Σ sin2 α = Σ cos2 α = 3/2 (f) cos3 ( θ + α) + cos3 (θ + β ) + cos3 (θ + γ ) = 3 cos (θ + α). cos (θ + β ). cos (θ + γ) where θ ∈ R. 27
If α, β, γ are roots of x 3 − 3 x 2 + 3 x + 7 = 0 (and ω is imaginary cube root of unity), then find the value of
12. 13.
14.
15. 16. 17.
α −1 β −1 γ −1 + + . β −1 γ −1 α −1
Given that, |z − 1| = 1, where ' z ' is a point on the argand plane. Show that
z−2 = i tan (arg z). z
28 of 38
11.
P is a point on the Argand diagram. On the circle with OP as diameter two points Q & R are taken such that ∠ POQ = ∠ QOR = θ. If ‘O’ is the origin & P, Q & R are represented by the complex numbers Z 1, Z 2 & Z3 respectively, show that : Z 22. cos 2 θ = Z 1. Z 3 cos² θ. Find an expression f or tan 7 θ in terms of tan θ , using com plex numbers. By considering tan 7θ = 0, show that x = tan2 (3 π/7) satisfies the cubic equation x 3 − 21x 2 + 35x − 7 = 0. nπ n −1 − 2 n / 2 cos 2 4 2nπ 4π 6π 2π 1 Prove that : cos = − When n ∈ N. + cos +..... + cos + cos 2 n + 1 2 2 n + 1 2 n + 1 2 n + 1 Show that all the roots of the equation a1z3 + a2z2 + a3z + a4 = 3, where |ai| ≤ 1, i = 1, 2, 3, 4 lie outside the circle with centre origin and radius 2/3.
If (1 + x)n = C0 + C1x + C2x² +.... + Cn x n (n ∈ N), prove that : C2 + C6 + C10 +..... =
1 2
n−1
∑ (n − k ) cos 2nkπ = – n2 , where n ≥ 3 is an integer
18.
Prove that
19.
Show that the equation
k =1
2
20.
21.
2
2
A1 A2 An + + ...... + = k has no imaginary root, given that : x − a1 x − a 2 x − an
a1, a2, a3.... an & A1, A2, A3..... An, k are all real numbers. Let z 1, z2, z 3 be three distinct complex numbers satisfying, ½z1-1½ = ½z 2-1½ = ½z3-1½. Let A, B & C be the points represented in the Argand plane corresponding to z1, z 2 and z 3 resp. Prove that z 1 + z 2 + z 3 = 3 if and only if D ABC is an equilateral triangle. Let α, β be fixed complex numbers and z is a variable complex number such that, 2 z − α + z − β = k. 2
Find out the limits for 'k' such that the locus of z is a circle. Find also the centre and radius of the circle. 22.
If 1, α1, α2, α3,......., αn − 1 are the n, nth roots of unity, then prove that (1 − α1) (1 − α2) (1 − α3)........ (1 − αn − 1) = n. Hence prove that sin
n 2π 3π (n − 1) π π . sin . sin ........ sin = n −1 . n n n n 2
23.
Find the real values of the parameter ‘a’ for which at least one complex number z = x + iy satisfies both the equality z − ai = a + 4 and the inequality z − 2 < 1.
24.
Prove that, with regard to the quadratic equation z2 + (p + ip′ ) z + q + iq′ = 0; where p, p′ , q, q′ are all real. (a) if the equation has one real root then q ′ 2 − pp ′ q ′ + qp ′ 2 = 0. (b) if the equation has two equal roots then p2 − p′ 2 = 4q & pp ′ = 2q ′ . State whether these equal roots are real or complex.
25.
The points A, B, C depict the complex numbers z1, z2, z 3 respectively on a complex plane & the angle 1 B & C of the triangle ABC are each equal to (π − α ) . Show that 2 α 2 (z2 − z 3)² = 4 (z3 − z1) (z1 − z2) sin . 2 If z 1, z 2 & z 3 are the affixes of three points A, B & C respectively and satisfy the condition |z1 – z 2| = |z1| + |z 2| and |(2 - i) z 1 + iz3 | = |z 1| + |(1 – i) z1 + iz 3| then prove that ∆ ABC in a right angled.
26. 27.
If 1, α1, α2, α3, α4 be the roots of x 5 − 1 = 0, then prove that ω − α1 . ω − α 2 . ω − α 3 . ω − α 4 = ω. 2 2 2 ω 2 − α1 ω − α 2 ω − α 3 ω − α 4
28.
If one the vertices of the square circumscribing the circle |z – 1| = the square.
28
2 is 2 +
3 i. Find the other vertices of [IIT – 2005, 4]
1.
A
2.
C
3.
D
4.
A
5.
A
6.
D
7.
C
8.
A
9.
A
11.
D
12.
A
13.
B
14.
D
15.
D
16.
A
17.
B
18.
A
19.
D
20.
C
21.
C
22.
B
23.
B
24.
D
25.
B
26.
C
27.
A
28.
ABCD 29.
30.
ACD 31.
AC
10.
AD
AB
EXERCISE-5 2 − 3, 2 − 3
i 3 or 1 – i 2 2
11.
3 ω2
1 2 α−β 2
23.
5 21 − , − 10 6
x = K, y =
4.
3–
21.
k>
28.
–i 3,1–
29
3K K∈R 2
3.
1.
3 + i, 1 +
3 –i
29 of 38
EXERCISE-4
Some questions (Assertion–Reason type) are given below. Each question contains Statement – 1 (Assertion) and Statement – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. So select the correct choice : Choices are : (A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B)Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1. (C) Statement – 1 is True, Statement – 2 is False. (D) Statement – 1 is False, Statement – 2 is True. 344. Let z = eiθ = cosθ + isinθ Statement 1: Value of eiA .eiB . eiC = –1 if A + B + C = π. Statement 2: arg(z) = θ and |z| = 1. 345 Let a1, a2, .... , an ∈R+ Statement–1 : Minimum value of
a1 a 2 a a + + .... + n −1 + n a2 a3 a n a1
Statement–2 : For positive real numbers, A.M ≥ G.M. 346.
5c 3b a , log and log then A.P., where a, b, c are in G.P. If a, b, c represents the sides of a a 5c 3b
Let log
triangle. Then : Statement–1 : Triangle represented by the sides a, b, c will be an isosceles triangle Statement–2 : b + c < a 347.
Let Z1, Z2 be two complex numbers represented by points on the curves |z| = Statement–1 : min |z1–z2| = 0 and max |z1 – z2| = 6
2 and |z – 3 – 3i| = 2 2 . Then
2
348.
Statement–2 : Two curves |z| = 2 and |z – 3 –3i| = 2 2 touch each other externally Statement–1 : If |z – i| ≤ 2 and z0 = 5 + 3i, then the maximum value of |iz + z0| is 7 Statement–2 : For the complex numbers z1 and z2 |z1 + z2| ≤ |z1| + |z2|
349.
Let z1 and z2 be complex number such that Statement–1
350.
z : arg 1 = 0 z2
Statement–2 : z1, z2 and origin are collinear and z1, z2 are on the same side of origin. Let fourth roots of unity be z1, z2, z3 and z4 respectively Statement–1
351.
z1 + z 2 =| z1 | + | z 2 |
: z12 + z 2 2 + z 32 + z 4 2 = 0
Let z1, z2, . . . , zn be the roots of z = 1, n ∈ N. Statement–1 : z1. z2 . . . zn = (– 1)n Statement–2 + an – 2 xn – 2 + . . . + a1x + a0 = 0, an ≠ 0, is (– 1)n.
352.
353.
354.
356.
:
z1 + z2 + z3 + z4 = 0.
:
Product of the roots of the equation anx n + an – 1x n – 1
a0 . an
Let z1, z2, z3 and z4 be the complex numbers satisfying z1 – z2 = z4 – z3. Statement–1 : z1, z2, z3, z4 are the vertices of a parallelogram
z1 + z3 z2 + z4 . = 2 2
Statement–2
:
Statement–1
: The minimum value of | z | + | z − i | | is 0.
Statement–2
: For any two complex number z1 and z2, z1 + z 2 ≤ z1 + z 2 .
Statement–1
: Let z1 and z2 are two complex numbers such that | z1 − z 2 |=| z1 + z 2 | then the orthocenter
of ∆AOB is 355.
Statement–2
n
z1 + z 2 . (where O is the origin) 2
Statement–2 : In case of right angled triangle, orthocenter is that point at which triangle is right angled. Statement–1 : If ω is complex cube root of unity then (x – y) (xω – y) (xω2 – y) is equal to x3 + y2 Statement–2 : If ω is complex cube root of unity then 1 + ω + ω2 = 0 and ω3 = 1 Statement-1 : If |z| ≤ 4, then greatest value of |z + 3 – 4i| is 9. Statement-2 : ∀Z1, Z2 ∈C, |Z1 + Z2| ≤ |Z1| + |Z2|
30 of 38 30
357.
Statement-1: The slope of line (2 – 3i) z + (2 + 3i) z − 1 = 0 is
2 3
Statement-2:: The slope of line az + az + b = 0 b∈R & a be any non-zero complex. Constant is − 6
358.
Statement-1: The value of
∑ sin k =1
2πk 2πk − i cos is i 7 7
Statement-2: The roots of the equation zn = 1
cos 2πk 2πk z= + i sin n n
359. 360. 361.
362.
Re(a) Im(a)
are called the nth roots of unity where
where k = 0, 1, 2, ... (n − 1)
Statement-1: |z1 – a| < a, |z2 – b| < b |z3 – c| < c, where a, b, c are +ve real nos, then |z1 + z2 + z3| is greater than 2|a Statement-2: |z1 ± z2| ≤ |z1| + |z2| + b + c| Statement-1: (cos2 + isin2)π = 1 Statement-2: (cosθ +isinθ)n = cosnθ + isin nθ it is not true when n is irrational number. Statement-1 : If α1, α2, α3 …. α 8 be the 8th root of unity, then α116 + α216 + α316 + … + α816 = 8 Statement-2 : In case of sum of pth power of nth roots of unity sum = 0 if p ≠ kn where p, k, n are integers sum = n if p = kn. Statement-1: Locus of z, satisfying the equation |z – 1| + |z – 8| = 16 is an ellipse of eccentricity 7/16 Statement-2:: Sum of focal distances of any point is constant for an ellipse
z2 1 n 2 = arg z2 – arg z1 & arg z = n(argz) Statement-2: If |z| = 1, then arg (z + z ) = arg z. 2 z1
363.
Statement-1: arg
364.
Statement-1: If |z − z + i| ≤ 2 then 5 − 2 ≤ | z | ≤ 5 + 2 Statement-2: If |z − 2 + i| ≤ 2 the z lies inside or on the circle having centre (2, −1) & radius 2.
365.
Statement-1: The area of the triangle on argand plane formed by the complex numbers z, iz and z + iz is Statement-2: The angle between the two complex numbers z and iz is
366.
Statement-1: If
zz1 − z 2 = k, (z1, z2 ≠ 0), then locus of z is circle. zz1 + z 2
Statement-2 : As,
367.
π . 2
1 2 |z| 2
z − z1 = λ represents a circle if, λ∉{0, 1} z − z2 z1 =0 . z2
Statement-1: If z1 and z2 are two complex numbers such that |z1| = |z2| + |z1 – z2|, then Im Statement-2: arg (z) = 0 ⇒ z is purely real.
368.
2π 2π 2 4 3 5 6 + i sin , p = α + α + α , q = α + α + α , then the equation whose roots 7 7
Statement-1: If α = cos
are p and q is x2 + x + 2 = 0 Statement-2: If α is a root of z7 = 1, then 1 + α + α2 + …. + α6 = 0. 369. 370.
371.
Statement-1: If |z| < 2 − 1 then |z2 + 2z cosα| is less than one. Statement-2: |z1 + z2| < |z1| + |z2| . Also |cosα| ≤ 1. Statement-1: The number of complex number satisfying the equation |z|2 + P|z| + q = 0 (p, q, ∈ R) is atmost 2. Statement-2 : A quadratic equation in which all the co-efficients are non-zero real can have exactly two roots. Statement-1: If β +
1 = 1(β ≠ 0) is a complex number, then the maximum value of |β| is β
Statement-2 :: On the locus β +
1 = 1 the farthest distance from origin is β
31 of 38 31
5 +1 . 2
5 +1 . 2
372.
344. 351. 358. 365. 372.
Statement-1: The locus of z moving in the Argand plane such that arg
B D A A A
z−2 π = is a circle. z+2 2
Statement-2: This is represent a circle, whose centre is origin and radius is 2. ANSWER 345. A 346. D 347. A 348. A 352. A 353. D 354. D 355. D 359. D 360. D 361. A 362. A 366. D 367. A 368. A 369. A
349. 356. 363. 370.
A A B D
350. B 357.. A 364. A 371. A
SOLUTION 1/ n
345.
a a a a a a a Using AM ≥ GM 1 + 2 + ... + n −1 + n ≥ n 1 . 2 .... n a 2 a3 a n a1 a 2 a 3 a1