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BANSAL CLASSES MATHEMATI CS 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 Each of 75 Minutes duration approx.
Question bank on Compound angles, Trigonometric eqn and ineqn, 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 a triangle ABC, b cos2 (A) in A.P.
Q.3
Q.4
(B) in G.P.
sin A (1 n ) cos A
Given a2 + 2a + cosec2
(B)
(n 1) cos A sin A
x I 2 (C) a R ; x
s2 3 3
The exact value of cos (A) – 1/2
Q.7
(C)
sin A (n 1) cos A
(D)
sin A (n 1) cos A
x I 2 (D) a , x are finite but not possible to find
(B) a = –1 ;
(B) A =
s2 2
(C) A >
s2 3
D None
2 3 6 9 18 27 cos ec cos cos ec cos cos ec is equal to 28 28 28 28 28 28 (B) 1/2 (C) 1 (D) 0
In any triangle ABC, (a + b)2 sin2 (A) c (a + b)
Q.8
None
If A is the area and 2s the sum of the 3 sides of a triangle, then : (A) A
Q.6
(D)
FG (a x)IJ = 0 then, which of the following holds good? H2 K
(A) a = 1 ;
Q.5
(C) in H.P.
n sin A cos A then tan(A + B) equals 1 n cos2 A
If tanB = (A)
A B 3 + a cos2 = c then a, b, c are : 2 2 2
C C + (a b)2 cos2 = 2 2
(B) b (c + a)
(C) a (b + c)
(D) c2
. cos 32 x sin3 72 x when simplified reduces to : cos x 2 . tan 32 x
tan x
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 triangle ABC, CH and CM are the lengths of the altitude and median to the base AB. If a = 10, b = 26, c = 32 then length (HM) (A) 5 (B) 7 (C) 9 (D) none
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[2]
Q.11
sin 2 sin cos The value of for all permissible vlaues of sin cos tan 2 1 (A) is less than – 1 (B) is greater than 1 (C) lies between – 1 and 1 including both (D) lies between – 2 and 2
Q.12
sin 3 = 4 sin sin 2 sin 4 in 0 has : (A) 2 real solutions (B) 4 real solutions (C) 6 real solutions (D) 8 real solutions.
Q.13
In a triangle ABC, CD is the bisector of the angle C. If cos
1 C has the value and l (CD) = 6, then 3 2
1 1 has the value equal to a b
(A) Q.14
1 9
(B)
1 12
(C)
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 (B) , , ,
Q.17
(D)
1 2
(B) 2
If cos ( + ) = 0 then sin ( + 2) = (A) sin (B) sin
12
12
RS , 7 , 19 , 23 UV T 12 12 12 12 W tan A has the value equal to tan B
1 2
(C) 2
(D)
(C) cos
(D) cos
With usual notations, in a triangle ABC, a cos(B – C) + b cos(C – A) + c cos(A – B) is equal to (A)
Q.18
12 12
If the median of a triangle ABC through A is perpendicular to AB then (A)
Q.16
(D) none
The set of angles btween 0 & 2 satisfying the equation 4 cos2 2 2 cos 1 = 0 is
(A)
Q.15
1 6
abc R2
(B)
sin 3 cos 3 sin cos
2
(A) 0 ,
abc 4R 2
cos 1 cot 2
(C)
4abc R2
(D)
abc 2R 2
2 tan cot = 1 if : 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 to (A) R2r (B) 2R2r (C) 4R2r (D) 16R2r
Q.20
In a triangle ABC, angle B < 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 : (A) /3 (B) 2/3 (C) /2 (D) 3/4
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[3]
Q.21
If cos =
2 cos 1 then tan cot has the value equal to, where(0 < < and 0 < < ) 2 2 2 cos
(A) 2 Q.22
2 , , 3 4 12
(B)
k 1 k1
(B)
The equation, sin2 (A) no root
Q.25
(C)
3 , , 2 8 8
(D)
3 , , 2 10 5
k1 k 1
(C)
k k1
(D)
k1 k
4 4 =1 has : 3 sin 1 sin 1 3
(C) two roots
(D) infinite roots
3 1 1 1 1 1 1 = KR where K has the value r1 r2 r2 r3 r3 r1 a 2 b 2 c2
With usual notation in a ABC
(B) 16
(C) 64
(D) 128
1 sin x 1 sin x 5 x 3 , then the value of the expression is 2 1 sin x 1 sin x x x x x (A) –cot (B) cot (C) tan (D) –tan 2 2 2 2
Q.26
If
Q.27
If x sin = y sin
(A) x + y + z = 0
2 4 = z sin then : 3 3
(B) xy + yz + zx = 0
In a ABC, the value of (A)
Q.29
(D) 3
C A B C tan = = k sin , then tan 2 2 2 2
(B) one root
equal to : (A) 1
Q.28
, , 2 3 6
If A + B + C = & sin A (A)
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)
r R
The value of cos (A)
1 32
Bansal Classes
(C) xyz + x + y + z = 1 (D) none
a cos A b cos B c cos C is equal to : abc
(B)
R 2r
(C)
R r
(D)
2r R
2 4 8 16 cos cos cos cos is : 10 10 10 10 10
(B)
1 16
(C)
cos / 10 16
(D)
Q. B. on -I, -II, -III & Binomial
10 2 5 64
[4]
Q.30
With usual notation in a ABC, if R = k (A) 1
(B) 2
r1 r2 r2 r3 r3 r1 r1 r2 r2 r3 r3 r1
(C) 1/4
where k has the value equal to: (D) 4
Q.31
If a 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) 2 a2 (B) 2 a1/3 (C) 2 a2/3 (D) 2 a3
Q.32
In a triangle ABC , AD is the altitude from A . Given b > c , angle C = 23° & AD = then angle B = (A) 157°
Q.33
(B) 113°
(C) 147°
[JEE ’94, 2] (D) none
The value of cot x + cot (60º + x) + cot (120º + x) is equal to : (A) cot 3x
(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 is acute angled (C) ABC is obtuse angled (D) nothing definite can be said about the nature of the .
Q.35
The value of
3 9 tan 2 x (D) 3 tan x tan 3 x
3 cot 76 cot 16 is : cot 76 cot 16
(A) cot 44º Q.36
(B) tan 44º
(C) tan 2º
(D) cot 46º
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
(B) r R2
(C)
1 R r2 2
(D)
1 r R2 2
Q.37
The number of solutions of tan (5 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
abc b 2 c2
A is identical to 2
(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 CF are the perpendiculars from the angular points of a ABC upon the opposite sides. The perimeters of the DEF and ABC are in the ratio : (A)
2r R
(B)
r 2R
(C)
r R
(D)
r 3R
where r is the in radius and R is the circum radius of the ABC
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[5]
Q.40
The value of cosec
– 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 C · cot has the value equal to : 2 2
(B) 3
(C) 2
(D) 1
Q.42
The set of values of ‘a’ for which the equation, cos 2x + a sin x = 2a 7 possess a solution is : (A) (, 2) (B) [2, 6] (C) (6, ) (D) ()
Q.43
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 (A)
Q.44
& 3 6
(B)
3 & 8 8
(C)
(A) 1/4
(B) 1/2
(C) 1
In ABC, the minimum value of
(A) 1
is
(C) 3
(D) non existent
(B)
1 3
(C) – 3
(D) –
The general 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 the angular points of a ABC is : (A) 4 R2 r
Q.49
A 2
If the orthocentre and circumcentre of a triangle ABC be at equal distances from the side BC and lie on the same side of BC then tanB tanC has the value equal to : (A) 3
Q.48
(B) 2
(D) 2
B
cot 2 2 . cot 2 2 cot 2
Q.47
3 & 5 10
abc a b c then the value of is : = fgh f g h
A
Q.46
(D)
Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ABC on the sides a, b and c respectively . If
Q.45
& 4 4
(B) 4 Rr2
Number of roots of the equation cos 2 x [] is (A) 2
Bansal Classes
(B) 4
(C)
a b c R s
(D)
a bc s R
3 1 3 sin x 1 0 which lie in the interval 2 4
(C) 6
(D) 8
Q. B. on -I, -II, -III & Binomial
[6]
Q.50
sec 8 1 sec 4 1 is equal to
(A) tan 2 cot 8
(B) tan 8 tan 2
(C) cot 8 cot 2
(D) tan 8 cot 2
Q.51
In a ABC if b = a 3 1 and C = 300 then the measure of the angle A is (A) 150 (B) 450 (C) 750 (D) 1050
Q.52
Number of values of [ 0 , 2 ] satisfying the equation cotx – cosx = 1 – cotx. cosx (A) 1 (B) 2 (C) 3 (D) 4
Q.53
The exact value of cos273º + cos247º + (cos73º . cos47º) is (A) 1/4 (B) 1/2 (C)3/4
Q.54
In a ABC, a = a1 = 2 , b = a2 , c = a3 such that ap+1 = where p = 1,2 then (A) r1 = r2
Q.55
(B) r3 = 2r1
3
3
(C) 1
(B) sin 2
3 5 cos when simplified 2 2
(C) 1 sin 2
(D) 1 + sin 2
If ‘O’ is the circumcentre of the ABC and R1, R2 and R3 are the radii of the circumcircles of triangles OBC, OCA and OAB respectively then a bc (A) 2 R3
a b c has the value equal to: R1 R 2 R 3
R3 (B) a bc
(C)
4 R2
(D)
4R 2
The maximum value of ( 7 cos + 24 sin) × ( 7 sin – 24 cos) for every R . (A) 25
Q.59
(D) none
The expression [1 sin (3) + cos (3 + )] 1 sin reduces to : (A) sin 2
Q.58
(D) r2 = 3r1
tan 2 cos 2 The expression, + cos sin () + cos (+) sin when 2 2 cos(2 )
Q.57
5p 4p 2 a 22 p ap 2p p 3 5p
(C) r2 = 2r1
simplified reduces to : (A) zero (B) 1 Q.56
(D) 1
(B) 625
(C)
625 2
(D)
625 4
4 sin50 sin550 sin650 has the values equal to (A)
31 2 2
Bansal Classes
(B)
31 2 2
(C)
31 2
d
i
3 3 1 (D)
Q. B. on -I, -II, -III & Binomial
2 2
[7]
Q.60
If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then a bc x y z is equal to (A)
A
tan 2
(B)
A
cot 2
(C)
A
tan 2
(D)
A
sin 2
Q.61
The medians 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
If x =
n x x x 3 , satisfies the equation sin cos = 1 sin x & the inequality , then: 2 2 2 2 2 4
(A) n = 1, 0, 3, 5 (C) n = 0, 2, 4 Q.63
FG H
The value of 1 cos (A)
9 16
(B) n = 1, 2, 4, 5 (D) n = 1, 1, 3, 5
9
IJ FG1 cos 3 IJ FG1 cos 5 IJ FG1 cos 7 IJ is KH 9KH 9K H 9K (B)
10 16
(C)
12 16
(D)
5 16
Q.64
The number of all possible triplets (a1 , a2 , a3) such that a1+ 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 semicircle is 2 2 (B) ab a bc Where is the area of the triangle ABC.
(A)
(C)
2 s
(D)
c 2
Q.66
For each natural number k , 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 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
If in a ABC,
cos A cos B cos C a b c
(A) right angled Q.68
Q.69
then the triangle is
(B) isosceles
(C) equilateral
If cos A + cosB + 2cosC = 2 then the 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 angles, then :
A 2
B 2
(A) 1 tan 1 tan = 2
Bansal Classes
A 2
B 2
(B) 1 cot 1 cot = 2
Q. B. on -I, -II, -III & Binomial
[8]
Q.70
2
B 2
(D) 1 tan 1 tan = 2
2
The value of , 3 cosec 20° sec 20° is : 2 sin 20
(A) 2 Q.71
A 2
A B (C) 1 sec 1 cos ec = 2
(B) sin 40
4 sin 20
(C) 4
(D) sin 40
If in a ABC, cosA·cosB + sinA sinB sin2C = 1 then, the statement which is incorrect, is (A) ABC is isosceles but not right angled (B) ABC is acute angled (C) ABC is right angled
(D) least angle of the triangle is
4
2 2 0.25sincosx2x 4 + 1 = 0, is :
tan x 4
Q.72
The set of values of x satisfying the equation, 2 (A) an empty set (B) a singleton (C) a set containing two values (D) an infinite set
Q.73
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 [ where is the area of the triangle ABC ]
Q.74
If in a 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,
cos(r x) = 0
lying in (0, p) is :
r 1
(A) 2
Q.76
If = 3 and sin = 1
(A)
Q.78
(B) 3
2
a b
a 2
a b2
If in a triangle ABC (A)
8
Bansal Classes
(C) a + b
(D) none
10 10 10 10 + tan 67 – cot 67 – tan7 is : 2 2 2 2
(A) a rational number (B) irrational number Q.79
(D) more than 5
. The value of the expression , a cosec b sec is
(B) 2 a 2 b 2
2
The value of cot 7
(C) 5
(C) 2(3 + 2 3 )
(D) 2 (3 – 3 )
2 cos A cos B 2 cos C a b then the value of the angle A is : a b c b c ca
(B)
4
(C)
3
(D)
Q. B. on -I, -II, -III & Binomial
2
[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 = 5800 then which one of the following is true
Q.82
A (A) 2 sin 1 sin A 1 sin A 2
A (B) 2 sin 1 sin A 1 sin A 2
A (C) 2 sin 1 sin A 1 sin A 2
A (D) 2 sin 2
With usual notations in a triangle ABC, if r1 = 2r2 = 2r3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a
1 sin A 1 sin A
(D) 2a = 3b
1 x2 x Q.83 If tan = 2 and tan = (x 0, 1), where 0 < , < , then tan 2 2 2 x 2x 1 x x 1 ( + ) has the value equal to : (A) 1
Q.84
Q.86
(C) 2
(D)
If r1, r2, r3 be the radii of excircles of the triangle ABC, then (A)
Q.85
(B) – 1
A
cot 2
(B)
A
B
cot 2 cot 2
(C)
A
tan 2
r1 r1r2
3 4
is equal to :
(D)
A
tan 2
Minimum value of 8cos2x + 18sec2x x R wherever it is defined, is : (A) 24 (B) 25 (C) 26 (D) 18 a2 b2 c 2 A B C In a ABC sin A sin B sin C . sin sin sin simplifies to 2 2 2
(A) 2
(B)
(C)
2
(D)
4
where is the area of the triangle Q.87
If is eliminated from the equations x = a cos( – ) and y = b cos ( – ) then
x2
y2
2xy cos( ) is equal to ab a b (A) cos2 ( – ) (B) sin2 ( – ) 2
Q.88
2
(C) sec2 ( – )
The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is (A) x = n (B) n ± (C) x = 2n 3 where n I
Bansal Classes
(D) cosec2 ( – )
(D) x =
Q. B. on -I, -II, -III & Binomial
n 3
[10]
Q.89
If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (logbc)3 + (logca)3 is (A) an odd prime (B) an even prime (C) an odd composite (D) an irrational number
Q.90
If the arcs of the same length in two circles S1 and S2 subtend angles 75° and 120° respectively at the S centre. The ratio 1 is equal to S2 (A)
Q.91
Q.92
1 5
81 16
(B)
(C)
Number of principal solution of the equation tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5
The expression
(D)
(C) 7
25 64
(D) more than 7
tan 2 20 sin 2 20 simplifies to tan 2 20 ·sin 2 20
(A) a rational which is not integral (C) a natural which is prime Q.93
64 25
(B) a surd (D) a natural which is not composite
The value of x that satisfies the relation x = 1 – x + x2 – x3 + x4 – x5 + ......... (A) 2 cos36° (B) 2 cos144° (C) 2 sin18°
(D) none
Select the correct alternatives : (More than one are correct) Q.94
If sin = sin then sin (A) sin
Q.95
3
= 3 3
3
3
(C) sin
3
3
(D) sin
Choose the INCORRECT statement(s).
Q.96
3
(B) sin
(A
1 1 1 1 sin 82 . cos 37 and sin 127 . sin 97 have the same value. 2 2 2 2
(B)
If tan A =
(C) (D)
The sign of the product sin 2 . sin 3 . sin 5 is positive. There exists a value of between 0 & 2 which satisfies the equation ; sin4 – sin2 – 1 = 0.
3 4 3
& tan B =
3 4 3
then tan (A B) must be irrational.
Which of the following functions have the maximum value unity ? (A) sin2 x cos2 x
(C)
sin 2x cos 2x
Bansal Classes
2
(B)
(D)
sin 2x cos 2x 2
6 1 sin x 1 cos x 3 5 2
Q. B. on -I, -II, -III & Binomial
[11]
Q.97
If the sides of a right angled triangle are { cos2cos2 + 2cos( + )} and
{sin2sin2 + 2sin( + )}, then the length of the hypotenuse is : (A) 2[1+cos()] (B) 2[1 cos()] (C) 4 cos2 Q.98
Q.99
2
(D) 4sin2
2
An extreme value of 1 + 4 sin + 3 cos is : (A) 3 (B) 4 (C) 5
The sines of two angles of a triangle are equal to (A) 245/1313
(B) 255/1313
(D) 6
5 99 & . The cosine of the third angle is : 13 101
(C) 735/1313
4 Q.100 It is known that sin = & 0 < < then the value of 5
(A) independent of for all in (0, /2) (C)
3 (7 24 cot ) for tan < 0 15
5
(B)
3
(D) 765/1313
3 sin ( ) cos2 cos ( ) 6 is: sin
for tan > 0
(D) none
Q.101 If x = sec tan & y = cosec + cot then : (A) x =
y 1 y 1
(B) y =
1 x 1 x
(C) x =
y 1 y 1
(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 If sin t + cos t = (A) 1
(B) –5
7 5
(D) –4
(C) 2
(D)
(C)
1 t then tan is equal to : 5 2
(*B) –
1 3
1 6
BINOMIAL There are 39 questions in this question bank. Q.104 Given that the term of the expansion (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 binomial (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 is 1/6 , then n = (A) 6 (B) 9 (C) 12 (D) 15
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[12]
x
n
Q.106 If the coefficients of x7 & x8 in the expansion of 2 are equal , then the value of n is : 3 (A) 15
(B) 45
(C) 55
(D) 56
1 1 1 Q.107 The coefficient of x49 in the expansion of (x – 1) x x 2 ..... x 49 is equal to 2 2 2 1 (A) – 2 1 50 2
(B) + ve coefficient of x
(C) – ve coefficient of x
1 (D) – 2 1 49 2
Q.108 The last digit of (3P + 2) is : (A) 1 (B) 2 4n where P = 3 and n N
(C) 4
(D) 5
1
n
Q.109 The sum of the binomial coefficients of 2 x is equal to 256 . The constant term in the x expansion is (A) 1120
Q.110 The coefficient of (A)
405 256
(B) 2110
x4
3 x in 2 2 x
(B)
(C) 1210
(D) none
10
is :
504 259
(C)
450 263
(D)
Q.111 The remainder, when (1523 + 2323) is divided by 19, is (A) 4 (B) 15 (C) 0
405 512
(D) 18
Q.112 Let (7 4 3 ) n = p + when n and p are positive integers and (0, 1) then (1 – ) (p + ) is (A) rational which is not an integer (B) a prime (C) a composite (D) none of these Q.113 If (11)27 + (21)27 when divided by 16 leaves the remainder (A) 0 (B) 1 (C) 2
(D) 14
Q.114 Last three digits of the number N = 7100 – 3100 are (A) 100 (B) 300 (C) 500
(D) 000
Q.115 The last two digits of the number 3400 are : (A) 81 (B) 43 (C) 29
(D) 01
Q.116 If (1 + x + x²)25 = a0 + a1x + a2x² + ..... + a50 . x50 then a0 + a2 + a4 + ..... + a50 is : (A) even (B) odd & of the form 3n (C) odd & of the form (3n 1) (D) odd & of the form (3n + 1)
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[13]
Q.117 The sum of the 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) none of 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 The expression
7 7 1 4x 1 1 4 x 1 is a polynomial in x of degree 2 2 4x 1
1
(A) 7
(B) 5
(C) 4
Q.120 If the second term of the expansion a 1/13
(A) 4
(D) 3
n
a is 14a5/2 then the value of a 1
(B) 3
(C) 12
n n
C3 is : C2
(D) 6
Q.121 If (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + x3 + ...... + xn) m
a0 + a1x + a2x2 + a3x3 + ...... + amxm then (A) n!
a r has the value equal to
r0
(B) (n + 1) !
(C) (n – 1)!
(D) none
Q.122 The value of 4 {nC1 + 4 . nC2 + 42 . nC3 + ...... + 4n 1} is : (A) 0 (B) 5n + 1 (C) 5n
(D) 5n 1
Q.123 If n be a positive integer such that n 3, then the value of the sum to n of the series 1.n
n 1 1!
(A) 0
(n 1) +
n 1 n 2 2!
n 1 n 2 n 3
(n 2) –
(B) 1
3!
(C) – 1
(n 3) + ...... is : (D) none of these
Q.124 In the expansion of (1 + x)43 if the coefficients of the (2r + 1)th and the (r + 2)th are equal, the value of r is : (A) 12 (B) 13 (C) 14 (D) 15
Q.125 The positive value of a so that the coefficient of x5 is equal to that of x15 in the expansion of x 2 (A)
1 2 3
1
(B)
(C) 1
3
x 1 x1 Q.126 In the expansion of 2/ 3 1/ 3 x x 1 x x1/ 2
(A)
10C
0
(B)
10C
7
a x3
10
is
(D) 2 3
10
, the term which does not contain x is : (C)
10C
(D) none
4 8
Q.127
If the 6th term in the expansion of the binomial
1 2 x 8/ 3 x log10 x is 5600, then x equals to
(A) 5
(C) 10
Bansal Classes
(B) 8
(D) 100
Q. B. on -I, -II, -III & Binomial
[14]
Q.128 Co-efficient of t in the expansion of, ( + p)m 1 + ( + p)m 2 ( + q) + ( + p)m 3 ( + q)2 + ...... ( + q)m 1 where q and p q is : m
(A)
Ct pt qt
m
(B)
pq m
(C)
Ct pt qt
pq m
(D)
pq
Ct pm t qm t
Ct pm t qm t
pq
Q.129 (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + ...... + x100) when written in the ascending power of x then the 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, f2 f + pf p is
(A) a natural number (C) a prime number
(B) a negative integer (D) are irrational number
Q.131 Number of rational in the expansion of (A) 25
(B) 26
2 43
100
is :
(C) 27
(D) 28 10
cos is Q.132 The greatest value of the term independent of x in the expansion of x sin x 10
(A)
10C
5
(B) 25
(C)
25
·
10C
5
(D)
C5
25
Q.133 If (1 + x – 3x2)2145 = a0 + a1x + a2x2 + ......... then a0 – a1 + a2 – a3 + ..... ends with (A) 1 (B) 3 (C) 7 (D) 9 9
4x 2 3 6 Q.134 Coefficient of x in the binomial expansion is 2 x 3 (A) 2438 (B) 2688 (C) 2868 1 Q.135 The term independent of ' x ' in the expansion of 9 x 3 x
(D) none 18
, x > 0 , is times the corresponding
binomial co-efficient . Then ' ' is : (A) 3
(B)
1 3
(C)
1 3
(D) 1
Q.136 The expression [x + (x31)1/2]5 + [x (x31)1/2]5 is a polynomial of degree : (A) 5 (B) 6 (C) 7 (D) 8 [JEE’92, 6 + 2]
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[15]
Q.137 Given (1 – 2x + 5x2 – 10x3) (1 + x)n = 1 + a1x + a2x2 + .... and that a12 = 2a2 then the value of n is (A) 6 (B) 2 (C) 5 (D) 3 Q.138 The sum of the series aC0 + (a + b)C1 + (a + 2b)C2 + ..... + (a + nb)Cn is where Cr's denotes combinatorial coefficient in the expansion of (1 + x)n, n N (A) (a + 2nb)2n (B) (2a + nb)2n (C) (a +nb)2n – 1 (D) (2a + nb)2n – 1 Q.139 The coefficient of the middle term in the binomial expansion in powers of x of (1 + x)4 and of (1 – x)6 is the same if equals 5 3
(A) –
10 3
(B)
(C) –
3 10
(D)
3 5
Q.140 (2n + 1) (2n + 3) (2n + 5) ....... (4n 1) is equal to : (A)
( 4 n) ! 2 . (2n) ! (2n) ! n
n
Q.141 If Sn = (A)
r 0
1 and Tn = n Cr
n 2
( 4 n) ! n ! 2 . (2n) ! (2n) !
(B)
n
n
r
nC r 0
(B)
n 1 2
r
(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 The coefficient 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 Classes
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.10 C Q.11 D Q.12 Q.15 C Q.16 A Q.17 A Q.18 B Q.19 Q.22 C Q.23 A Q.24 D Q.25 C Q.26 Q.29 D Q.30 C Q.31 C Q.32 B Q.33 Q.36 C Q.37 A Q.38 D Q.39 C Q.40 Q.43 B Q.44 A Q.45 A Q.46 A Q.47 Q.50 D Q.51 D Q.52 B Q.53 C Q.54 Q.57 C Q.58 C Q.59 B Q.60 B Q.61 Q.64 D Q.65 A Q.66 B Q.67 C Q.68 Q.71 C Q.72 A Q.73 B Q.74 A Q.75 Q.79 D Q.80 B Q.81 C Q.82 C Q.83 Q.86 B Q.87 B Q.88 D Q.89 A Q.90 Q.93 C Select the correct alternatives : (More than one are 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 Classes
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.14 Q.21 Q.28 Q.35 Q.42 Q.49 Q.56 Q.63 Q.70 Q.78 Q.85 Q.92
Q.98
BD
Q.99
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]
QUESTION BANK ON PH-1 (COMPOUND ANGLES) PH-2 (TRIGONOMETRIC EQUATIONS & INEQUATIONS) PH-3 (SOLUTIONS OF TRIANGLE) & BINOMIAL
Time Limit : 7 Sitting Each of 75 Minutes duration approx.
Question bank on Compound angles, Trigonometric eqn and ineqn, 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 a triangle ABC, b cos2 (A) in A.P.
Q.3
Q.4
(B) in G.P.
sin A (1 n ) cos A
Given a2 + 2a + cosec2
(B)
(n 1) cos A sin A
x I 2 (C) a R ; x
s2 3 3
The exact value of cos (A) – 1/2
Q.7
(C)
sin A (n 1) cos A
(D)
sin A (n 1) cos A
x I 2 (D) a , x are finite but not possible to find
(B) a = –1 ;
(B) A =
s2 2
(C) A >
s2 3
D None
2 3 6 9 18 27 cos ec cos cos ec cos cos ec is equal to 28 28 28 28 28 28 (B) 1/2 (C) 1 (D) 0
In any triangle ABC, (a + b)2 sin2 (A) c (a + b)
Q.8
None
If A is the area and 2s the sum of the 3 sides of a triangle, then : (A) A
Q.6
(D)
FG (a x)IJ = 0 then, which of the following holds good? H2 K
(A) a = 1 ;
Q.5
(C) in H.P.
n sin A cos A then tan(A + B) equals 1 n cos2 A
If tanB = (A)
A B 3 + a cos2 = c then a, b, c are : 2 2 2
C C + (a b)2 cos2 = 2 2
(B) b (c + a)
(C) a (b + c)
(D) c2
. cos 32 x sin3 72 x when simplified reduces to : cos x 2 . tan 32 x
tan x
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 triangle ABC, CH and CM are the lengths of the altitude and median to the base AB. If a = 10, b = 26, c = 32 then length (HM) (A) 5 (B) 7 (C) 9 (D) none
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[2]
Q.11
sin 2 sin cos The value of for all permissible vlaues of sin cos tan 2 1 (A) is less than – 1 (B) is greater than 1 (C) lies between – 1 and 1 including both (D) lies between – 2 and 2
Q.12
sin 3 = 4 sin sin 2 sin 4 in 0 has : (A) 2 real solutions (B) 4 real solutions (C) 6 real solutions (D) 8 real solutions.
Q.13
In a triangle ABC, CD is the bisector of the angle C. If cos
1 C has the value and l (CD) = 6, then 3 2
1 1 has the value equal to a b
(A) Q.14
1 9
(B)
1 12
(C)
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 (B) , , ,
Q.17
(D)
1 2
(B) 2
If cos ( + ) = 0 then sin ( + 2) = (A) sin (B) sin
12
12
RS , 7 , 19 , 23 UV T 12 12 12 12 W tan A has the value equal to tan B
1 2
(C) 2
(D)
(C) cos
(D) cos
With usual notations, in a triangle ABC, a cos(B – C) + b cos(C – A) + c cos(A – B) is equal to (A)
Q.18
12 12
If the median of a triangle ABC through A is perpendicular to AB then (A)
Q.16
(D) none
The set of angles btween 0 & 2 satisfying the equation 4 cos2 2 2 cos 1 = 0 is
(A)
Q.15
1 6
abc R2
(B)
sin 3 cos 3 sin cos
2
(A) 0 ,
abc 4R 2
cos 1 cot 2
(C)
4abc R2
(D)
abc 2R 2
2 tan cot = 1 if : 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 to (A) R2r (B) 2R2r (C) 4R2r (D) 16R2r
Q.20
In a triangle ABC, angle B < 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 : (A) /3 (B) 2/3 (C) /2 (D) 3/4
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[3]
Q.21
If cos =
2 cos 1 then tan cot has the value equal to, where(0 < < and 0 < < ) 2 2 2 cos
(A) 2 Q.22
2 , , 3 4 12
(B)
k 1 k1
(B)
The equation, sin2 (A) no root
Q.25
(C)
3 , , 2 8 8
(D)
3 , , 2 10 5
k1 k 1
(C)
k k1
(D)
k1 k
4 4 =1 has : 3 sin 1 sin 1 3
(C) two roots
(D) infinite roots
3 1 1 1 1 1 1 = KR where K has the value r1 r2 r2 r3 r3 r1 a 2 b 2 c2
With usual notation in a ABC
(B) 16
(C) 64
(D) 128
1 sin x 1 sin x 5 x 3 , then the value of the expression is 2 1 sin x 1 sin x x x x x (A) –cot (B) cot (C) tan (D) –tan 2 2 2 2
Q.26
If
Q.27
If x sin = y sin
(A) x + y + z = 0
2 4 = z sin then : 3 3
(B) xy + yz + zx = 0
In a ABC, the value of (A)
Q.29
(D) 3
C A B C tan = = k sin , then tan 2 2 2 2
(B) one root
equal to : (A) 1
Q.28
, , 2 3 6
If A + B + C = & sin A (A)
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)
r R
The value of cos (A)
1 32
Bansal Classes
(C) xyz + x + y + z = 1 (D) none
a cos A b cos B c cos C is equal to : abc
(B)
R 2r
(C)
R r
(D)
2r R
2 4 8 16 cos cos cos cos is : 10 10 10 10 10
(B)
1 16
(C)
cos / 10 16
(D)
Q. B. on -I, -II, -III & Binomial
10 2 5 64
[4]
Q.30
With usual notation in a ABC, if R = k (A) 1
(B) 2
r1 r2 r2 r3 r3 r1 r1 r2 r2 r3 r3 r1
(C) 1/4
where k has the value equal to: (D) 4
Q.31
If a 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) 2 a2 (B) 2 a1/3 (C) 2 a2/3 (D) 2 a3
Q.32
In a triangle ABC , AD is the altitude from A . Given b > c , angle C = 23° & AD = then angle B = (A) 157°
Q.33
(B) 113°
(C) 147°
[JEE ’94, 2] (D) none
The value of cot x + cot (60º + x) + cot (120º + x) is equal to : (A) cot 3x
(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 is acute angled (C) ABC is obtuse angled (D) nothing definite can be said about the nature of the .
Q.35
The value of
3 9 tan 2 x (D) 3 tan x tan 3 x
3 cot 76 cot 16 is : cot 76 cot 16
(A) cot 44º Q.36
(B) tan 44º
(C) tan 2º
(D) cot 46º
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
(B) r R2
(C)
1 R r2 2
(D)
1 r R2 2
Q.37
The number of solutions of tan (5 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
abc b 2 c2
A is identical to 2
(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 CF are the perpendiculars from the angular points of a ABC upon the opposite sides. The perimeters of the DEF and ABC are in the ratio : (A)
2r R
(B)
r 2R
(C)
r R
(D)
r 3R
where r is the in radius and R is the circum radius of the ABC
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[5]
Q.40
The value of cosec
– 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 C · cot has the value equal to : 2 2
(B) 3
(C) 2
(D) 1
Q.42
The set of values of ‘a’ for which the equation, cos 2x + a sin x = 2a 7 possess a solution is : (A) (, 2) (B) [2, 6] (C) (6, ) (D) ()
Q.43
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 (A)
Q.44
& 3 6
(B)
3 & 8 8
(C)
(A) 1/4
(B) 1/2
(C) 1
In ABC, the minimum value of
(A) 1
is
(C) 3
(D) non existent
(B)
1 3
(C) – 3
(D) –
The general 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 the angular points of a ABC is : (A) 4 R2 r
Q.49
A 2
If the orthocentre and circumcentre of a triangle ABC be at equal distances from the side BC and lie on the same side of BC then tanB tanC has the value equal to : (A) 3
Q.48
(B) 2
(D) 2
B
cot 2 2 . cot 2 2 cot 2
Q.47
3 & 5 10
abc a b c then the value of is : = fgh f g h
A
Q.46
(D)
Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ABC on the sides a, b and c respectively . If
Q.45
& 4 4
(B) 4 Rr2
Number of roots of the equation cos 2 x [] is (A) 2
Bansal Classes
(B) 4
(C)
a b c R s
(D)
a bc s R
3 1 3 sin x 1 0 which lie in the interval 2 4
(C) 6
(D) 8
Q. B. on -I, -II, -III & Binomial
[6]
Q.50
sec 8 1 sec 4 1 is equal to
(A) tan 2 cot 8
(B) tan 8 tan 2
(C) cot 8 cot 2
(D) tan 8 cot 2
Q.51
In a ABC if b = a 3 1 and C = 300 then the measure of the angle A is (A) 150 (B) 450 (C) 750 (D) 1050
Q.52
Number of values of [ 0 , 2 ] satisfying the equation cotx – cosx = 1 – cotx. cosx (A) 1 (B) 2 (C) 3 (D) 4
Q.53
The exact value of cos273º + cos247º + (cos73º . cos47º) is (A) 1/4 (B) 1/2 (C)3/4
Q.54
In a ABC, a = a1 = 2 , b = a2 , c = a3 such that ap+1 = where p = 1,2 then (A) r1 = r2
Q.55
(B) r3 = 2r1
3
3
(C) 1
(B) sin 2
3 5 cos when simplified 2 2
(C) 1 sin 2
(D) 1 + sin 2
If ‘O’ is the circumcentre of the ABC and R1, R2 and R3 are the radii of the circumcircles of triangles OBC, OCA and OAB respectively then a bc (A) 2 R3
a b c has the value equal to: R1 R 2 R 3
R3 (B) a bc
(C)
4 R2
(D)
4R 2
The maximum value of ( 7 cos + 24 sin) × ( 7 sin – 24 cos) for every R . (A) 25
Q.59
(D) none
The expression [1 sin (3) + cos (3 + )] 1 sin reduces to : (A) sin 2
Q.58
(D) r2 = 3r1
tan 2 cos 2 The expression, + cos sin () + cos (+) sin when 2 2 cos(2 )
Q.57
5p 4p 2 a 22 p ap 2p p 3 5p
(C) r2 = 2r1
simplified reduces to : (A) zero (B) 1 Q.56
(D) 1
(B) 625
(C)
625 2
(D)
625 4
4 sin50 sin550 sin650 has the values equal to (A)
31 2 2
Bansal Classes
(B)
31 2 2
(C)
31 2
d
i
3 3 1 (D)
Q. B. on -I, -II, -III & Binomial
2 2
[7]
Q.60
If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then a bc x y z is equal to (A)
A
tan 2
(B)
A
cot 2
(C)
A
tan 2
(D)
A
sin 2
Q.61
The medians 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
If x =
n x x x 3 , satisfies the equation sin cos = 1 sin x & the inequality , then: 2 2 2 2 2 4
(A) n = 1, 0, 3, 5 (C) n = 0, 2, 4 Q.63
FG H
The value of 1 cos (A)
9 16
(B) n = 1, 2, 4, 5 (D) n = 1, 1, 3, 5
9
IJ FG1 cos 3 IJ FG1 cos 5 IJ FG1 cos 7 IJ is KH 9KH 9K H 9K (B)
10 16
(C)
12 16
(D)
5 16
Q.64
The number of all possible triplets (a1 , a2 , a3) such that a1+ 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 semicircle is 2 2 (B) ab a bc Where is the area of the triangle ABC.
(A)
(C)
2 s
(D)
c 2
Q.66
For each natural number k , 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 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
If in a ABC,
cos A cos B cos C a b c
(A) right angled Q.68
Q.69
then the triangle is
(B) isosceles
(C) equilateral
If cos A + cosB + 2cosC = 2 then the 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 angles, then :
A 2
B 2
(A) 1 tan 1 tan = 2
Bansal Classes
A 2
B 2
(B) 1 cot 1 cot = 2
Q. B. on -I, -II, -III & Binomial
[8]
Q.70
2
B 2
(D) 1 tan 1 tan = 2
2
The value of , 3 cosec 20° sec 20° is : 2 sin 20
(A) 2 Q.71
A 2
A B (C) 1 sec 1 cos ec = 2
(B) sin 40
4 sin 20
(C) 4
(D) sin 40
If in a ABC, cosA·cosB + sinA sinB sin2C = 1 then, the statement which is incorrect, is (A) ABC is isosceles but not right angled (B) ABC is acute angled (C) ABC is right angled
(D) least angle of the triangle is
4
2 2 0.25sincosx2x 4 + 1 = 0, is :
tan x 4
Q.72
The set of values of x satisfying the equation, 2 (A) an empty set (B) a singleton (C) a set containing two values (D) an infinite set
Q.73
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 [ where is the area of the triangle ABC ]
Q.74
If in a 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,
cos(r x) = 0
lying in (0, p) is :
r 1
(A) 2
Q.76
If = 3 and sin = 1
(A)
Q.78
(B) 3
2
a b
a 2
a b2
If in a triangle ABC (A)
8
Bansal Classes
(C) a + b
(D) none
10 10 10 10 + tan 67 – cot 67 – tan7 is : 2 2 2 2
(A) a rational number (B) irrational number Q.79
(D) more than 5
. The value of the expression , a cosec b sec is
(B) 2 a 2 b 2
2
The value of cot 7
(C) 5
(C) 2(3 + 2 3 )
(D) 2 (3 – 3 )
2 cos A cos B 2 cos C a b then the value of the angle A is : a b c b c ca
(B)
4
(C)
3
(D)
Q. B. on -I, -II, -III & Binomial
2
[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 = 5800 then which one of the following is true
Q.82
A (A) 2 sin 1 sin A 1 sin A 2
A (B) 2 sin 1 sin A 1 sin A 2
A (C) 2 sin 1 sin A 1 sin A 2
A (D) 2 sin 2
With usual notations in a triangle ABC, if r1 = 2r2 = 2r3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a
1 sin A 1 sin A
(D) 2a = 3b
1 x2 x Q.83 If tan = 2 and tan = (x 0, 1), where 0 < , < , then tan 2 2 2 x 2x 1 x x 1 ( + ) has the value equal to : (A) 1
Q.84
Q.86
(C) 2
(D)
If r1, r2, r3 be the radii of excircles of the triangle ABC, then (A)
Q.85
(B) – 1
A
cot 2
(B)
A
B
cot 2 cot 2
(C)
A
tan 2
r1 r1r2
3 4
is equal to :
(D)
A
tan 2
Minimum value of 8cos2x + 18sec2x x R wherever it is defined, is : (A) 24 (B) 25 (C) 26 (D) 18 a2 b2 c 2 A B C In a ABC sin A sin B sin C . sin sin sin simplifies to 2 2 2
(A) 2
(B)
(C)
2
(D)
4
where is the area of the triangle Q.87
If is eliminated from the equations x = a cos( – ) and y = b cos ( – ) then
x2
y2
2xy cos( ) is equal to ab a b (A) cos2 ( – ) (B) sin2 ( – ) 2
Q.88
2
(C) sec2 ( – )
The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is (A) x = n (B) n ± (C) x = 2n 3 where n I
Bansal Classes
(D) cosec2 ( – )
(D) x =
Q. B. on -I, -II, -III & Binomial
n 3
[10]
Q.89
If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (logbc)3 + (logca)3 is (A) an odd prime (B) an even prime (C) an odd composite (D) an irrational number
Q.90
If the arcs of the same length in two circles S1 and S2 subtend angles 75° and 120° respectively at the S centre. The ratio 1 is equal to S2 (A)
Q.91
Q.92
1 5
81 16
(B)
(C)
Number of principal solution of the equation tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5
The expression
(D)
(C) 7
25 64
(D) more than 7
tan 2 20 sin 2 20 simplifies to tan 2 20 ·sin 2 20
(A) a rational which is not integral (C) a natural which is prime Q.93
64 25
(B) a surd (D) a natural which is not composite
The value of x that satisfies the relation x = 1 – x + x2 – x3 + x4 – x5 + ......... (A) 2 cos36° (B) 2 cos144° (C) 2 sin18°
(D) none
Select the correct alternatives : (More than one are correct) Q.94
If sin = sin then sin (A) sin
Q.95
3
= 3 3
3
3
(C) sin
3
3
(D) sin
Choose the INCORRECT statement(s).
Q.96
3
(B) sin
(A
1 1 1 1 sin 82 . cos 37 and sin 127 . sin 97 have the same value. 2 2 2 2
(B)
If tan A =
(C) (D)
The sign of the product sin 2 . sin 3 . sin 5 is positive. There exists a value of between 0 & 2 which satisfies the equation ; sin4 – sin2 – 1 = 0.
3 4 3
& tan B =
3 4 3
then tan (A B) must be irrational.
Which of the following functions have the maximum value unity ? (A) sin2 x cos2 x
(C)
sin 2x cos 2x
Bansal Classes
2
(B)
(D)
sin 2x cos 2x 2
6 1 sin x 1 cos x 3 5 2
Q. B. on -I, -II, -III & Binomial
[11]
Q.97
If the sides of a right angled triangle are { cos2cos2 + 2cos( + )} and
{sin2sin2 + 2sin( + )}, then the length of the hypotenuse is : (A) 2[1+cos()] (B) 2[1 cos()] (C) 4 cos2 Q.98
Q.99
2
(D) 4sin2
2
An extreme value of 1 + 4 sin + 3 cos is : (A) 3 (B) 4 (C) 5
The sines of two angles of a triangle are equal to (A) 245/1313
(B) 255/1313
(D) 6
5 99 & . The cosine of the third angle is : 13 101
(C) 735/1313
4 Q.100 It is known that sin = & 0 < < then the value of 5
(A) independent of for all in (0, /2) (C)
3 (7 24 cot ) for tan < 0 15
5
(B)
3
(D) 765/1313
3 sin ( ) cos2 cos ( ) 6 is: sin
for tan > 0
(D) none
Q.101 If x = sec tan & y = cosec + cot then : (A) x =
y 1 y 1
(B) y =
1 x 1 x
(C) x =
y 1 y 1
(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 If sin t + cos t = (A) 1
(B) –5
7 5
(D) –4
(C) 2
(D)
(C)
1 t then tan is equal to : 5 2
(*B) –
1 3
1 6
BINOMIAL There are 39 questions in this question bank. Q.104 Given that the term of the expansion (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 binomial (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 is 1/6 , then n = (A) 6 (B) 9 (C) 12 (D) 15
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[12]
x
n
Q.106 If the coefficients of x7 & x8 in the expansion of 2 are equal , then the value of n is : 3 (A) 15
(B) 45
(C) 55
(D) 56
1 1 1 Q.107 The coefficient of x49 in the expansion of (x – 1) x x 2 ..... x 49 is equal to 2 2 2 1 (A) – 2 1 50 2
(B) + ve coefficient of x
(C) – ve coefficient of x
1 (D) – 2 1 49 2
Q.108 The last digit of (3P + 2) is : (A) 1 (B) 2 4n where P = 3 and n N
(C) 4
(D) 5
1
n
Q.109 The sum of the binomial coefficients of 2 x is equal to 256 . The constant term in the x expansion is (A) 1120
Q.110 The coefficient of (A)
405 256
(B) 2110
x4
3 x in 2 2 x
(B)
(C) 1210
(D) none
10
is :
504 259
(C)
450 263
(D)
Q.111 The remainder, when (1523 + 2323) is divided by 19, is (A) 4 (B) 15 (C) 0
405 512
(D) 18
Q.112 Let (7 4 3 ) n = p + when n and p are positive integers and (0, 1) then (1 – ) (p + ) is (A) rational which is not an integer (B) a prime (C) a composite (D) none of these Q.113 If (11)27 + (21)27 when divided by 16 leaves the remainder (A) 0 (B) 1 (C) 2
(D) 14
Q.114 Last three digits of the number N = 7100 – 3100 are (A) 100 (B) 300 (C) 500
(D) 000
Q.115 The last two digits of the number 3400 are : (A) 81 (B) 43 (C) 29
(D) 01
Q.116 If (1 + x + x²)25 = a0 + a1x + a2x² + ..... + a50 . x50 then a0 + a2 + a4 + ..... + a50 is : (A) even (B) odd & of the form 3n (C) odd & of the form (3n 1) (D) odd & of the form (3n + 1)
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[13]
Q.117 The sum of the 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) none of 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 The expression
7 7 1 4x 1 1 4 x 1 is a polynomial in x of degree 2 2 4x 1
1
(A) 7
(B) 5
(C) 4
Q.120 If the second term of the expansion a 1/13
(A) 4
(D) 3
n
a is 14a5/2 then the value of a 1
(B) 3
(C) 12
n n
C3 is : C2
(D) 6
Q.121 If (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + x3 + ...... + xn) m
a0 + a1x + a2x2 + a3x3 + ...... + amxm then (A) n!
a r has the value equal to
r0
(B) (n + 1) !
(C) (n – 1)!
(D) none
Q.122 The value of 4 {nC1 + 4 . nC2 + 42 . nC3 + ...... + 4n 1} is : (A) 0 (B) 5n + 1 (C) 5n
(D) 5n 1
Q.123 If n be a positive integer such that n 3, then the value of the sum to n of the series 1.n
n 1 1!
(A) 0
(n 1) +
n 1 n 2 2!
n 1 n 2 n 3
(n 2) –
(B) 1
3!
(C) – 1
(n 3) + ...... is : (D) none of these
Q.124 In the expansion of (1 + x)43 if the coefficients of the (2r + 1)th and the (r + 2)th are equal, the value of r is : (A) 12 (B) 13 (C) 14 (D) 15
Q.125 The positive value of a so that the coefficient of x5 is equal to that of x15 in the expansion of x 2 (A)
1 2 3
1
(B)
(C) 1
3
x 1 x1 Q.126 In the expansion of 2/ 3 1/ 3 x x 1 x x1/ 2
(A)
10C
0
(B)
10C
7
a x3
10
is
(D) 2 3
10
, the term which does not contain x is : (C)
10C
(D) none
4 8
Q.127
If the 6th term in the expansion of the binomial
1 2 x 8/ 3 x log10 x is 5600, then x equals to
(A) 5
(C) 10
Bansal Classes
(B) 8
(D) 100
Q. B. on -I, -II, -III & Binomial
[14]
Q.128 Co-efficient of t in the expansion of, ( + p)m 1 + ( + p)m 2 ( + q) + ( + p)m 3 ( + q)2 + ...... ( + q)m 1 where q and p q is : m
(A)
Ct pt qt
m
(B)
pq m
(C)
Ct pt qt
pq m
(D)
pq
Ct pm t qm t
Ct pm t qm t
pq
Q.129 (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + ...... + x100) when written in the ascending power of x then the 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, f2 f + pf p is
(A) a natural number (C) a prime number
(B) a negative integer (D) are irrational number
Q.131 Number of rational in the expansion of (A) 25
(B) 26
2 43
100
is :
(C) 27
(D) 28 10
cos is Q.132 The greatest value of the term independent of x in the expansion of x sin x 10
(A)
10C
5
(B) 25
(C)
25
·
10C
5
(D)
C5
25
Q.133 If (1 + x – 3x2)2145 = a0 + a1x + a2x2 + ......... then a0 – a1 + a2 – a3 + ..... ends with (A) 1 (B) 3 (C) 7 (D) 9 9
4x 2 3 6 Q.134 Coefficient of x in the binomial expansion is 2 x 3 (A) 2438 (B) 2688 (C) 2868 1 Q.135 The term independent of ' x ' in the expansion of 9 x 3 x
(D) none 18
, x > 0 , is times the corresponding
binomial co-efficient . Then ' ' is : (A) 3
(B)
1 3
(C)
1 3
(D) 1
Q.136 The expression [x + (x31)1/2]5 + [x (x31)1/2]5 is a polynomial of degree : (A) 5 (B) 6 (C) 7 (D) 8 [JEE’92, 6 + 2]
Bansal Classes
Q. B. on -I, -II, -III & Binomial
[15]
Q.137 Given (1 – 2x + 5x2 – 10x3) (1 + x)n = 1 + a1x + a2x2 + .... and that a12 = 2a2 then the value of n is (A) 6 (B) 2 (C) 5 (D) 3 Q.138 The sum of the series aC0 + (a + b)C1 + (a + 2b)C2 + ..... + (a + nb)Cn is where Cr's denotes combinatorial coefficient in the expansion of (1 + x)n, n N (A) (a + 2nb)2n (B) (2a + nb)2n (C) (a +nb)2n – 1 (D) (2a + nb)2n – 1 Q.139 The coefficient of the middle term in the binomial expansion in powers of x of (1 + x)4 and of (1 – x)6 is the same if equals 5 3
(A) –
10 3
(B)
(C) –
3 10
(D)
3 5
Q.140 (2n + 1) (2n + 3) (2n + 5) ....... (4n 1) is equal to : (A)
( 4 n) ! 2 . (2n) ! (2n) ! n
n
Q.141 If Sn = (A)
r 0
1 and Tn = n Cr
n 2
( 4 n) ! n ! 2 . (2n) ! (2n) !
(B)
n
n
r
nC r 0
(B)
n 1 2
r
(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 The coefficient 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 Classes
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.10 C Q.11 D Q.12 Q.15 C Q.16 A Q.17 A Q.18 B Q.19 Q.22 C Q.23 A Q.24 D Q.25 C Q.26 Q.29 D Q.30 C Q.31 C Q.32 B Q.33 Q.36 C Q.37 A Q.38 D Q.39 C Q.40 Q.43 B Q.44 A Q.45 A Q.46 A Q.47 Q.50 D Q.51 D Q.52 B Q.53 C Q.54 Q.57 C Q.58 C Q.59 B Q.60 B Q.61 Q.64 D Q.65 A Q.66 B Q.67 C Q.68 Q.71 C Q.72 A Q.73 B Q.74 A Q.75 Q.79 D Q.80 B Q.81 C Q.82 C Q.83 Q.86 B Q.87 B Q.88 D Q.89 A Q.90 Q.93 C Select the correct alternatives : (More than one are 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 Classes
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.14 Q.21 Q.28 Q.35 Q.42 Q.49 Q.56 Q.63 Q.70 Q.78 Q.85 Q.92
Q.98
BD
Q.99
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]
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