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TARGET IIT JEE
MATHEMATICS
COMPOUND ANGLES TRIGONOMETRIC EQUATIONS & INEQUA INEQUATIONS TIONS SOLUTION OF TRIANGLES SEQUENCES & PROGRESSION
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Question bank on Compound angles, Trigonometric eq n and ineqn, Solutions of Triangle, Sequence & Progression There are 132 questions in this question bank. Select the correct alternative : (Only one is correct) Q.1
Q.2
If x + y = 3 – cos4θ
and x – y = 4 sin2θ 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
(A)
Q.4
1 − n cos 2 A
sin A
B 2
=
3 2
(C) in H.P.
Given a2 + 2a + cosec 2 x
(B)
(n − 1) cos A
(C)
sin A
s2 3 3
The exact value of cos
(D)
( n − 1) cos A
sin A ( n + 1) cos A
x
∈I 2 (D) a , x are finite but not possible to find (B) a = –1 ;
(B) A =
2π
s2
cos ec
28 (B) 1/2
3π 28 C 2
6π 28
cos ec
+ (a − b)2 cos2
(B) b (c + a)
π
2
(A) sin x cos x
+ cos
9π 28
s2
(D) None
3
+ cos
18π 28
cos ec
(C) 1
) . cos ( 32π + x) − sin3 ( 72π cos ( x − π2 ) . tan ( 32π + x)
tan ( x −
(C) A >
2
In any triangle ABC, (a + b)2 sin2 (A) c (a + b)
Q.8
sin A
F G π (a + x)J I H 2 K = 0 then, which of the following holds good?
∈I
(A) – 1/2 Q.7
None
If A is the area and 2s the sum of the 3 sides of a triangle, then : (A) A ≤
Q.6
(D)
then tan(A + B) equals
2 (C) a ∈ R ; x ∈φ Q.5
c then a, b, c are :
Quest
(1 − n ) cos A
(A) a = 1 ;
2
+ a cos 2
(B) in G.P.
n sin A cos A
If tanB =
A
(B) − sin2 x
− x)
C 2
27 π 28
is equal to
(D) 0 =
(C) a (b + c)
(D) c2
when simplified reduces to : (C) − sinx 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) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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(A) 5 Q.11
(B) 7 2 sin θ
The value of
sin θ − cos θ
(C) 9
sin θ + cos θ
−
tan 2 θ − 1
(D) none
for all permissible vlaues of θ
(A) is less than – 1
(B) is greater than 1
(C) lies between – 1 and 1 including both
(D) lies between –
Q.12
sin 3θ = 4 sin θ sin 2θ sin 4θ in 0 ≤ θ ≤ π 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
C 2
2 and
2
has the value
1
and l (CD) = 6, then
3
1 1 + has the value equal to a b (A) Q.14
1
(B)
9
1
(C)
12
Quest
RS π , 5π , 19π , 23π UV T12 12 12 12 W R 5π , 13π , 19π U (C) S T 12 12 12 VW
Q.17
(D)
1
(B) 2
2
If cos (α + β) = 0 then sin (α + 2β) = (A) sin α (B) − sin α
ST 12 , 12
,
,
12
12
tan A tan B
VW
has the value equal to 1
(C) − 2
(D) −
(C) cos β
(D) − cos β
2
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
π 7 π 17 π 23π , , , 12 12 12 12 R π 7 π 19 π 23π U
(B)
If the median of a triangle ABC through A is perpendicular to AB then (A)
Q.16
(D) none
6
The set of angles btween 0 & 2π satisfying the equation 4 cos2 θ − 2 2 cos θ − 1 = 0 is (A)
Q.15
1
abc
(B)
R2
sin 3 θ − cos 3 θ sin θ − cos θ
(A) θ ∈ 0 ,
−
π
2
abc 4R 2
cos θ 2 1 + cot θ
(C)
4abc
(D)
R2
abc 2R 2
− 2 tan θ cot θ = − 1 if : π , π 2
(B) θ ∈
(C) θ ∈ π ,
3π
2
3π , 2π 2
(D) θ ∈
Q.19
With usual notations in a triangle ABC, ( I I1 ) · ( I I 2 ) · ( I I 3 ) 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 + tan 2 x) = 0 where (0 < k < 1) . Then the measure of angle A is :
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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(A) π /3 Q.21
(B) 2π /3
If cos α =
2 cos β − 1 2 − cos β
(A) 2 Q.22
2π 3
,
π
,
π
(B)
4 12
k − 1
(B)
k + 1
(D)
3
2
3
,
6
C
π 3π π
(C)
2
C
,
8
,
8
A
(D)
π 3π π 2
,
10
,
5
B
= k sin , then tan tan = 2 2 2 2
k + 1
(C)
k − 1
= 1 −
3 sin θ − 1
k
(D)
k + 1
4 3 sin θ − 1
k + 1 k
has :
Quest (C) two roots
(D) infinite roots
3 1 1 1 1 1 1 + + + = K R where K has the value r1 r2 r2 r3 r3 r1 a 2 b2 c2
If
5π 2
(B) 16
x
(B) cot
2
(D) 128
1 − sin x + 1 + sin x
32
2
π 10
x
is
(D) –tan
2
x 2
4 π = z sin θ + then : 3 3
cos
(C) xyz + x + y + z = 1 (D) none
a cos A + b cos B + c cos C
is equal to :
a+b+c
(B)
R
1
(C) tan
(B) xy + yz + zx = 0
r
The value of cos
x
1 − sin x − 1 + sin x
2π
In a ∆ ABC, the value of
(A)
(C) 64
< x < 3π , then the value of the expression
If x sin θ = y sin θ +
(A)
Q.29
,
(B) one root
(A) x + y + z = 0
Q.28
has the value equal to, where(0 < α < π and 0 < β < π)
With usual notation in a ∆ ABC
(A) –cot
Q.27
2
(C) 3
4
The equation, sin2 θ −
equal to : (A) 1
Q.26
β
π π π
(A) no root Q.25
2
cot
2
If A + B + C = π & sin A +
(A) Q.24
(B)
α
(D) 3π /4
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
then tan
(C) π /2
2π 10
(B)
R
(C)
2r
cos 1 16
4π 10
cos
8π 10
cos
R r
16 π 10
(C)
(D)
2r R
is :
cos ( π / 10 ) 16
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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(D) −
10 + 2 5 64
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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
where k has the value equal to:
(C) 1/4
(D) 4
Q.31
If a cos3 α + 3a cos α sin2 α = m and a sin3 α + 3a cos2 α sin α = 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°
[JEE ’94, 2] (D) none
(C) 147°
(B) tan 3x
(C) 3 tan 3x
(D)
3 − 9 tan 2 x 3 tan x − tan 3 x
Quest
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 + cot 76° cot 16° cot 76°
+
(A) cot 44º
is :
cot 16°
(B) tan44º
(C) tan2º
(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 2
R r2
(D)
1 2
r R2
Q.37
The number of solutions of tan (5π cos θ) = cot (5 π sin θ) for θ in (0, 2π) is : (A) 28 (B) 14 (C) 4 (D) 2
Q.38
If A = 3400 then 2 sin
Q.39
2 2 b −c
The value of cot x + cot (60º + x) + cot (120º + x) is equal to : (A) cot 3x
Q.36
a bc
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 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
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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Q.40
The value of cosec
π 18
–
3 sec
π 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 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
π 3
&
π
(B)
6
π 8
&
3π 8
π 4
&
π
(D)
4
π 5
&
3π 10
Quest
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 (A) 1/4
a
f
+
b
+
g
c
h
= λ
abc f gh
then the value of λ is :
(B) 1/2
(C) 1
∑
Q.45
(C)
In ∆ ABC, the minimum value of
cot
2
A 2
. cot
2
(D) 2
B
2 is
A
∏ cot 2 2
(A) 1 Q.46
Q.48
(D) non existent
(B)
1
(C) – 3
3
(D) –
The general solution of sin x + sin 5x = sin 2x + sin 4x is : (A) 2nπ (B) nπ (C) nπ /3 where n ∈ I
1 3
(D) 2 nπ /3
The product of the distances of the incentre from the angular points of a ∆ ABC is : (A) 4 R2 r
Q.49
(C) 3
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.47
(B) 2
(B) 4 Rr2
(C)
2 Number of roots of the equation cos x +
[−π, π] is (A) 2
(B) 4
(a
3 +1 2
b c) R
(D)
s
sin x −
3 4
R
− 1 = 0 which lie in the interval
(C) 6
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(a bc )s
(D) 8
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sec 8θ − 1 Q.50
sec 4θ − 1 is equal to (A) tan 2θ cot 8θ
Q.51
(B) tan 8θ tan 2θ
In a ∆ABC if b = a (A) 150
(C) cot 8θ cot 2θ
(D) tan 8θ cot 2θ
3 − 1 and ∠C = 300 then the measure of the angle A is
(B) 450
(C) 750
(D) 1050
Q.52
Number of values of θ ∈ [ 0 , 2 π] satisfying the equation cotx – cosx = 1 – cotx. cosx (A) 1 (B) 2 (C) 3 (D) 4
Q.53
The exact value of cos 273º + 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
Q.57
2− p
a p 22 − p −
(
)
3π
(
3π
4p − 2 5
p
cos(2 π − α )
ap
(D) r2 = 3r1
) + cos α − π sin(π − α) + cos (π + α) sin α − π when
2
(C) − 1
2
(D) none
3 π − α The expression [1 − sin (3π − α) + cos (3π + α)] 1 − sin 2 reduces to : (A) sin 2α (B) − sin 2α (C) 1 − sin 2α
+
5 π 2
cos
when simplified
− α
(D) 1 + sin 2α
If ‘O’ is the circumcentre of the ∆ ABC and R1, R2 and R3 are the radii of the circumcircles of triangles
(A)
a bc
(B)
2 R3
The maximum value of (A) 25
Q.59
3
p
(C) r2 = 2r1
tan 2 − α cos 2 − α
OBC, OCA and OAB respectively then
Q.58
5
Quest
The expression,
(B) r3 = 2r1
simplified reduces to : (A) zero (B) 1 Q.56
(D) 1
a R1
R3
+
b R2
+
(C)
a bc
c
has the value equal to:
R3 4∆ R
2
(D)
∆ 4R 2
( 7 cosθ + 24 sinθ ) × ( 7 sinθ – 24 cos θ ) for every θ ∈ R . (B) 625
(C)
625 2
(D)
625 4
4 sin50 sin550 sin650 has the values equal to (A)
3 +1 2 2
(B)
3 −1 2 2
(C)
3 −1 2
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(D)
3d 3 − 1i 2 2
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Q.60
If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then abc
is equal to
x yz
(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 =
nπ 2
, satisfies the equation sin
x 2
(A) n = −1, 0, 3, 5 (C) n = 0, 2, 4 Q.63
F
The value of GH 1 + cos (A)
9 16
x
x
2
2
− cos = 1 − sinx & the inequality
−
π 2
≤
3π 4
, then:
(B) n = 1, 2, 4, 5 (D) n = −1, 1, 3, 5
π I
F G1 + cos 3π J I F G 1 + cos 5π J I F G1 + cos 7π J I J is 9 K H 9 K H 9 K H 9 K
Quest (B)
10
(C)
16
12
16
(D)
5
16
Q.64
The number of all possible triplets (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 semicircle is (A)
2∆
(B)
2∆
(C)
a+b a + b−c Where ∆ is the area of the triangle ABC.
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 C k+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, (A) right angled
Q.68
Q.69
cos A a
=
cos B b
=
cos C c
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 B (A) 1 + tan 1 + tan = 2 2 2
A B (B) 1 + cot 1 + cot = 2 2 2 Quest Tutorials
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Q.70
B 1 + cos ec = 2 2
The value of ,
3 cosec 20° − sec 20°
is :
2 sin 20°
(A) 2 Q.71
A B (D) 1 − tan 1 − tan = 2 2 2
A (C) 1 + sec 2
(B) sin 40°
(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
Q.72
4 sin 20°
(C) 4
(D) least angle of the triangle is
The set of values of x satisfying the equation, 2 (A) an empty set (C) a set containing two values
(
tan x − π 4
) − 2 (0.25)
sin
2
(x − π4 )
cos 2 x
π 4
+ 1 = 0, is :
(B) a singleton (D) an infinite set
Quest
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 θ = 3 α and sin θ = 1
(A)
Q.78
(B) 3
2
a +b
a a 2 + b2
1
0
2
+ tan 67
1 2
(C) a + b
0
– cot 67
1 2
0
– tan7
(A) a rational number (B) irrational number
Q.79
If in a triangle ABC (A)
π 8
2 cos A a
(B)
π 4
+
(D) more than 5
. The value of the expression , a cosec α − b sec α is
(B) 2 a 2 + b 2
2
The value of cot 7
(C) 5
cos B b
+
2 cos C c
1 2
(D) none
0
is :
(C) 2(3 + 2 3 )
=
a bc
(C)
+
b ca
then the value of the angle A is :
π 3
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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(D) 2 (3 – 3 )
(D)
π 2
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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
A = − 1 + sin A + 1 − sin 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
If
tan α =
Q.86
x2 − x 2
and tan β =
x − x +1 (α + β) has the value equal to :
1 2x 2 − 2 x + 1
π 2
, then tan
Quest (B) – 1
∑
(D) 2a = 3b
(x ≠ 0, 1), where 0 < α, β <
(C) 2
(D)
If r1, r2, r3 be the radii of excircles of the triangle ABC, then
(A) Q.85
1 + sin A + 1 − sin A
With usual notations in a triangle ABC, if r1 = 2r2 = 2r3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a
(A) 1
Q.84
A = 2
A
cot
(B)
2
∑
cot
A 2
cot
B 2
(C)
∑
tan
∑ r1 ∑ r1r2
A 2
3 4
is equal to :
(D)
∏ tan
A 2
Minimum value of 8cos2x + 18sec2x ∀ x ∈ R wherever it is defined, is : (A) 24 (B) 25 (C) 26 (D) 18
In a ∆ABC
2 2 a 2 b c . sin A sin B sin C simplifies to + + sin A sin B sin C 2 2 2
(A) 2∆
(B) ∆
(C)
∆ 2
(D)
∆ 4
where ∆ is the area of the triangle Q.87
If θ is eliminated from the equations x = a cos( θ – α) and y = b cos (θ – β) then x2
+
y2
−
2xy
2 2 ab a b (A) cos2 ( α – β)
Q.88
cos(α − β) is equal to (B) sin2 (α – β)
(C) sec2 ( α – β)
(D) cosec2 (α – β)
The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is (A) x = nπ
(B) nπ ±
π 3
(C) x = 2nπ
(D) x =
nπ 3
where n ∈ I Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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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 centre. The ratio
(A) Q.91
S1 S2
is equal to
1
81
(B)
5
16
Number of principal solution of the equation tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5 2
Q.92
(D)
25
(C) 7
25 64
(D) more than 7
2
tan 20° − sin 20°
Quest
The expression
tan 2 20° · sin 2 20°
simplifies to
(A) a rational which is not integral (C) a natural which is prime Q.93
64
(C)
(B) a surd (D) a natural which is not composite
The value of x that satisfies the relation x = 1 – x + x 2 – 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 θ = sin α then sin (A) sin
Q.95
Q.96
θ 3
=
π α − 3 3
α
π α + 3 3
(B) sin
3
(C) sin
π α + 3 3
(D) − sin
Choose the INCORRECT statement(s). 1
°
°
°
°
(B)
If tan A =
(C) (D)
The sign of the product sin 2 . sin 3 . sin 5 is positive. There exists a value of θ between 0 & 2 π which satisfies the equation ; sin4 θ – sin2 θ – 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 of the following functions have the maximum value unity ? (A) sin2 x − cos2 x
(C) −
sin 2x
−
cos 2x
2
(B)
(D)
sin 2x
−
cos 2x
2
6 1
sin x + 5 2
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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1 3
cos x
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Q.97
If the sides of a right angled triangle are {cos2α + cos2β + 2cos(α + β)} and {sin2α + sin2β + 2sin(α + β)}, then the length of the hypotenuse is : (A) 2[1+cos(α − β)] (B) 2[1 − cos(α + β)] (C) 4 cos2
Q.98
Q.99
α−β
(D) 4sin2
2
α+β 2
An extreme value of 1 + 4 sin θ + 3 cos θ is : (A) − 3 (B) − 4 (C) 5
The sines of two angles of a triangle are equal to (A) 245/1313
(B) 255/1313
Q.100 It is known that sin β =
4 5
13
&
99 101
. The cosine of the third angle is :
(C) 735/1313
5
(D) 765/1313
2 cos(α + β ) 3 sin (α + β) − cos π 6
& 0 < β < π then the value of
(A) independent of α for all β in (0, π /2)
(C)
5
(D) 6
sin α
is:
for tan β > 0
Quest
3 (7 + 24 cot α ) 15
(B)
for tan β < 0
3
(D) none
Q.101 If x = sec φ − tan φ & y = cosec φ + cot φ then : (A) x =
y
+
1
y
−
1
(B) y =
1+ x
(C) x =
1− x
y
−
1
y
+
1
(D) xy + x − y + 1 = 0
Q.102 If 2 cosθ + sinθ = 1, then the value of 4 cosθ + 3sinθ is equal to (A) 3 Q.103 If sin t + 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
SEQUENCE & PROGRESSION Select the correct alternative : (Only one is correct) Q.104 If a, b, c be in A.P., b, c, d in G.P. & c, d, e in H.P., then a, c, e will 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., 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 numbers a , b, c are in 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
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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Q.107 The sum
∞
1
r=2
r2 − 1
∑
is equal to :
(A) 1
(B) 3/4
(C) 4/3
(D) none
Q.108 In a potato race , 8 potatoes are placed 6 metres apart on a straight line, the first being 6 metres from the basket which is also placed in the same line. A contestant starts from the basket and puts one 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 of the cubic x3 – 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 lies an odd number of stones placed at intervals of 10 m. These 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
Quest
Q.111 If log
( 5 . 2 x +1)
2 ; log
4
( 21− x +1)
and 1 are in Harmonical Progression then
(A) x is a positive real (C) x is rational which is not integral
(B) x is a negative real (D) x is an integer
Q.112 If a, b, c are in G.P., then the equations, ax2 + 2bx + c = 0 & dx2 + 2ex + f = 0 have a common root, if d a
,
e f
,
b c
are in :
(A) A.P.
(B) G.P.
(C) H.P.
(D) none
Q.113 If the sum of the roots of the quadratic equation, ax2 + bx + c = 0 is equal to sum of the squares of their a
reciprocals, then
c
b
,
a
(A) A.P.
c
,
are in :
b
(B) G.P.
(C) H.P.
Q.114 If for an A.P. a1 , a2 , a3 ,.... , a n ,.... a1 + a3 + a5 = – 12 and a1 a2 a3 = 8 then the value of a2 + a4 + a6 equals (A) – 12 (B) – 16
Q
. 1 1
5
G
a
i v e n
G
. P .
f o u r
,
p o
t h e n
c o
s it i v e
w
h i c h
m
o n
( A
)
t h e
m
( B
)
c o m
m
o n
r a t i o
( C
)
c o m
m
o n
d
( D
)
c o m
m
o
n
o
n u
f
f
b e r
t h e
r a t i o
o
m
o
G
in
A
f o l lo w
f
G
. P .
i n g
. P . i s
i s
. P .
I f
5
(D) none
(C) – 18 (D) – 21 [ Apex : Q.62 of Test - 1 Scr. 2004 ] ,
6
h o
l d s ?
,
9
a n d
1
5
a r e
a d d e d
r e s p e c t i v e l y
t o
t h e s e
n u m
b e r s
,
w
e
g e t
3 / 2
2 / 3
i f f e r e n c e
o f
t h e
A
. P .
i s
3 / 2
d i f f e r e n c e
o f
t h e
A
. P . i s
2 / 3
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
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Q.116 Consider an A.P. with first term 'a' and the common difference d. Let Sk denote the sum of the first K terms. Let
Skx Sx
is independent of x, then
(A) a = d/2
(B) a = d
(C) a = 2d
(D) none
Q.117 Concentric circles of radii 1, 2, 3......100 cms are drawn. 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) 1000π (B) 5050π (C) 4950π (D) 5151π Q.118 Consider
the A.P. a1 , a2 ,..... , a n ,.... the G.P. b1 , b2 ,....., b n ,..... 9
such that a1 = b1 = 1 ; a 9 = 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 ]
Quest
Q.119 For an increasing A.P. a1, 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 G.P. : g1, g2, g3, ...... g n ....... such that g 1 + g2 + g3 = 13 and g1 + g 2 + g 3 =91
then which of the following does not hold? (A) The greatest term of the G.P. is 9. (C) g1 = 1
(B) 3g4 = g3 (D) g2 = 3
Q.121 If p , q, r in H.P. and p & r be different having same sign then the roots of the equation px2 + qx + r = 0 are (A) real & equal (B) real & distinct (C) irrational (D) imaginary Q.122 The point A(x1, y1) ; B(x2, y2) and C(x3, y3) lie on the parabola y = 3x2. 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 +
(C) 3 –
2
2
(D) 3 – 2 2
Q.123 If a, b, c are in A.P., then a2 (b + c) + b2 (c + a) + c2 (a + b) is equal to : (A)
(a + b + c )
Q.124 If Sn =
3
(B)
8
1 3
1
(A) 1/2
+
1+ 2 3
1
+2
3
+...... +
2 9
(a + b + c) 3
(C)
1 + 2 + 3 + ...... + n 3
1
(B) 1
3
3
+ 2 + 3 + ...... + n
3
3 10
(a + b + c) 3
(D)
1 9
(a + b + c)3
, 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. Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
JEEMAIN.GURU
[14]
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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
+ ................∞ is equal to
1
1
(C)
3
(D) 1
2
Q.127 Consider an A.P. a1 , a 2 , a 3 ,......... such that a 3 + a5 + a8 = 11 and a 4 + a2 = –2, then the value of a1 + a6 + a7 is (A) –8 (B) 5 (C) 7 (D) 9 Q.128 A circle of radius r is inscribed in a square. The mid points of sides 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, t hen the radius of the circle inscribed in the nth square is
1−n 2 (A) 2 r
3−3n 2 r (B) 2
−n 2 (C) 2 r
− 5−3n 2 r (D) 2
Quest ∞
Q.129 The sum
∑
2
k =1
k + 2
3k
equal to
(A) 12
(B) 8
∞
Q.130 The sum 5
(C) 6
(D) 4
(C) 320
(D) 388
2n +2
∑ 4n −2 is equal to n =1
(A) 1372
(B) 440
Q.131 Given am+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? (B) a n = 2 n A n B n
(A) a m = AB
A B
(C) a m = a1
m 2 − m − n − mn
A B
m+ n
(D) a n = a1
2
Q.132 The sum of the infinite series, 1 − (A)
1 2
(B)
25 24
2
2
5
2
+
3 5
2
−
4
2
5
3
+
5
2
5
4
(C)
−
6
2
5
5
m 2 −m − n − n 2 m+n
+........ is :
25 54
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
JEEMAIN.GURU
(D)
125 252
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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
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 B D A A B B B A C B C D
Quest
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
Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439
JEEMAIN.GURU
B A B A C C
Q.108 Q.113 Q.118 Q.123 Q.128
C C B B A
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