BANSAL CLASSES MATHEMATICS TARGET IIT JEE 2007 STERLING
QUESTION BANK ON PH-1 (COMPOUND ANGLES) PH-2 (TRIGONOMETRIC EQUATIONS & INEQUATIONS) PH-3 (SOLUTIONS OF TRIANGLE) SEQUENCE & PROGRESSION
To be discussed w.e.f. 1st January 2007, 35 problems in each turn.
n n Question bank on Compound angles, Trigonometric eq and ineq , Solutions of Triangle, Sequence & Progression There are 132 questions in this question bank. Select the correct alternative : (Only one is correct)
Q.1 Q.1
Q.2
If x + y = 3 – cos4 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 a triang triangle le ABC, ABC, b cos2 (A) in A.P.
Q.3
If tan 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 tan(A + B) equals
(B)
( n 1) cos A
(C)
sin A
sin A
( n 1) cos A
(D)
sin A ( n 1) cos A
F G (a x)J I good? 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 The exac exactt valu valuee of cos
2
s2
tan x
(C) A >
2
cos ec
28 (B) 1/2
3 28
cos
6 28
co cos ec
s2
9 28
D None
3
cos
18 28
(C ) 1
In any triang triangle le ABC, ABC, (a + b)2 sin2 (A) c (a + b)
Q.8
2
=
I
(A) – 1/2 Q.7
B
If A is the the area and 2s the sum of the 3 sides of a triangle, triangle, then : (A) A
Q.6
2
+ a cos2
(B) in G.P.
n sin A cos A
(A) a = 1 ;
A
C 2
(B) b (c + a)
+ (a b)2 cos2
C 2
cos ec
27 28
is equal to
(D) 0 =
(C) a (b + c)
(D) c2
x sin3 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 sca scalen lene tria riangle (B) ABC is a right triangle (C) ABC ABC is an obtu obtuse se angl angled ed tria triang ngle le (D) (D) ABC is an equilateral triangle
Q.10
In a triangle ABC, CH and CM are the lengths of the altitude and median to the base AB. If If a = 10, b = 26, c = 32 then length (HM)
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[2]
(A) 5 Q.11 Q.11
(B) 7
sin 2
The The value value of
sin cos (A) is less than – 1
(C) 9
sin cos tan 2 1
(D) none
for all permissible vlaues of (B) is greater than 1
(C) lies lies betwee between n – 1 and 1 includ including ing both both
(D) lies between between –
2 and
Q.12 Q.12
sin 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 triangle triangle ABC, CD is the the bisector bisector of the angle C. If If cos
2
has
C 2
1
has the value and l (CD) = 6, then 3
1 1 has the value equal to a b (A) Q.14
1
9
(B)
1
12
RS , 5 , 19 , 23 UV T12 12 12 12 W R 5 , 13 , 19 UV (C) S T 12 12 12 W
Q.17
(B)
7 17 23 ,
12 12
,
12
,
12
1
R , 7 , 19 , 23 UV T 12 12 12 12 W
(B) 2
2
(D) S
(C)
If cos ( + ) = 0 then sin ( + 2) = (A) sin (B) sin
2
(C) cos
tan A tan B
has the value equal to 1
(D)
(D)
cos
2
With usual usual notations, notations, in a triangle triangle ABC, a cos(B cos(B – C) + b cos(C – A) + c cos(A – B) is equal equal to (A)
Q.18
(D) none
6
If the median of a triangle ABC through through A is perpendicular perpendicular to AB then (A)
Q.16 .16
1
The set of angles angles btween btween 0 & 2 satisfying the equation 4 cos2 2 2 cos 1 = 0 is (A)
Q.15
(C)
abc
(B)
R 2
abc
(C)
4R 2
4abc
(D)
R 2
abc 2R 2
cos cos3 2 tan cot = 1 if : sin cos 1 cot 2
sin3
2
(A) 0 ,
, 2
(B)
(C) ,
3
2
3 , 2 2
(D)
Q.19
With usual notations in a triangle 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 triangle ABC, angle angle B < angle angle C and the values of B & C satisfy the equation 2 tan x - k (1 + tan2 x) = 0 where (0 < k < 1) . Then the measure of angle A is :
Bansal C lasses
Q. B. on --II, -I -II, -I -III, Sequence & Progression
[3]
(A) Q.21 Q.21
/3
(B) 2/3
If cos cos =
1 then tan cot has the value equal equal to, where(0 < < and 0 < < ) 2 2 2 cos
2 3
,
4
,
12 12
k 1
(B)
k 1
The equati equation, on, sin2
If
5
(D)
2
x 2
3
(C)
3
2
,
C
1 32
Bansal C lasses
10
(D)
3 ,
2
10
,
5
B
k 1
1
=1
k
1 k
4 sin 3
(D)
(C) two roots
1 r1
1 1 1 1 = r2 r2 r3 r3 r1
K R 3 a 2 b 2 c2
(C) 64
where K has the value
(D) 128
1 sin x
1 sin x x
(C) tan
2
k
(D) infinite roots
1
x
k 1
has :
1
ABC
1 sin x 1 sin x
is
(D) –tan
2
x 2
2
4 then : = z sin 3 3
a cos A
(B)
R
8
A
(C)
(B) (B) xy + yz + zx = 0
r
8
,
= k sin 2 , then tan 2 tan 2 = 2
k 1
sin 3
ABC, the value of
The value value of cos
(A)
6
C
(B) cot
In a
3
x 3 , then the value of the expression
If x sin sin = y sin
(A)
Q.29
2
,
(B) 16
(A) (A) x + y + z = 0
Q.28 Q.28
,
4
With With usual notation notation in a
(A) –cot
Q.27 Q.27
(B) one root
equal to : (A) 1
Q.26 .26
(B)
(A) no root Q.25
(C ) 3
2
If A + B + C = & sin A (A)
Q.24
(B)
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 Q.23
(D) 3/4
2 cos
(A) 2 Q.22 Q.22
/2
(C )
cos
2 10
(B)
b cos B c cos C is equal to : abc
R
2 r
cos 1 16
(C) (C) xyz xyz + x + y + z = 1 (D) (D) none none
4 10
cos
(C)
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 10 2 5 64
[4]
Q.30
With With usual usual notation notation 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 equal to:
(C) 1/4
(D) 4
Q.31 Q.31
If a cos cos3 + 3a cos sin2 = m and a sin3 + 3a cos2 sin = n . Then (m + n)2/3 + (m n)2/3 is equal to : (A) (A) 2 a2 (B) 2 a1/3 (C) (C) 2 a2/3 (D) (D) 2 a3
Q.32
In a triangle ABC , AD is the altitude a ltitude from A . Given b > c , angle C = 23° & AD = then angle B = (A) 157°
Q.33
(B) 113°
c2
[JEE ’94, 2] (D) none
(C) 147°
(B) tan 3x
(C) 3 tan 3x
Q.34 Q.34
In a ABC, cos 3A + cos 3B + cos 3C = 1 then : (A) ABC is right angled (B) ABC is acute angled (C) ABC is obtuse angled (D) nothing definite can be said about about the nature of the .
Q.35
The The value value 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 incirc incircle le 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 x yz is equal to : (A) (A) R r 2
(B) rR 2
(C)
1 2
R r 2
(D)
1 2
r R 2
Q.37
The number number of solutions solutions of tan (5 cos ) = cot (5 sin ) for in (0, 2) is : (A) 28 (B) 14 (C) 4 (D) 2
Q.38 Q.38
If A = 340 340 0 then 2 sin
Q.39
b
The value value of cotx cot x + cot (60º (60º + x) + cot cot (120º (120º + x) x) is equa equall to : (A) cot 3x
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 perpendiculars perpendiculars from the angular angular points points of a ABC upon the opposite sides. 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 --II, -I -II, -I -III, Sequence & Progression
[5]
Q.40
The value value of cosec cosec
18
–
3 sec
18
is a
(A) surd (C) negative natural number Q.41 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 the value value equal to :
(B) 3
(C ) 2
(D) 1
Q.42
The set of values of ‘a’ for which which the the equation, equation, 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 angle angled d triang triangle le the the hypote hypotenuse nuse is 2 2 times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are (A)
Q.44
3
&
6
(B)
8
&
3
(C)
8
&
4
4
(D)
5
&
3 10
Let f, g, g, h be be the lengths lengths of the the perpendic perpendiculars ulars from from the circumcentre circumcentre of of the ABC on the sides a, b and c respectively . If (A) 1/4
a f
b g
c h
=
abc fgh
then the value value of is :
(B) 1/2
(C ) 1
cot A2 . cot cot A2 2
Q.45 .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 general general solution solution 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 of the distances of the incentre from from the angular points of a ABC is : (A) 4 R 2 r
Q.49
(C ) 3
If the orthocentre orthocentre and circumcentre circumcentre of a triangle ABC be at equal distances distances from the the side BC and and lie on the same side of BC then tanB tanC has the value equal to : (A) 3
Q.47
(B) 2
(B) 4 Rr2
Number Number of of roots roots of of the equation equation [ ] is (A) 2
Bansal C lasses
(B) 4
(C) cos 2 x
a
31 2 (C ) 6
b c R
(D)
s
sin x
3 4
1 0
a bcs R
which lie in the interval interval
(D) 8
Q. B. on -I, -II, -III & Binomial
[6]
sec 8
Q.50
Q.51 Q.51
1 is equal to sec 4 1 (A) tan 2 cot 8 (B) tan 8 tan 2 In a
ABC
3 1
if b = a
(A) 150 Q.52
Q.53
Q.54 Q.54
C = 300
(B) 450
Number Number of values values of (A) 1
(B) 2
(C) 3
(B) r 3 = 2r 1
3 tan 2
The The expres expressio sion, n,
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
ap
(D) r 2 = 3r 1
+ cos sin( ) + cos cos ( + ) sin when
2
1
(C)
2
(D) none
3 5 cos when simplified 2 2
The The expres expressio sion n [1 sin sin (3 ) + cos cos (3 + )] 1 sin
(B) sin 2
(C) 1 sin 2
(D) 1 + sin 2
If ‘O’ is the circumcen circumcentre tre of 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
R 2
(D)
4R 2
The maximum maximum value value of ( 7 cos + 24 sin ) × ( 7 sin – 24 cos ) for every (A) 25
Q.59 Q.59
(D) 4
In a ABC, a = a1 = 2 , b = a2 , c = a3 such that a p+1 =
OBC, OCA and OAB respectively then
Q.58
(D) 1050
The exact exact value value of 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
(D) tan 8 cot 2
equation cotx – cosx = 1 – cotx. cosx [ 0 , 2 ] satisfying the equation
where p = 1,2 then (A) r 1 = r 2
Q.55
and
(C) cot 8 cot 2
(B) 625
(C)
625
(D)
2
R .
625 4
4 sin5 in50 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)
3d 3
1i
2 2
Q. B. on --II, -I -II, -I -III, Sequence & Progression
[7]
Q.60
If x, y and z are the distances of incentre from the vertices of the triangle ABC 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
The The median medianss of a ABC are 9 cm, 12 cm and 15 cm respectively . Then the area of the triangle is (A) 96 sq cm (B) 84 sq cm (C) 72 sq cm (D) 60 sq cm
Q.62 Q.62
If x =
n
x
2
2
, satisfies the equation sin
cos
(A) n = 1, 0, 3, 5 (C) n = 0, 2, 4 Q.63
F
G1 cos The The value value of H (A)
9
2
x
= 1 sin x & the inequality 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
16
x
(B)
10
(C)
16
12
16
(D)
5 16
Q.64
The number number of all possible possible triplets triplets (a 1 , 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 Q.65
In a ABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius of the semicircle 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 each natural natural number number k , let C k 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 in this manner .The particle starts at (1, 0).If the particle crosses the 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 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 + 2cosC 2cosC = 2 then the sides sides of the ABC are in (A) A.P. (B) G.P (C) H.P.
(D) obtuse
(D) none
If A and B are complimentary complimentary angles, 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
The value value of ,
2
3 cosec 20° sec20°
Q.71 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 ABC is isosce isosceles les but but not right right angled angled (B) ABC is acute angled (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 set of values values of of x satisfyin satisfying g the equation, equation, 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 of the arithmetic arithmetic mean of the lengths lengths of of the sides sides of a triangle triangle and and harmonic harmonic mean 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 ABC ]
Q.74
If in a triangle triangle sin A : sin C = sin (A 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 number of solution of the equation, equation,
cos(r x) = 0 lying in (0, p) is : r 1
(A) 2
Q.76 .76
If
= 3 and sin
=
1
(A)
Q.78
(B) 3
a
2
b
(C) 5 a
a2
b2
. The The value of the expression , a cosec b sec is
(B) 2 a2 b2
2
1
0
1
(C) a + b
0
0
(D) none
0
1 1 The value value of cot 7 + tan 67 – cot 67 – tan7 is : 2 2 2 2 (A) a rationa rationall number number (B) (B) irrati irration onal al numb number er
Q.79
(D) more than 5
If in a triang triangle le ABC ABC (A)
8
Bansal C lasses
2 cos A a
(B)
4
cos B b
(C) 2(3 + 2 3 )
2 cos C c
a b c
(C)
3
b ca
(D) 2 (3 – 3 )
then the value of the angle A is : (D)
2
Q. B. on --II, -I -II, -I -III, Sequence & Progression
[9]
Q.80
The value of the expression (sinx + cosecx)2 + (cosx + secx)2 – ( tanx + cotx)2 wherever defined is equal to (A) 0 (B) 5 (C ) 7 (D) 9
Q.81
If A = 580 580 0 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 Q.83
A 2
1 sin A 1 sin A
With With usual notations notations in a triangle triangle ABC, if r 1 = 2r 2 = 2r 3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a If
tan tan
x2 x
=
and tan
x2 x 1 ( + ) has the value value equal to : (A) 1
=
1 2 x 2 2x 1
(B) – 1
(x
1 sin A
1 sin A
(D) 2a = 3b
0, 1), where 0 < ,
(C ) 2
(D)
r r r
<
2
, then tan
3 4
1
Q.84 .84
If r 1, r 2, r 3 be the radii of of excircles of the triangle triangle ABC, ABC, then
(A) Q.85
Q.86 Q.86
A
cot
2
(B)
cot
A 2
cot
(A) 2
2
(C)
tan
A 2
(D)
tan
A 2
x R
Minimum Minimum value value of 8cos 8cos2x + 18sec2x (A) 24 (B) 25
In a ABC
B
is equal to :
1 2
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
(B)
(C)
2
(D)
4
where is the area of the triangle Q.87 .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 general general solution solution of the the trigon trigonometr ometric ic equat equation ion 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 Q.89
If log loga b + log bc + logca vanishes where where a, b and c are positive positive 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 arcs of the the same same length length in two two circles S1 and S2 subtend angles 75° and 120° respectively at the S1
centre. The ratio
(A) Q.91
Q.92
S2
is equal to
1
81
(B)
5
16
Number Number of principal principal solution solution of the equation equation tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5
The The expr expres essio sion n
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
(C)
(B) a surd surd (D) a natural which is not composite
The value of x that satisfies satisfies the relation relation 2 3 x = 1 – x + x – x + x4 – x5 + ......... (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 Q.94
(A) sin
Q.95
Q.96 Q.9 6
If sin sin = sin then sin
3
=
3 3
3 3
(B) sin
3
(C) sin
(D)
sin 3
3
Choose Choose the INCORREC INCORRECT T statement(s). statement(s). 1
1
(B)
If tanA =
(C) (D)
The sign sign of the produc productt sin2 sin 2 . sin3 sin 3 . sin 5 is positi positive. ve. Ther Theree exist existss a valu valuee 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
(A
2
. cos 37
1
3 4
3
2
have the same value.
then tan (A B) must be irrational.
Which of the the followi following ng functions functions have have the maximum value unity ? (A) sin2 x cos2 x
(C)
sin 2x
Bansal C lasses
cos2x 2
(B)
(D)
sin 2x
cos 2x 2
1 1 cos x sin x 3 5 2
6
Q. B. on -I - I, -I - II, -I - III, Sequence & Progression
[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 extreme extreme value value of 1 + 4 sin + 3 cos is : (A) 3 (B) 4 (C ) 5
The sines of two angles angles of a triangle triangle are equal to (A) 245/1313
(B) 255/1313
Q.100 It is known known 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
(B)
5 3
(D) 765/1313
3 sin ( )
2 cos 6
cos ( )
sin
is:
for tan > 0
(D) none
Q.101 If x = 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.102 If 2 cos + sin = 1, then the value of 4 cos + 3sin is equal to (A) 3 Q.103 Q.103 If sint sin t + cost = (A)
1
(B) –5 1 5
then tan
t 2
(C )
7 5
(D) –4
is equal equal to : 1
(*B) – 3
(C ) 2
(D)
1 6
SEQUENCE & PROGRESSION Select the correct alternative : (Only one is correct)
Q.104 If a, b, c be in A.P., A.P., b, c, d in G.P. G.P. & c, d, e in H.P., then a, c, e will wil l be in : (A) A.P. (B) G.P. (C) H.P. (D) none of these Q.105 If a, b, c are in H.P., H.P., then a, a c, a b are in : (A) A.P. (B) G.P. (C) H.P.
(D) none of these
Q.106 If three positive positive numbers a , b, b, c are in H.P. H.P. then e n ( a c) n ( a 2 b c) simplifies to (A) (a – c)2 (B) zero (C ) ( a – c ) (D) 1
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[12]
Q.107 The The sum
r 2
1 r 2
1
is equal to :
(A) 1
(B) 3/4
(C) 4/3
(D) none
Q.108 In a potato race race , 8 potatoes are placed placed 6 metres apart on a straight straight line, the first being 6 metres metres from the basket basket which which is also placed placed in the the same line. A contestant contestant starts from the the basket basket and puts puts one one potato potato at a time into the basket. Find the total distance he must run in order to finish the race. (A) 420 (B) 210 (C) 432 (D) none Q.109 If the roots roots of of the cubic x 3 – px2 + qx – r = 0 are in G.P. then (A) q3 = p3r (B) p3 = q3r (C) pq = r
(D) pr = q
Q.110 Along a road road lies an odd number number of stones placed at intervals intervals of 10 m. These stones stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Then the number of stones is (A) 15 (B) 29 (C) 31 (D) 35 Q.111 Q.111 If log
( 5 . 2 x 1)
2 ; log
( 21 x 1)
4
and 1 are in Harmonical Progression then
(A) x is a positive real (C) (C) x is rati ratio onal nal whic which h is not not inte integr gral al
(B) x is a negative real (D) (D) x is an inte intege ger r
Q.112 If a, b, c are in G.P G.P., ., then the equations, ax 2 + 2bx + c = 0 & dx2 + 2ex + f = 0 have a common root, if d
,
e f
,
a b c
are in :
(A) A.P.
(B) G.P.
(C) H.P.
(D) none
Q.113 If the sum of the roots of of the quadratic quadratic equation, ax 2 + bx + c = 0 is equal to sum of the squares of their a b c
reciprocals, then , c
(A) A.P.
,
a b
are in :
(B) G.P.
Q.114 If for an A.P. A.P. a 1 , a2 , a3 ,.... , an ,.... a1 + a3 + a5 = – 12 and a1 a2 a3 = 8 then the value of a 2 + a4 + a6 equals (A) – 12 (B) – 16
(C) H.P.
(D) none
(C) – 18 (D) – 21 [ Apex : Q.62 of Test - 1 Scr. 2004 ]
Q.115 Given four positive number in A.P A.P.. If 5 , 6 , 9 and 15 are added respectively to these numbers , we get a G.P. , then which of the following holds? (A) the common ratio of G.P. is 3/2 (B) common ratio of G.P. is 2/3 (C) common difference of the A.P. is 3/2 (D) common difference of the A.P. is 2/3
Bansal C lasses
Q. B. on -I - I, -I - II, -I - III, Sequence & Progression
[13]
Q.116 Consider an A.P. A.P. with first term 'a' and the common common difference d. Let Let S k denote the sum of the first K Skx terms. Let S is independent of x, then x
(A) a = d/2
(B) a = d
(C) a = 2d
(D) none
Q.117 Concentric circles of radii 1, 2, 3......100 3......100 cms are drawn. The The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green regions in sq. cm is equal to (A) 1000 (B) 5050 (C) 4950 (D) 5151 Q.11 .118 Consider
the the A.P A.P. a 1 , a2 ,..... , an ,.... the th e G.P. b1 , b2 ,....., bn ,..... 9
such that a1 = b1 = 1 ; a9 = b9 and (A) b6 = 27
a
r
369 then
r 1
(B) b7 = 27
(C) b8 = 81 (D) b9 = 18 [ Apex : Q.68 of Test - 1 Scr. 2004 ]
Q.119 For an increasing increasing A.P. A.P. a 1, a2, ...... a n if a1 + a3 + a5 = – 12 : a1a3a5 = 80 then which of the following does not hold? (A) a1 = – 10 (B) a2 = – 1 (C) a3 = – 4 (D) a5 = 2 2 2 2 Q.120 Consider a decreasing decreasin g G.P. .P. : g 1, g2, g3, ...... gn ....... such that g1 + g2 + g3 = 13 and g1 g 2 g3 =91
then which of the following does not hold? (A) (A) The great reates estt term term of the the G.P. .P. is 9. (C) g1 = 1
(B) (B) 3g4 = g3 (D) g2 = 3
Q.121 If p , q, r in H.P. H.P. and p & r be different different having same sign sign then the roots of the equation equation px 2 + qx + r = 0 are (A) real & equal (B) real & disti stinct (C) irrational (D) imaginary Q.122 The point A(x 1, y1) ; B(x2, y2) and C(x3, y3) lie on the parabola y = 3x 2. If x1, x2, x3 are in A.P. and y1, y2, y3 are in G.P. then the common ratio of the G.P. is (A) 3 + 2 2
(B) 3 +
2
(C) 3 –
2
(D) 3 – 2 2
Q.123 If a, b, c are in A.P., A.P., then a 2 (b + c) + b2 (c + a) + c2 (a + b) is equal to : (A)
(a
Q.124 Q.124 If Sn =
b c) 3
(B)
8
1 3
1
1 2 3
1
(A) 1/2
2
3
+...... +
2 9
(a + b + c)3
(C)
1 2 3 ...... n 3
1
(B) 1
23 33 ...... n3
3 10
(a + b + c)3
1
(D) (a + b + c)3 9
, n = 1, 2, 3,...... Then S n is not greater than:
(C ) 2
(D) 4
Q.125 If Sn denotes the sum of the first n terms of a G.P. , with the first term and the common ratio both positive, then (A) Sn , S2n , S3n form a G.P. (B) Sn , S2n , – Sn , S3n , –S2n form a G.P. (C) S2n – Sn , S3n – S2n , S3n – Sn form a G.P. (D) S2n –Sn , S3n –S2n , S3n –Sn form a G.P.
Bansal C lasses
Q. B. on -I, -II, -III & Binomial
[14]
Q.126
1 2 .4 (A)
1.3 2.4.6
1.3.5 2.4.6.8
1
1.3.5.7 2.4.6.8.10
(B)
4
................
1
(C)
3
is equal to 1
(D) 1
2
Q.127 Consider an A.P. A.P. a 1 , a2 , a3 ,......... such that a3 + a5 + a8 = 11 and a4 + a2 = –2, then the value of a1 + a6 + a7 is (A) –8 (B) 5 (C) 7 (D) 9 Q.128 Q.1 28 A circle of radius radius r is inscribed in a square. The mid points of sides of of the square have been connected by line segment and a new square resulted. The sides of the resulting square were also connected by segments so that a new square was obtained and so on, then the radius of the circle inscribed in the n th square is
1n 2 (A) 2 r
Q.129 The sum
33n 2 r (B) 2 2 k 2
k 1
k
(C)
equal to
3
(A) 12
(B) 8
Q.130 The sum 5
5 3 n 2 r (D) 2
n 2 2 r
2n 2
4
n 2
n 1
(A) 1372
(C) 6
(D) 4
(C) 320
(D) 388
is equal to (B) 440
Q.131 Given a m+n = A ; am–n = B as the terms of the G.P. a 1 , a2 , a3 ,............. then for A 0 which of the following holds? (A) a m AB (C) a m
A a1 B
1 2
Bansal C lasses
2 n A n Bn
m 2 m n mn m n
Q.132 The sum sum of the the infinite infinite series, series, 1 (A)
(B) a n
(B)
25 24
(D) a n
2
22 5
32 52
42 53
(C)
52 54 25 54
A a1 B
62 55
m 2 mn n 2 m n
+........ is :
(D)
125 252
Q. B. on -I - I, -I - II, -I - III, Sequence & Progression
[15]
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.10 C Q.11 D Q.12 Q.15 C Q .1 6 A Q.17 A Q .1 8 B Q.19 Q.22 C Q .2 3 A Q.24 D Q .2 5 C Q.26 Q.29 D Q .3 0 C Q.31 C Q .3 2 B Q.33 Q.36 C Q .3 7 A Q.38 D Q .3 9 C Q.40 Q.43 B Q .4 4 A Q.45 A Q .4 6 A Q.47 Q.50 D Q .5 1 D Q.52 B Q .5 3 C Q.54 Q.57 C Q .5 8 C Q.59 B Q .6 0 B Q.61 Q.64 D Q .6 5 A Q.66 B Q .6 7 C Q.68 Q.71 C Q .72 A Q.73 B Q .7 4 A Q.75 Q.79 D Q .8 0 B Q.81 C Q .8 2 C Q.83 Q.86 B Q .8 7 B Q.88 D Q .8 9 A Q.90 Q.93 C Select the correct correct alternatives : (More than 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
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 BD
Q.99
BC
SEQUENCE & PROGRESSION Select the correct alternative : (Only one is correct) Q.104 B Q.105 C Q.106 A Q.107 Q.109 A Q.110 C Q.111 B Q.112 Q.114 D Q.115 A Q.116 A Q.117 Q.119 B Q.120 C Q.121 D Q.122 Q.124 C Q.125 B Q.126 C Q.127 Q.129 B Q.130 c Q.131 A Q.132
Bansal C lasses
B A B A C C
Q. B. on -I, -II, -III & Binomial
Q.108 Q.113 Q.118 Q.123 Q.128
D B D A A B B B A C B C D
C C B B A
[16]
