15 - TRIGONOMETRY
Page 1
( Answers at the end of all questions )
2
(b) c = a + b
1
(d) c + a
[ AIEEE 2005 ]
y 2 2 is equal to = α, then 4x - 4xy cos α + y 2 ( c ) 4 sin
2
α
(
ra
(b) 4
) - 4 sin
2
α
( c ) Arithmetic-Geometric Progression
[ AIEEE 2005 ]
( d ) H.P.
xa
( b ) A.P.
.e
α - β
If
3
(b)
130
= sin
3
(c)
130
6 65
a 2 cos 2 θ + b 2 sin 2 θ +
the maximum and minimum values of u
(a) 2(a
2
[ AIEEE 2005 ]
21 , then the value of 65
is
2
w w w
1
(c) a + b+c
Let α, β be such that π < α - β < 3π. If sin α + sin β = -
(a) -
(7)
[ AIEEE 2005 ]
If in triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in
cos
(6
(d) b = a + c
ce .c
(b) a + b
If cos - x - cos -
( a ) G.P.
(5)
(c) b = c
π . If r is the inradius and R is the circumradius of the 2 triangle ABC, then 2 ( r + R ) equals
( a ) 2 sin 2α
(4)
are the roots of the equation
In triangle ABC, let ∠ C =
(a) b + c
(3)
Q and tan 2
a ≠ 0, then
+ b x + c = 0,
(a) a = b + c
P If tan 2
om
ax
(2)
π . 2
In triangle PQR, ∠ R =
m
(1)
2
+ b )
(b) 2
The sides of a triangle are
a2 + b2
sin α,
(d) -
6 65
[ AIEEE 2004 ]
a 2 sin 2 θ + b 2 sin 2 θ , then difference between 2
is given by
(c) (a + b)
cos α and
2
(d) (a - b)
1 + sin α cos α
2
[ AIEEE 2004 ]
for some 0 < α <
π . 2
Then the greatest angle of the triangle is ( a ) 60°
( b ) 90°
( c ) 120°
( d ) 150°
[ AIEEE 2004 ]
15 - TRIGONOMETRY
Page 2
( Answers at the end of all questions )
A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of a river is 60° and when he retires 40 m away from the tree, the angle of elevation becomes 30°. The breadth of the river is
(9)
( b ) 30 m
( c ) 40 m
2 C If in a triangle a cos 2
( a ) in A. P.
( d ) 60 m
2 A + c cos 2
( b ) in G. P.
3b , then the sides a, b and c are 2
=
( c ) in H. P.
[ AIEEE 2004 ]
om
( a ) 20 m
( d ) satisfy a + b = c
ce .c
(8)
[ AIEEE 2003 ]
( 10 ) The sum of the radii of inscribed and circumscribed ci cles, for an n sided regular polygon of side a, is π ( b ) b cot n
(
)
ra
π ( a ) a cot 2n
2
π cot 2n
(d)
a π cot 4 2n [ AIEEE 2003 ]
m
3 3 th portion of a vertic l pole subtends an angle tan - 1 at a point in 4 5 the horizontal plane through it fo t and at a distance 40 m from the foot. The height of the vertical pole is
xa
( 11 ) The upper
( b ) 40 m
.e
( a ) 20 m
2
( c ) 60 m
( d ) 80 m
2
[ AIEEE 2003 ]
2
( 12 ) The value of cos α + cos ( α + 120° ) + cos ( α - 120° ) is 3 2
w w
(a)
b)
1 2
(c) 1
(d) 0
[ AIEEE 2003 ]
( 13 ) The trigonometric equation sin - x = 2 sin - a has a solution for 1 1 1 1 (a) lal< (b) lal ≥ (c) ( d ) all real values of a < lal < 2 2 2 2 [ AIEEE 2003 ] 1
w
1
θ - φ ( 14 ) If sin θ + sin φ = a and cos θ + cos φ = b, then the value of tan 2 (a)
(c)
a2 + b2 4 - a2 - b2 a2 + b2 4 + a2 + b2
(b)
(d)
is
4 - a2 - b2 a2 + b2 4 + a2 + b2 a2 + b2
[ AIEEE 2002 ]
15 - TRIGONOMETRY
Page 3
( Answers at the end of all questions )
1
2
(a)
(b) 3
( 16 ) The value of tan -
(a)
π 2
(b)
1
2π , then the value of x is 3
(c)
(d)
3
3 - 1 3 + 1
[ AIEEE 2002 ]
1 -1 1 -1 1 -1 + tan + tan + … + tan 3 7 13
π 4
(c)
2π 3
ce .c
1
om
( 15 ) If tan - ( x ) + 2 cot - ( x ) =
(d) 0
n
+ n + 1 1
is
[ AIEEE 2002 ]
(b)
3
a
3
3 - 1
(c)
a(3 + 2
m
(a) a
ra
( 17 ) The angles of elevation of the top of a tower ( A ) from the top ( B ) and bottom ( D ) at a building of height a are 30° and 45° resp ctively. If the tower and the building stand at the same level, then the height f the tower is
( 18 ) If cos ( α - β ) = 1 and co ( α + β ) =
xa
ordered pairs ( α, β ) = (b) 1
c) 2
1 , e
(d) a(
3
- 1)
[ AIEEE 2002 ]
- π ≤ α, β ≤ π, then the number of
(d) 4
[ IIT 2005 ]
.e
(a) 0
3)
w w
( 19 ) Which of the following is correct for triangle ABC having sides a, b, c opposite to the angles A, B, C respectively ( a ) a sin
A B - C = ( b - c ) cos 2 2
A B - C ( d ) sin = a cos 2 2
w
A B + C ( c ) ( b + c ) sin = a cos 2 2
A B + C ( b ) a sin = ( b + c ) cos 2 2
( 20 ) Three circles of unit radii are inscribed in an equilateral triangle touching the sides of the triangle as shown in the figure. Then, the area of the triangle is (a) 6 + 4
3
(c) 7 + 4
3
( b ) 12 + 8 3 7 (d) 4 + 3 2 [ IIT 2005 ]
[ IIT 2005 ]
15 - TRIGONOMETRY
Page 4
( Answers at the end of all questions ) If θ and φ are acute angles such that sin θ = lies in π π (a) , 3 2
π 2π (b) , 3 2
1 2
and cos θ =
2π 5π (c) , 3 3
1 , then θ and φ 3
5π (d) , π 6
[ IIT 2004 ]
om
( 21 )
( 22 ) For which value of x, sin [ cot - ( x + 1 ) ] = cos ( tan - x ) ? 1
(b) 0
(d) -
(c) 1
[ IIT 2004 ]
If a, b, c are the sides of a triangle such that a : b : then A : B : C is (a) 3 : 2 : 1
( 24 ) Value of
(b) 3 : 1 : 2
tan 2 α
x2 + x +
(c) 1 : 3 : 2
,
x2 + x
(b)
5 2
( c ) 2 tan α
:
3 : 2,
π 2
[ IIT 2004 ]
is always greater than or
( d ) sec α
[ IIT 2003 ]
xa
(a) 2
=
(d) 1 : 2 : 3
α ∈ 0,
x >
m
equal to
If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the largest side to the perimeter is equal to
.e
( 25 )
1 2
ra
( 23 )
1 2
ce .c
(a)
1
(a) 1:1 +
(b) 2:3
w w
3
( 26 ) The natural domain of
sin - 1 ( 2x ) +
1 1 (b) - , 4 4
3 :2 +
π 6
3
(d) 1:2 +
3
[ IIT 2003 ]
for all x ∈ R, is
1 1 (c) - , 2 2
w
1 1 (a) - , 4 2
(c)
1 1 (d) - , 2 4
[ IIT 2003 ]
3
( 7 ) The length of a longest interval in which the function 3 sin x - 4 sin x is increasing is (a)
( 28 )
π 3
(b)
π 2
(c)
3π 2
(d)
π
[ IIT 2002 ]
Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC ( R being the radius of the circumcircle ) ? ( a ) a sin A, sin B
( b ) a, b, c
( c ) a, sin B, R
( d ) a, sin A, R
[ IIT 2002 ]
15 - TRIGONOMETRY
Page 5
( Answers at the end of all questions )
The number of integral values of k for which the equation 7 cos x + 5 sin x = 2k + 1 has a solution is (a) 4
( 30 )
(b) 8
( c ) 10
( d ) 12
[ IIT 2002 ]
om
( 29 )
π be a fixed angle. If P = ( cos θ, sin θ ) 2 sin (α - θ ) ], then Q is obtained from P by Let
0 < α <
and
Q = [ cos ( α - θ ),
Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a po t X on the circumference of the circle, then 2r equals
α + β =
(b)
3
w w w ( 34 )
(c)
2 PQ ⋅ RS PQ + RS
(d)
PQ 2 + RS 2 2
[ IIT 2001 ]
A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depress on of 30°. After some time, the angle of depression becomes 60°. The distance n ( metres ) traveled by the car during this time is ( a ) 100
( 33 )
PQ + RS 2
(b)
.e
( 32 )
PQ ⋅ RS
xa
(a)
m
( 31 )
[ IIT 2002 ]
ra
ce .c
( a ) clockwise rotation around origin through an angle α ( b ) anticlockwise rotation around origin through an angle α ( c ) reflection in the line through origin with slope tan α α ( d ) reflection in the line through origin with slope t 2
π 2
and
( a ) 2 ( tan β + tan γ ) ( c ) tan β + 2tan γ
200 3 3
(c)
100 3 3
( d ) 200
3
[ IIT 2001 ]
β + γ = α , then tan α equals ( b ) tan β + tan γ ( d ) 2tan β + tan γ
[ IIT 2001 ]
π x2 x3 x4 x6 + + - ... + cos - 1 x 2 - ... = sin - 1 x for 0 < l x l < 2 4 2 4 2 then x equals If
(a)
1 2
(b) 1
(c) -
1 2
(d) - 1
2,
[ IIT 2001 ]
15 - TRIGONOMETRY
Page 6
( Answers at the end of all questions )
(b)
n
2
2 2
(c)
n
1 2n
( 36 ) The number of distinct real roots of π π ≤ x ≤ 4 4
(a) 0
is
(b) 2
( cos αn ),
under the restrictions
(c) 1
(d) 1
sin x cos x cos x
cos x sin x cos x
[ IIT 2001 ]
cos x cos x sin x
= 0 in the interval
(d) 3
[ IIT 2001 ]
ra
-
…..
( cos α1 ) ⋅ ( cos α2 ) ….. ( cos αn ) = 1 is
and
1
( cos α2 )
om
1
(a)
⋅
( cos α1 )
The maximum value of π 0 ≤ α1, α2, ….. αn ≤ 2
ce .c
( 35 )
( 37 ) If f ( θ ) = sin θ ( sin θ + sin 3θ ), then f ( θ )
( b ) ≤ 0 for all real θ ( d ) ≤ 0 only when θ ≤ 0
xa
m
( a ) ≥ 0 only when θ ≥ 0 ( c ) ≥ 0 for all real θ
( 38 ) In a triangle ABC, 2ac sin 2
2
2
(b)
w w
.e
(a) a +b -c
2
2
2
w
(a) a + b
( 40 )
(A - B + C) = 2
+ a - b
( 39 ) In a triangle ABC, if ∠ C = (b) b + c
[ IIT 2000 ]
2
2
(c) b - c - a
2
2
2
2
(d) c - a - b
[ IIT 2000 ]
π , r = inradius and R = circum-radius, then 2 ( r + R ) = 2
(c) c + a
(d) a + b + c
[ IIT 2000 ]
A pole stands vertically inside a triangular park ∆ ABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in ∆ ABC, the foot of the pole is at the ( a ) centroid
( b ) circumcentre
( c ) incentre
( d ) orthocentre
[ IIT 2000 ]
15 - TRIGONOMETRY
Page 7
( Answers at the end of all questions )
P If tan 2 2 equation ax + bx + c = 0 ( a ≠ 0 ), then PQR, ∠ R =
(a) a+b =c
(b) b+c =a
(c) c+a =b
tan - 1
( 42 ) The number of real solutions of ( a ) zero
( b ) one
( c ) two
and
x ( x + 1 ) + sin - 1
(c) 2
[ IIT 1999 ]
x2 + x +
( d ) infinite
= cos x + cos (
( d ) infinit
=
π
2
is
[ IIT 1999 ]
2x ) attains its
[ IIT 1998 ]
ra
(b) 1
are the roots of the
(d) b = c
( 43 ) The number of values of x where the function f ( x maximum is (a) 0
Q tan 2
om
π . 2
In a triangle
ce .c
( 41 )
π f2 = 1 16
(c)
π f4 = 1 64
(b
π f3 = 1 32
d)
π f5 = 1 128
.e
xa
(a)
m
( 44 ) If, for a positive integer n, θ fn ( θ ) = tan ( 1 + sec θ ) ( 1 + sec 2θ ) ... ( 1 + sec 2 n θ ) , then 2
sin P, sin Q, sin R are in A. P., then
w w
( 45 ) If in a triangle PQR,
( a ) he altitudes are in A. P. ( c ) the medians are in G. P.
( b ) the altitudes are in H. P. ( d ) the medians are in A. P.
w
46 ) The number of values of 2 3 sin x - 7 sin x + 2 = 0 is (a) 0
(b ) 5
[ IIT 1999 ]
(c)
x
6
[ IIT 1998 ]
in the interval [ 0, 5π ] satisfying the equation
( d ) 10
[ IIT 1998 ]
( 47 ) Which of the following number( s ) is / are rational ? ( a ) sin 15°
( b ) cos 15°
( c ) sin 15° cos 15°
( d ) sin 15° cos 75°
[ IIT 1998 ]
15 - TRIGONOMETRY
Page 8
( Answers at the end of all questions )
n
( 48 )
∑ br sinr θ,
Let n be an odd integer. If sin nθ =
for every value of θ, then b0 and
r =0
( a ) 1, 3
( c ) - 1, n
( b ) 0, n
( d ) 0, n
2
1 cos ( p - d ) x sin ( p - d ) x
(b) p
a2 cos ( p + d ) x sin ( p + d ) x (c) d
[ IIT 1998 ]
does not depend upon is
(d) x
ra
(a) a
a cos px sin px
- 3n + 3
ce .c
( 49 ) The parameter, on which the value of the determinant
om
b1 respectively are
[ IIT 1997 ]
2
2 ) - cos ( x + 1 ) is
( 50 ) The graph of the function cos x cos ( x
xa
m
π ( a ) a straight line passing through the point , - sin 2 1 and parallel to the X-axis 2 2 ( b ) a straight line passing th oug ( 0, - sin 1 ) with slope 2 ( c ) a straight line passing through ( 0, 0 ) 2
( d ) a parabola with ve t x ( 1, - sin 1 )
.e
[ IIT 1997 ]
w w
( 51 ) If A0 A1 A A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius, then the product f the lengths of the line segments A0 A1, A0 A2 and A0 A4 is 3 4
(a)
w
( 52
( 53 )
(b) 3
2
sec θ =
3
4 xy
( x + y )2
(a) x + y ≠ 0
(c) 3
(d)
3
3
[ IIT 1998 ]
2
is true if and only if
( b ) x = y, x ≠ 0
(c) x = y
( d ) x ≠ 0, y ≠ 0
[ IIT 1996 ]
The minimum value of the expression sin α + sin β + sin γ, where α, β, γ are the real numbers satisfying α + β + γ = π is ( a ) positive
( b ) zero
( c ) negative
(D) -3
[ IIT 1995 ]
15 - TRIGONOMETRY
Page 9
( Answers at the end of all questions )
sin ∠ BAD sin ∠ CAD
equals
1 3
(c)
1
(a)
(b)
6
( 55 )
If x =
(b) 1
∞ ∑
n=0
(c) 2
cos 2n φ ,
∞ ∑
y =
.e
w w w
2 3
[ IIT 1995 ]
tan x + sec x = 2 cos x
lying in the interval
n=0
sin 2n φ ,
∞ ∑
n=0
[ IIT 1993 ]
cos 2n φ sin 2n φ , for 0 < φ <
m
z =
( b ) xyz xy + z ( d ) xyz = yz + x
If f ( x ) = cos [ π ] function, then π (a) f = -1 2
( 58 )
(d)
(d) 3
xa
( a ) xyz = xz + y ( c ) xyz = x + y + z
π . If D divides BC internally in the ratio 4
3
then
( 57 )
1
Number of solutions of the equation [ 0, 2π ], is (a) 0
( 56 )
and ∠ C =
ce .c
1 : 3, then
π 3
om
In a triangle ABC, ∠ B =
ra
( 54 )
π , 2
[ IIT 1993 ]
cos [ - π ] x , where [ x ] stands for the greatest integer 2
(b) f(π) = 1
(c) f(-π) = 0
π (d) f = 2 4
[ IIT 1991 ]
2
he equation ( cos p - 1 ) x + ( cos p ) x + sin p = 0 in the variable x has real roots. Then p can take any value in the interval
( a ) ( 0, 2π )
π π (c) - , 2 2
( b ) ( - π, 0 )
( d ) ( 0, π )
[ IIT 1990 ]
( 59 ) In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B 3 satisfy the equation 3 sin x - 4 sin x - k = 0, 0 < k < 1, then the measure of angle C is (a)
π 3
(b)
π 2
(c)
2π 3
(d)
5π 6
[ IIT 1990 ]
15 - TRIGONOMETRY
Page 10
( Answers at the end of all questions )
x
( 60 ) The number of real solutions of the equation sin ( e ) = 5 (b) 1
(c) 2
+ 5–
x
is
( d ) infinitely many
[ IIT 1990 ]
om
(a) 0
x
( 61 ) The general solution of sin x - 3 sin 2x + sin 3x = cos x - cos 2x + cos 3x is π nπ + 2 8
(b)
( d ) 2 n π + cos -
( 62 ) The value of the expression
(b) 4
(c)
3 cosec 20°
2 sin 20 o
1 + cos 2 θ
sin 2 θ 7π 24
w w w
(
)
4 s n 4θ
π 2
[ IIT 1988 ]
and satisfying the equation
= 0
are
4 sin 4θ
1
5π 24
nd θ =
sin 40 o
(c)
11 π 24
(d)
π 24
[ IIT 1988 ]
In a tr angle, the lengths of the two larger sides are 10 and 9 respectively. If the angle a e in A. P., then the lengths of the third side can be a) 5 -
65
cos 2 θ
.e
(a)
( 64 )
m
sin 2 θ
4 sin 20 o
4 sin 4θ
xa
cos 2 θ
[ IIT 1989 ]
- se 20° is equal to
(d)
sin 40 o
( 63 ) The values of θ lying between θ = 0 1 + sin 2 θ
3 2
1
ra
(a) 2
ce .c
π 8 π n nπ + (c) (-1) 2 8 (a) nπ +
6
(b) 3
3
(c) 5
(d) 5 +
The smallest positive root of the equation tan x = x lies π 3π π (b) , π ( c ) π, (d) ( a ) 0, 2 2 2
6
in 3π π, 2
[ IIT 1987 ]
[ IIT 1987 ]
( 66 ) The number of all triplets ( a1, a2, a3 ) such that 2 a1 + a2 cos 2x + a3 sin x = 0 for all x is (a) 0
(b) 1
(c) 3
( d ) infinite
( e ) none of these
[ IIT 1987 ]
15 - TRIGONOMETRY
Page 11
( Answers at the end of all questions )
2π 3
(a) -
2π 3
(c)
2π sin 3 4π 3
is
(d)
5π 3
( e ) none of these
[ IIT 1986 ]
The expression 3π π + α + sin 6 ( 5 π - α ) is equal to - α + sin 4 ( 3π + α ) - 2 sin 6 3 sin 4 2 2 (a) 0
(b) 1
ce .c
( 68 )
(b)
1
om
( 67 ) The principal value of sin –
( d ) sin 4α + cos 4α
(c)3
( e ) none
f these
[ IIT 1986 ]
( 69 ) There exists a triangle ABC satisfying the conditi ns
xa
( e ) b sin A < a,
3π π ( 70 ) 1 + cos 1 + cos 1 8 8
.e
1 2
(b
cos
π 8
w w
(a)
π 2 π A < , b > a 2
ra
( c ) b sin A > a,
π ( b ) b sin A > a 2 π A < ( d b s n A < a, 2 π A > , b = a 2 A <
A >
m
( a ) b sin A = a,
cos
(c)
[ IIT 1986 ]
5π 7π 1 + cos 8 8 1 8
is equal to
1+
(d)
2
2
[ IIT 1984 ]
2
w
( 71 ) From the top of a light-house 60 m high with its base at the sea-level, the angle of depr ssion of a boat is 15°. The distance of the boat from the foot of the lighthouse is 2
(a)
3 - 1 60 metres 3 + 1
(b)
(c)
3 + 1 60 metres 3 - 1
( d ) None of these
3 + 1 3 - 1
4 -1 2 ( 72 ) The value of tan cos - 1 + tan 5 3 (a)
6 17
(b)
7 16
(c)
16 7
metres
[ IIT 1983 ]
is
( d ) None of these
[ IIT 1983 ]
15 - TRIGONOMETRY
Page 12
( Answers at the end of all questions ) ( 73 ) If f ( x ) = cos ( ln x ), then f ( x ) f ( y ) -
(b)
1 2
(c) -2
has the value
( d ) none of these
[ IT 1983 ]
om
(a) -1
1 x + f ( xy ) f 2 y
( 74 ) The general solution of the trigonometric equation sin x + cos x = 1 is giv n by
(c) x + nπ + (-1)
2
n
π π - , n = 0, ± 1, ± 2, … 4 4
4
π , n = 0, ± 1 ± 2, … 2
ce .c
( a ) x = 2 n π, n = 0, ± 1, ± 2, … ( b ) x = 2 n π +
( d ) none of these
[ IIT 1981 ]
( 75 ) If A = sin θ + cos θ, then for all real values of θ 3 ≤ A ≤ 1 4 1 3 ≤ A (d) 16 4
(a) 1 ≤ A ≤ 2
ra
13 ≤ A ≤ 1 16
m
(c)
(b)
xa
π 2 1 2 2 2 ( 76 ) The equation 2 cos x sin x = x + x - , 0 < x ≤ 2 2 ( a ) no real solution ( b one real solution ( c ) more than one rea sol ti n
.e u not
w w
-4 5 4 5
(c)
w
( 78
has [ IIT 1980 ]
-4 , then sin θ is 3
( 77 ) If tan θ =
(a)
[ IIT 1980 ]
but not
4 5 -4 5
(b)
-4 5
or
4 5
( d ) none of these
[ IIT 1979 ]
If α + β + γ = 2 π, then
γ β α + tan + tan 2 2 2 β β α ( b ) tan tan + tan tan 2 2 2 γ β α ( c ) tan + tan + tan 2 2 2 ( d ) none of these ( a ) tan
β γ α tan tan 2 2 2 γ γ α + tan tan = 1 2 2 2 β γ α = - tan tan tan 2 2 2 = tan
[ IIT 1979 ]
15 - TRIGONOMETRY
Page 13
( Answers at the end of all questions )
2 b
3 c
4 b
5 a
6 d
7 c
8 a
9 a
10 c
11 b
12 a
13 a
14 b
21 b
22 d
23 d
24 c
25 c
26 a
27 a
28 d
29 b
30 d
31 a
32 b
33 c
34 b
41 a
42 c
43 a
61 b
62 b
63 a,c
44 a,b,c,d 65 a
66 d
46 c
47 c 67 e
48 b 68 b
49 b 69 a,d
50 a 70 c
51 c
71 c
52 b
72
w
w w
.e
xa
m
ra
64 a,c
45 d
15 c
16 b
1 c
18 d
19 a
20 a
35 a
36 c
37 c
38 b
39 a
40 b
ce .c
1 b
om
Answers
53 c
73 d
54 a
74 c
55 d
56 b
57 a,c
58 b
59 c
60 0
75 b
76 a
77 a
78 a
79
80
