BANSAL CLASSES MATHEMATICS TARGET IIT JEE 2007 NUCLEUS (F)
QUESTION BANK ON PH-1 (COMPOUND ANGLES) PH-2 (TRIGONOMETRIC EQUATIONS & INEQUATIONS) PH-3 (SOLUTIONS OF TRIANGLE) & BINOMIAL
Time Limit : 7 Sitting Sitting Each of 75 Minutes Minutes duration approx. approx.
Question bank on Compound angles, Trigonometric eq n and ineq n, Solutions of Triangle & Binomial
There are 142 questions in this question bank. Select the correct alternative : (Only one is correct)
Q. 1
Q.2
If x + y = 3 – cos4
and x – y = 4 sin2 then
(A) x4 + y4 = 9
(B)
x y 16
(C) x3 + y3 = 2(x2 + y2)
(D)
x y 2
If in in a tria riangle ABC, b cos2 (A) in A.P.
Q. 3
If tanB =
(A)
Q. 4
1 n cos2 A
sin A (1 n ) cos A
Given a2 + 2a + cosec2 x
2 (C) a R ; x
Q.5
3 2
c then a, b, c are : (C) in H.P.
(D)
None
then then tan(A + B) B) equals
(B)
( n 1) cos A
(C)
sin A
sin A
(D)
( n 1) cos A
sin A ( n 1) cos A
F G (a x)J I H 2 K = 0 then, which of the following holds good? x
I 2 (D) a , x are finite but not possible to find
(B) a = –1 ;
s2
(B) A =
3 3
The ex exact va value of of cos
2
s2
tan x
(C) A >
2
cos ec
28 (B) 1/2
In any any tria trian ngle ABC ABC, (a (a + b) b)2 sin2 (A) c (a + b)
Q.8
2
=
I
(A) – 1/2 Q.7 Q.7
2
B
If A is the the area area and and 2s 2s the the sum sum of of the the 3 sides sides of of a tria triang ngle le,, then then : (A) A
Q.6
+ a cos2
(B) in G.P.
n sin A cos A
(A) a = 1 ;
A
3
cos
28
6 28
cos ec
9 28
s2
D None
3
cos
18 28
(C) 1 C 2
+ (a b) b)2 cos2
(B) b (c + a)
C 2
cos ec
27 28
is equal to
(D) 0 =
(C) a (b + c)
(D) c2
x sin 3 72 x when simplified reduces to : cos x 2 . tan 32 x
. cos 3 2 2
(A) sin x cos x
(B)
sin2 x
(C )
sin x cos x
(D) sin2x
Q. 9
If in a ABC, sin3A + sin3B + sin3C = 3 sinA · sinB · sinC then (A) ABC may be a scalene triangle (B) ABC is a right triangle (C) ABC is an obtuse angled triangle (D) ABC is an equilateral triangle
Q.10
In a triang triangle le ABC, CH and CM are the the lengths lengths of the the altitude altitude and and median median to the the base AB. If If a = 10, 10, b = 26, c = 32 then length (HM) (A) 5 (B) 7 (C) 9 (D) none
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[2]
Q.1 Q.11
The value of
sin 2
sin cos (A) is less than – 1
sin cos tan 2 1
for all permissible vlaues of (B) is greater than 1
(C) (C) lie liess bet betwe ween en – 1 and and 1 inc includ luding ing bo both
(D) (D) lies lies bet betwe ween en –
Q.12
sin 3 = 4 sin sin 2 sin 4 in 0 (A) 2 real solutions (C) 6 real solutions
: (B) 4 real solutions (D) 8 real solutions.
Q.13
In a triang triangle le ABC, ABC, CD is the the bisec bisector tor of of the angle angle C. If If cos
2 and
2
has
C 2
has the value
1 3
and l (CD) l (CD) = 6, then
1 1 has the value equal to a b (A) Q.14 Q.14
1
(B)
9
1
(C)
12
RS , 5 , 19 , 23 UV T12 12 12 12 W R 5 , 13 , 19 UV (C) S T 12 12 12 W
7 17 23 , , , 12 12 12 12
Q.17
(D)
1
(B) 2
2
(C)
If cos ( + ) = 0 then sin ( + 2) = (A) sin (B) sin
RS , 7 , 19 , 23 UV T 12 12 12 12 W
2
(C) cos
tan A tan B
has the value equal to 1
(D)
(D)
cos
2
With With usual usual notations notations,, in a triangle triangle ABC, ABC, a cos(B – C) + b cos(C – A) + c cos(A – B) is equal equal to (A)
Q.18
(B)
If the media median n of a triangle triangle ABC throug through h A is is perpendicu perpendicular lar to AB then then (A)
Q.16
(D) none
6
The The set set of angl angles es btwe btween en 0 & 2 satisfying the equation 4 cos2 2 2 cos cos 1 = 0 is (A)
Q.15
1
abc
(B)
R 2
abc 4R 2
(C)
4abc
(D)
R 2
abc 2R 2
cos cos3 2 tan cot = 1 if : sin cos 1 cot 2
sin 3
2
(A) 0 ,
, 2
(B)
(C) ,
3
2
3 , 2 2
(D)
Q.19
With With usua usuall notat notation ionss in a tria triang ngle le ABC, ABC, ( I I1 ) · ( I I2 ) · ( I I3 ) has the value equal equal to 2 2 2 (A) R r (B) 2R r (C) 4R r (D) 16R2r
Q.20
In a triang triangle le ABC, angle B < angle angle C and and the values values of B & C satisfy satisfy the the equation equation 2 2 tan x - k (1 + tan x) = 0 where (0 < k < 1) . Then the measure of angle A is : (A) /3 (B) 2/3 (C) /2 (D) 3/4
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[3]
Q.21
If cos =
1 then tan cot has the value equal to, where(0 < < and 0 < < ) 2 2 2 cos
2 cos
(A) 2 Q.22
2 3
,
,
(B)
4 12
k 1
(B)
k 1
The equa equati tion on,, sin2 (A) no root
,
2
3
,
(D)
3
C
8
,
8
A
(C)
k 1
4 sin
2
,
(D)
3 2
,
10
,
5
B
= k sin 2 , then tan 2 tan 2 = 2
k 1
3
3
(C)
6
C
If A + B + C = & sin A (A)
Q.24 Q.24
(C) 3
2
In a ABC, if the median, bisector and altitude drawn from the vertex A divide the angle at the vertex into four equal parts then the angles of the ABC are : (A)
Q.23
(B)
1
=1
k
1 k
4 sin
(B) one root
3
(D)
1
k 1 k
has :
(C) two roots
(D) infinite roots
1 1 1 1 1 1 K R 3 Q.25 Q.25 With ith usu usual al not notat atio ion n in in a ABC = 2 2 2 where K has the value r1 r2 r2 r3 r3 r1 a b c equal to : (A) 1
Q.26
If
5 2
x
(B) cot
2
If x sin = y sin (A) x + y + z = 0
Q.28
In a (A)
Q.29 .29
(D) 128
1 sin x
1 sin x
32
Bansal C lasses
2
10
1 sin x (D) –tan
2
is x 2
4 then : = z sin 3 3
a cos A
(B)
R
x
(C) tan
(B) xy + yz + zx = 0
r
1
x
1 sin x
2
ABC, the value of
The valu alue of cos cos
(A)
(C) 64
x 3 , then the value of the expression
(A) –cot
Q.27
(B) 16
cos cos
2 10
(B)
b cos B c cos C is equal to : abc
R
(C)
2 r
cos 1 16
(C) xyz + x + y + z = 1 (D) non none
4 10
cos
8 10
cos
R r
16 10
(C)
(D)
2 r R
is :
cos / 10 16
(D)
Q. B. on -I, -II, -III & Binomial
10
2 5 64
[4]
Q.30 Q.30
With ith usu usual al nota notati tion on in a ABC, if R = k (A) 1
(B) 2
r1 r2 r2 r3 r3 r1 r1 r2
r2 r3 r3 r1
where k has the value value equal to:
(C) 1/4
(D) 4
Q.31
If a cos3 + 3a cos sin2 = m and a sin3 + 3a cos 2 sin = n . Then (m + n)2/3 + (m n)2/3 is equal to : (A) 2 a2 (B) 2 a1/3 (C) 2 a2/3 (D) 2 a3
Q.32
In a triangle triangle ABC , AD is the altitude from A . Given b > c , angle angle C = 23° & AD = then angle B = (A) 157°
Q.33 .33
(B) 113°
c2
[JEE ’94, 2] (D) none
(C) 147°
(B) tan 3x
(C) 3 tan 3x
Q.34
In a ABC, cos 3A + cos 3B + cos 3C = 1 then : (A) ABC is right angled (B) ABC ABC is acute angled (C) ABC is obtuse angled (D) nothing definite can be said about about the nature of the .
Q.35 Q.35
The value lue of
3 c ot 7 6 c ot 1 6 cot 7 6
(A) cot 44º
cot 1 6
(D)
3 9 tan 2 x 3 tan x
tan 3 x
is :
(B) tan 44º
(C) tan 2º
(D) cot 46º
If the the inci incirc rcle le of the the ABC touches its sides s ides respectively respec tively at L, M and N and if x, y, y, z be the circumradii circumra dii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz x yz is equal to : (A) R r R r 2
(B) r R r R 2
(C)
1 2
R r R r 2
(D)
1 2
r R r R 2
Q.37 Q.37
The The numb number er of solu soluti tion onss of tan (5 cos cos ) = cot (5 sin ) for in (0, 2) is : (A) 28 (B) 14 (C) 4 (D) 2
Q.38
If A = 3400 then 2 sin
Q.39
b
The value of cot x + cot (60º + x) + cot (120º + x) is equal to : (A) cot 3x
Q.36 Q.36
abc 2
A 2
is identical to
(A)
1 sin A
1 sin A
(B)
1 sin A
1 sin A
(C)
1 sin A
1 sin A
(D)
1 sin A
1 sin A
AD, BE and and CF are the the perpen perpendicu diculars lars from from the the angul angular ar points points of of a ABC ABC upon the opposite sides. sid es. The perimeters of the DEF and ABC are in the ratio : (A)
2 r R
(B)
r 2 R
(C)
r R
(D)
r 3 R
where r is the in radius and R is the circum radius of the ABC
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[5]
Q.40 Q.40
The valu alue of of co cosec sec
18
–
3 sec
18
is a
(A) surd (C) negative natural number Q.41
(B) rational which is not integral (D) natural number
In a ABC if b + c = 3a then cot (A) 4
B 2
· cot
C 2
has the value equal to :
(B) 3
(C) 2
(D) 1
Q.42
The set of of values values of ‘a’ ‘a’ for for which which the the equa equation tion,, cos 2x 2x + a sin x = 2a 7 possess a solution is : (A) ( , 2) (B) [2, 6] (C) (6, ) (D) ( )
Q.43
In a right right angl angled ed tria triang ngle le the the hypot hypotenu enuse se is 2 2 times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are (A)
Q.44
3
&
(B)
6
8
&
3
(C)
8
4
&
(D)
4
5
&
3 10
Let Let f, g, g, h be the the length lengthss of the the perpen perpendicu diculars lars from from the the circu circumce mcentre ntre of of the ABC on the sides a, b and c respectively respectivel y . If If (A) 1/4
a f
b g
c h
=
abc fgh
then the value of is :
(B) 1/2
(C) 1
cot A2 . cot cot A2 2
Q.45
In
ABC, the minimum value of
2
(D) 2
B 2 is
2
(A) 1 Q.46
Q.48
(D) non existent
(B)
1
(C) – 3
3
(D) –
The The gene general ral soluti solution on of sin x + sin 5x = sin 2x + sin 4x is : (A) 2n (B) n (C) n/3 where n I
1 3
(D) 2 n/3
The product product of of the distances distances of the incent incentre re from from the angular angular points points of a ABC is : (A) 4 R 2 r
Q.49 Q.49
(C) 3
If the orthoc orthocentr entree and circumcen circumcentre tre of a triangle triangle ABC be at equal equal distance distancess from the side side BC and lie on the same side of BC then tanB tanC has the t he value equal to : (A) 3
Q.47
(B) 2
(B) 4 Rr2
Numbe Numberr of of roo roots ts of of the the equat equation ion [ ] is (A) 2
Bansal C lasses
(B) 4
(C) cos 2 x
a
3 1 2 (C) 6
b c R
(D)
s
sin x
3 4
a bcs R
1 0 which
lie in the interval
(D) 8
Q. B. on -I, -II, -III & Binomial
[6]
sec 8
Q.50
Q.51
1 is equal equal to sec 4 1 (A) tan 2 cot 8 (B) tan 8 tan 2 In a
ABC
if b = a
(A) 150 Q.52 Q.52
Q.53 Q.53
Q.54
C = 300
(B) 450
Numbe umberr of of val value uess of of (A) 1
(B) 2
(C) 3
The expr xpressi essio on,
(B) r 3 = 2r 1
3 tan 2
cos (2
)
5 p
2 p a 2 p 2 p 3
(C) r 2 = 2r 1
cos 32
(D) 1 4 p 2 p
5
a p
(D) r 2 = 3r 1
+ cos sin( sin ( ) + cos ( + ) sin when
2
1
(C)
2
(D) none
3 5 cos when simplified 2 2
The expr xpressi essio on [1 sin (3 ) + cos (3 + )] 1 sin
(B)
sin 2
(C) 1 sin 2
(D) 1 + sin 2
If ‘O’ ‘O’ is the the cir circu cumc mcen entr tree of the the ABC and R 1, R 2 and R 3 are the radii of the circumcircles of triangles
(A)
a bc 2 R 3
(B)
a R 1
R 3 a bc
b R 2
(C)
c
has the value equal to:
R 3 4
(D)
R 2
4R 2
The The maxi maximu mum m valu valuee of ( 7 cos cos + 24 sin ) × ( 7 sin – 24 cos ) for every (A) 25
Q.59
(D) 4
In a ABC, a = a1 = 2 , b = a2 , c = a 3 such that a p+1 =
OBC, OCA and OAB respectively then
Q.58 Q.58
(D) 1050
The The exa exact ct valu valuee of of cos cos273º + cos247º + (cos73º . cos47º) is (A) 1/4 (B) 1/2 (C)3/4
reduces to : (A) sin 2 Q.57
then the measure of the angle angle A is
(C) 750
simplified reduces to : (A) zero (B) 1 Q.56 Q.56
(D) tan 8 cot 2
[ 0 , 2 ] satisfying the equation cotx – cosx = 1 – cotx. cosx
where p = 1,2 then (A) r 1 = r 2
Q.55 Q.55
3 1 and
(C) cot 8 cot 2
(B) 625
(C)
625
(D)
2
R .
625 4
4 sin50 sin550 sin650 has the values equal to (A)
3
1
2 2
Bansal C lasses
(B)
3
1
2 2
(C)
3
1 2
(D)
Q. B. on -I, -II, -III & Binomial
3d 3
1i
2 2
[7]
Q.60
If x, y and z are are the distances distances of incentre incentre from from the vertices vertices of the triangle triangle ABC respectively respectively then a b c
is equal to
x yz
(A)
A
tan 2
(B)
A
cot 2
(C)
A
tan 2
(D)
A
sin 2
Q.61 Q.61
The medi edians ans of of a ABC are 9 cm, 12 cm and 15 cm respectively re spectively . Then the area of the triangle trian gle is (A) 96 sq cm (B) 84 sq cm (C) 72 sq cm (D) 60 sq cm
Q.62
If x =
n 2
, satisfies the equation sin
x 2
cos
(A) n = 1, 0, 3, 5 (C) n = 0, 2, 4 Q.63 Q.63
F
G1 cos The value lue of H (A)
x 2
x
= 1 sin x & the inequality
2
2
3 4
, then:
(B) n = 1, 2, 4, 5 (D) n = 1, 1, 3, 5
I F
3 I F 5 I F 7 I 1 cos J G 1 cos J G1 cos J is J G 9 K H 9 K H 9 K H 9 K
9
(B)
16
10
(C)
16
12
(D)
16
5 16
Q.64 Q.64
The The numb number er of all possi possibl blee trip triplet letss (a1 , a2 , a3) such that a 1+ a2 cos 2x + a3 sin² x = 0 for all x is (A) 0 (B) 1 (C) 3 (D) infinite
Q.65
In a ABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius of the s emicircle is (A)
2
(B)
2
a b a b c Where is the area of the triangle ABC.
(C)
2
(D)
s
c 2
Q.66
For For eac each h natu natura rall numb number er k , let let Ck denotes the circle with radius k centimeters and centre at the origin. On the circle Ck , a particle moves k centimeters in the counter- clockwise direction. After completing its motion on Ck , the particle moves to Ck+1 in the radial direction. The motion of the particle continues continue s in this manner .The particle starts at (1, 0).If the particle crosses the t he positive direction of the x- axis for the first time on the circle Cn then n equal to (A) 6 (B) 7 (C) 8 (D) 9
Q.67
If in a ABC,
cos A a
(A) right angled Q.68
Q.69
cos B b
cos C c
then the triangle is
(B) isosceles
(C) equilateral
If cos cos A + cosB cosB + 2cos 2cosC C = 2 then then the the sides sides of the the ABC are in (A) A.P. (B) G.P (C) H.P.
(D) obtuse
(D) none
If A and and B are are comp complim liment entary ary angles angles,, then then :
A
2
B
(A) 1 tan 1 tan = 2
Bansal C lasses
2
(B) 1 cot
A
B 1 cot = 2 2 2
Q. B. on -I, -II, -III & Binomial
[8]
A
2
B
Q.70 .70
The val value of of ,
2
sec 20° 3 cosec 20° sec20°
Q.71
(B) sin 40
B 2
4 sin 20 (D) sin 40
(C) 4
If in a ABC, cosA·cosB + sinA sinB sin2C = 1 then, the statement which is incorrect, is (A) ABC is isos isosce cele less but but not not rig right ang angled led (B) (B) ABC is acute angled a ngled (C) ABC is right angled
Q.72
2
is :
2 sin 20
(A) 2
A
(D) 1 tan 1 tan = 2
(C) 1 sec 1 cos ec = 2
(D) least angle of the triangle is
The The set set of of valu values es of x satisf satisfyi ying ng the equatio equation, n, 2 (A) an empty set (C) a set containing two values
2 0.25
tan x 4
sin
2
x 4
cos 2 x
4
+ 1 = 0, is :
(B) a singleton (D) an infinite set
Q.73
The product product of the arithmetic arithmetic mean of the lengths lengths of the the sides of of a triangle triangle and and harmonic harmonic mean of of the lengths of the altitudes of the triangle is equal to : (A) (B) 2 (C) 3 (D) 4 [ where is the area of the triangle t riangle ABC ]
Q.74
If in a triang triangle le sin A : sin C = sin (A B) B) : sin (B C) then a2 : b2 : c2 (A) are in A.P. (B) are in G.P. (C) are in H.P. (D) none of these [ Y G ‘99 Tier - I ] 5
Q.75
The The numbe numberr of solu solutio tion n of the the equ equat ation ion,,
cos(r x) = 0 lying in (0, p) is : r 1
(A) 2
Q.76
If
= 3 and
sin =
1
(A)
Q.78 Q.78
(B) 3
a
2
b
1 2
If in in a tria triang ngle le ABC ABC (A)
8
Bansal C lasses
a2
b2
0
(A) (A) a rati ratio onal nal num numbe berr
Q.79 Q.79
a
+ tan 67
1 2
(C) a + b
0
– cot 67
1 2
0
– tan7
a
(B)
4
cos B b
2 cos C c
1 2
(D) none
0
is :
(C) (C) 2(3 + 2 3 )
(B) irr irraatio tional nu number 2 cos A
(D) more than 5
. The value of the expression , a cosec b sec is
(B) 2 a 2 b 2
2
The value alue of cot cot 7
(C) 5
a b c
(C)
3
b ca
(D) 2 (3 – 3 )
then the value of the angle A is : (D)
Q. B. on -I, -II, -III & Binomial
2
[9]
Q.80
The The value value of the expres expressio sion n (sinx (sinx + cose cosecx) cx)2 + (cosx + secx)2 – ( tanx + cotx) 2 wherever defined is equal to (A) 0 (B) 5 (C) 7 (D) 9
Q.81
If A = 5800 then which one of the following is true
A 2
A 1 sin A 1 sin A 2
(B) 2 sin
A 1 sin A 1 sin A 2
(D) 2 sin
(A) 2 sin (C) 2 sin Q.82
Q.83
A 2
1 sin A 1 sin A
With usual usual notatio notations ns in a triang triangle le ABC, ABC, if r 1 = 2r 2 = 2r 3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a If
tan
=
x2 x
and tan
x2 x 1 ( + ) has the value value equal to : (A) 1
=
1 2 x 2 2x 1
(B) – 1
1 sin A
1 sin A
(D) 2a = 3b
(x 0, 1), where 0 <
(C) 2
(D)
r r r
,
<
2
, then tan
3 4
1
Q.84
If r1 , r 2, r 3 be the radii of excircles of the triangle ABC, then
(A) Q.85
Q.86
A
cot
(B)
2
cot
A 2
cot
Mini Minimu mum m valu valuee of 8cos cos2x + 18sec2x (A) 24 (B) 25
In a ABC
B 2
(C)
tan
A 2
is equal to :
1 2
(D)
tan
A 2
x R
wherever it is defined, is : (C) 26 (D) 18
a 2 b 2 c 2 . sin A sin B sin C simplifies to 2 2 2 sin A sin B sin C
(A) 2
(B)
(C)
2
(D)
4
where is the area of the triangle Q.87
If is eliminated from the the equations x = a cos( – ) and y = b cos ( – ) then
x2
y2
2xy
ab a 2 b 2 (A) cos2 ( – ) Q.88
cos( ) is equal to (B) sin2 ( – )
(C) sec2 ( – )
(D) cosec2 ( – )
The The genera generall solutio solution n of the the trigono trigonomet metric ric equ equati ation on tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is (A) x = n
(B) n ±
3
(C) x = 2n
(D) x =
n 3
where n I
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[10]
Q.89
If loga b + log bc + logca vanishes where where a, b and c are positiv positivee reals different than than unity then then the value 3 3 3 of (loga b) + (log bc) + (logca) is (A) an odd prime (B) an even prime (C) an odd composite (D) an irrational number
Q.90
If the the arcs arcs of the the same same leng length th in in two two circ circles les S1 and S2 subtend angles 75° and 120° respectively at the S1
centre. The ratio
(A) Q.91
Q.92 Q.92
S2
is equal to
1
(B)
5
81
(C)
16
Numbe Numberr of princi principal pal solutio solution n of of the the equat equation ion tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5
The ex expres ressio sion
tan 2 20 sin 2 20 tan 2 20 ·sin 2 20
(D)
25
(C) 7
25 64
(D) more than 7
simplifies to
(A) a rational which is not integral (C) a natural which is prime Q.93
64
(B) a surd (D) a natural which is not composite
The The valu valuee of x that that sat satisf isfies ies the the rela relatio tion n 2 3 4 5 x = 1 – x + x – x + x – x + ......... (A) 2 cos36° (B) 2 cos144° (C) 2 sin18°
(D) none
Select the correct correct alternatives : (More than one one are correct) correct)
Q.94
(A) sin
Q.95
Q.96
If sin = sin then sin
3
=
3 3
3 3
(B) sin
3
(C) sin
(D)
sin 3
3
Choo Choose se the the INCO INCORR RREC ECT T state stateme ment nt(s (s). ). 1
(B)
If tan A =
(C) (D)
The sig sign of the product sin sin 2 . sin 3 . sin 5 is positive. There ex exists a value of of between 0 & 2 which satisfies the equation ; 4 2 sin – sin – 1 = 0.
3 4
3
& tan B =
2
. sin 97
1
sin 82
2
and sin 127
1
(A
2
. cos 37
1
3 4
3
2
have the same value.
then tan (A B) must be irrational.
Which Which of the the followi following ng func functio tions ns have have the the maximu maximum m value value unity unity ? (A) sin2 x cos cos2 x
(C)
sin 2x
Bansal C lasses
cos 2x 2
(B)
(D)
sin 2x
cos 2x 2
1 1 cos x sin x 3 5 2
6
Q. B. on -I, -II, -III & Binomial
[11]
Q.97
If the sides of a ri ght angled angled tri angle are { cos2 cos2 cos2 + 2cos( + )} and
{sin2 sin2 + 2sin( + )}, then the length of the hypotenuse is : (A) 2[1+cos( )] (B) 2[1 cos( )] (C) 4 cos2 Q.98
Q.99
(D) 4sin2
2
2
An extr extrem emee val value ue of 1 + 4 sin sin + 3 cos is : (A) 3 (B) 4 (C) 5
The sines sines of of two two angle angless of a trian triangle gle are equal equal to to (A) 245/1313
(B) 255/1313
Q.100 Q.100 It is known known that that sin =
4 5
3 (7 2 4 c ot ) 15
13
&
99 101
for tan < 0
. The cosine of the third angle is :
(C) 735/1313
& 0 < < then the value of
(A) independent of for all in (0, /2)
(C)
5
(D) 6
5
(B)
3
(D) 765/1313
3 sin ( )
2 cos 6
sin
cos ( )
is:
for tan > 0
(D) none
Q.10 Q.101 1 If x = sec sec tan & y = cosec + cot then : (A) x =
1 y 1 y
(B) y =
1 x 1 x
(C) x =
1 y 1 y
(D) xy + x y + 1 = 0
Q.10 Q.102 2 If 2 cos cos + sin = 1, then the value of 4 cos + 3sin is equal to (A) 3 Q.10 Q.103 3 If sin sintt + cos t = (A)
1
(B) –5 1 5
then tan
t 2
(C )
7 5
(D) –4
is equal to :
(*B) –
1 3
(C) 2
(D)
1 6
BINOMIAL There are are 39 questions in this question bank. bank.
Q.104 Given Given that the term term of the expansio expansion n (x1/3 x1/2)15 which does not contain x is 5 m where m N , then m = (A) 1100 (B) 1010 (C) 1001 (D) none Q.105 In the the bino binomia miall (21/3 + 31/3)n, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end end is 1/6 , then n = (A) 6 (B) 9 (C) 12 (D) 15
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[12]
Q.106 Q.106 If the coeffic coefficien ients ts of x & x in the expansion expansion of 2 7
(A) 15
8
(B) 45
Q.107 The coeffi coefficien cientt of of x in the expansion of (x – 1)
n
are equal , then the value of n is :
3
(C) 55
49
(A) – 2 1
x
250 1
(D) 56
x 1 x 1 x 1 ..... is equal to 2 22 249
(B) + ve coefficient of x (D) – 2 1
(C) – ve coefficient of x
Q.108 Q.108 The last digit digit of of (3P + 2) is : (A) 1 (B) 2 4n where P = 3 and n N
1 2
49
(C) 4
(D) 5
n
1 Q.109 The sum sum of the the binomial binomial coefficients coefficients of of 2 x is equal to 256 . The constant term in the x
expansion is (A) 1120
(B) 2110
(C) 1210
(D) none
10
x 3 Q.110 Q.110 The coeffic coefficien ientt of of x4 in 2 is : 2 x (A)
405 256
(B)
504 259
(C)
450
(D)
263
Q.111 Q.111 The remaind remainder er,, when when (15 (1523 + 2323) is divided by 19, is (A) 4 (B) 15 (C) 0
512
(D) 18
Q.112 Let (7 4 3 ) n = p = p + + when n and p and p are are positive integers and (A) rational which is not an integer (C) a composite
405
(0, 1) then (1 – ) (p + ) is
(B) a prime (D) none of these
Q.1 Q.113 If (11) (11)27 + (21)27 when divided by 16 leaves the remainder (A) 0 (B) 1 (C) 2
(D) 14
Q.114 Q.114 Last three three digits digits of of the number number N = 7100 – 3100 are (A) 100 (B) 300 (C) 500
(D) 000
Q.115 Q.115 The last two digits digits of the number number 3400 are : (A) 81 (B) 43 (C) 29
(D) 01
Q.116 If (1 (1 + x + x²)25 = a0 + a1x + a2x² + ..... + a50 . x50 then a0 + a2 + a4 + ..... + a 50 is : (A) even (B) odd & of the form 3n (C) odd & of of the form (3n 1) (D) odd & of the form (3n + 1)
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[13]
Q.117 Q.117 The sum of the series series (1² + 1).1! + (2² + 1).2! + (3² + 1). 3! + ..... + (n² + 1). n! is : (A) (n + 1). (n+2)! (B) n.(n+1)! (C) (n + 1). (n+1)! (D) no none of of th these Q.118 Let Pm stand for nPm . Then the expression 1 . P1 + 2 . P2 + 3 . P3 + ..... + n . Pn = (A) (n + 1) ! 1 (B) (n + 1) ! + 1 (C) (n + 1) ! (D) none of these Q.119 Q.119 The The expre expressi ssion on
1 4x 1 7 1 4x 1 7 is a polynomial in x of degree 2 2 4x 1 1
(A) 7
(B) 5
(C) 4
(D) 3
n
n 1/13 a C 5/2 Q.120 If the second second term of the the expansion expansion a is 14a then the value of n 3 is : C2 a 1
(A) 4
(B) 3
(C) 12
(D) 6
Q.121 Q.121 If (1 + x) x) (1 + x + x2) (1 + x + x 2 + x3) ...... (1 + x + x 2 + x3 + ...... + x n)
m
2
3
a
m
a0 + a1x + a2x + a3x + ...... + a mx then
(A) n!
r 0
(B) (n + 1) !
r
has the value equal to
(C) (n – 1)!
(D) none
Q.122 Q.122 The The val value ue of of 4 {nC1 + 4 . nC2 + 42 . nC3 + ...... + 4 n 1} is : (A) 0 (B) 5n + 1 (C) 5n
(D) 5n 1
Q.123 If n be a positive positive integer integer such that n 3, then the value of the sum to n terms of the series 1.n
n 1 1!
(A) 0
(n 1) +
n 1 n 2 2!
(B) 1
n 1 n 2 n 3
(n 2) –
3!
(C) – 1
(n 3) + ...... is : (D) none of these
Q.124 In the expan expansion sion of (1 (1 + x)43 if the coefficients of the (2r + 1)th and the (r + 2)th terms are equal, the value of r is : (A) 12 (B) 13 (C) 14 (D) 15 10
a Q.125 The The positiv positivee value value of of a so that that the the coefficient of x is equal to that of x in the expansion of x 2 3 is x 5
(A)
1 2 3
1
(B)
3
15
(C) 1
(D) 2 3
10
x 1 x1 , the term which does not contain x is : Q.126 In the expans expansion ion of 2/ 3 x x1/ 3 1 x x1/ 2 (A)
10
C0
(B)
10
C7
(C)
10
C4
(D) none 8
1 2 Q.127 127 If the the 6 term in the expansion of the binomial 8/ 3 x log10 x is 5600, then x equals to x th
(A) 5
Bansal C lasses
(B) 8
(C) 10
(D) 100
Q. B. on -I, -II, -III & Binomial
[14]
Q.128 Co-e Co-eff ffici icien entt of of t in the expansion expansion of, of, m 1 m 2 ( + p) + ( + p) ( + q) + ( + p)m 3 ( + q)2 + ...... ( + q)m 1 where q and p q is : m
(A)
Ct pt qt
m
(B)
p q m
(C)
Ct p
t
qt
p q m
(D)
p q
Ct pm t q m t
m t m t Ct p q
p q
Q.129 129 (1 + x) x) (1 (1 + x + x 2) (1 + x + x 2 + x3) ...... (1 + x + x 2 + ...... + x100) when written in the ascending ascending power of x then the highest highest exponent of x is ______ . (A) 4950 (B) 5050 (C) 5150 (D) none
Q.130 Let 5 2 6
n
= p + f where n N and p N and 0 < f < 1 then the value of, f 2 f + pf p is
(A) a natural number (C) a prime number
(B) a negative integer (D) are irrational number
Q.131 Number Number of ration rational al terms terms in the expan expansion sion of of (A) 25
(B) 26
2
4 3
100
is :
(C) 27
(D) 28 10
Q.132
cos The greatest greatest value value of the the term independent independent of x in the expansion expansion of x sin x 10
(A)
10
C5
(B) 2
5
5
(C) 2 ·
10
C5
(D)
is
C5
2
5
Q.133 Q.133 If (1 + x – 3x 3x 2)2145 = a0 + a1x + a2x2 + ......... then a 0 – a1 + a2 – a3 + ..... ends with (A) 1 (B) 3 (C) 7 (D) 9 Q.134
4x 2 3 6 Coef Coeffi fici cien entt of of x in the binomial expansion 3 2x (A) 2438
(B) 2688
9
is
(C) 2868
(D) none 18
1 , x > 0 , is times the corresponding Q.135 The term indepe independe ndent nt of ' x ' in the expansion expansion of 9 x 3 x binomial binomial co-efficie co-efficient nt . Then ' ' is : (A) 3
(B)
1 3
(C)
1 3
(D) 1
Q.136 Q.136 The expressi expression on [x + (x31)1/2]5 + [x (x31)1/2]5 is a polynomial of degree : [JEE’92, 6 + 2] (A) 5 (B) 6 (C) 7 (D) 8
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[15]
Q.137 137 Give Given n (1 – 2x + 5x2 – 10x 3) (1 + x)n = 1 + a 1x + a2x2 + .... and that a12 = 2a2 then the value of n is (A) 6
(B) 2
(C) 5
(D) 3
Q.138 Q.138 The sum of the the series series aC0 + (a + b)C1 + (a + 2b)C2 + ..... + (a + nb)C n is where Cr's denotes combinatorial coefficient in the expansion of of (1 + x)n, n N (A) (a + 2nb)2 n (B) (2a + nb)2 n (C) (a +nb)2n – 1 (D) (2a + nb)2 n – 1 Q.139 The coefficien coefficientt of the middle term term in the binomial binomial expansion expansion in powers powers of x of (1 + x)4 and of (1 – x)6 is the same if equals 5
(A) –
10
(B)
3
(C) –
3
3
(D)
10
3 5
Q.140 (2n + 1) (2n + 3) (2n + 5) ....... (4n 1) is equal to : (A)
(4 n) ! n
2 . (2n) ! (2n) ! n
Q.141 If S = n
(A)
r 0
1 n
n 2
Cr
( 4 n) ! n !
(B)
n
2 . (2n) ! (2n) !
n
and T = n
r 0
(B)
n 2
r n
C r
1
(C)
( 4 n) ! n ! (2n) ! (2n) !
(D)
(4 n) ! n ! 2 n ! (2n) !
Tn
then S is equal to n (C) n – 1
(D)
2n 1 2
Q.142 Q.142 The The coeff coeffici icien entt of xr (0 r n 1) in the expression : (x + 2)n1 + (x + 2)n2. (x + 1) + (x + 2)n3 . (x + 1)² + ...... + (x + 1)n1 is : (A) nCr (2r 1) (B) nCr (2nr 1) (C) nCr (2r + 1) (D) nCr (2nr + 1)
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[16]
Answers Select the correct alternative : (Only one is correct) Q. 1 D Q. 2 D Q.3 A Q. 4 B Q.5 Q. 8 D Q. 9 D Q .1 0 C Q.11 D Q .1 2 Q.15 C Q.16 A Q .1 7 A Q.18 B Q .1 9 Q.22 C Q.23 A Q .2 4 D Q.25 C Q .2 6 Q.29 D Q.30 C Q .3 1 C Q.32 B Q .3 3 Q.36 C Q.37 A Q .3 8 D Q.39 C Q .4 0 Q.43 B Q.44 A Q .4 5 A Q.46 A Q .4 7 Q.50 D Q.51 D Q .5 2 B Q.53 C Q .5 4 Q.57 C Q.58 C Q .5 9 B Q.60 B Q .6 1 Q.64 D Q.65 A Q .6 6 B Q.67 C Q .6 8 Q.71 C Q .7 2 A Q .7 3 B Q.74 A Q .7 5 Q.79 D Q.80 B Q .8 1 C Q.82 C Q .8 3 Q.86 B Q.87 B Q .8 8 D Q.89 A Q .9 0 Q.93 C Select the correct correct alternatives : (More than one one are correct) correct) Q.94 ABD Q.95 BCD Q.96 ABCD Q.97 AC Q.100 ABC Q.101 BCD Q.102 AC Q.103 BC BINOMIAL Select the correct alternative : (Only one is correct) Q.104 C Q.105 B Q.106 C Q.107 A Q.108 Q.111 C Q.112 D Q.113 A Q.114 D Q.115 Q.118 A Q.119 D Q.120 A Q.121 B Q.122 Q.125 A Q.126 C Q .127 C Q.128 B Q.129 Q.132 D Q.133 B Q.134 B Q.135 D Q.136 Q.139 C Q.140 B Q.141 A Q.142 B
Bansal C lasses
A D D D D D C D C A C A C
Q. 6 Q.13 Q.20 Q.27 Q.34 Q.41 Q.48 Q.55 Q.62 Q.69 Q.76 Q.84 Q.91
D A C B C C B A B A B C C
Q.7 Q .1 4 Q .2 1 Q .2 8 Q .3 5 Q .4 2 Q .4 9 Q .5 6 Q .6 3 Q .7 0 Q .7 8 Q .85 Q .9 2
Q .98
BD
Q .9 9
BC
D D D B C
Q.109 Q.116 Q.123 Q.130 Q .137
A A A B A
Q.110 Q.117 Q.124 Q.131 Q.138
Q. B. on -I, -II, -III & Binomial
D B D A A B B B A C B C D
A B C B D
[17]
