15 - Trigonometry

15 - TRIGONOMETRY( Answers at the end of all questions ) (1) In triangle PQR, ∠ R = ax 2 Page 1 π . 2 ⎛P⎞ If tan ⎜ ⎟ ⎝2⎠ ⎛Q⎞ and tan ⎜ ⎟ ⎝2⎠ are the roots of the equation + b x + c = 0, a ≠ 0, then (c) b = c (d) b = a + c [ AIEEE 2005 ] (a) a = b + c (b) c = a + b (2) π . If r is the inradius and R is the circumradius of the 2 triangle ABC, then 2 ( r + R ) equals In triangle ABC, let ∠ C = (a) b + c (b) a + b (c) a + b+c (d) c + a [ AIEEE 2005 ] (3) If cos - x - cos ( a ) 2 sin 2α 1 1 y 2 2 = α, then 4x - 4xy cos α + y is equal to 2 (b) 4 ( c ) 4 sin 2 α ( d ) - 4 sin 2 α [ AIEEE 2005 ] (4) 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 ( a ) G.P. ( b ) A.P. ( c ) Arithmetic-Geometric Progression ( d ) H.P. [ AIEEE 2005 ] (5) Let α, β be such that π < α - β < 3π. If sin α + sin β = cos α - β 2 3 130 21 , then the value of 65 is 3 130 6 65 6 65 (a) - (b) (c) (d) - [ AIEEE 2004 ] (6) If u = sin a 2 cos 2 θ + b 2 sin 2 θ + a 2 sin 2 θ + b 2 sin 2 θ , then difference between 2 the maximum and minimum values of u (a) 2(a 2 is given by 2 + b ) 2 (b) 2 a2 + b2 (c) (a + b) (d) (a - b) 2 [ AIEEE 2004 ] (7) The sides of a triangle are sin α, cos α and 1 + sin α cos α 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 ( Answers at the end of all questions ) (8) Page 2 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 ( a ) 20 m ( b ) 30 m ( c ) 40 m ( d ) 60 m [ AIEEE 2004 ] (9) 2 ⎛ C ⎞ If in a triangle a cos ⎜ ⎟ ⎝ 2 ⎠ 2 ⎛ A ⎞ + c cos ⎜ ⎟ ⎝ 2 ⎠ = 3b , then the sides a, b and c are 2 ( a ) in A. P. ( b ) in G. P. ( c ) in H. P. ( d ) satisfy a + b = c [ AIEEE 2003 ] ( 10 ) The sum of the radii of inscribed and circumscribed circles, for an n sided regular polygon of side a, is ⎛ π ⎞ ( a ) a cot ⎜ ⎟ ⎝ 2n ⎠ ⎛ π⎞ ( b ) b cot ⎜ ⎟ ⎝ n⎠ (c) a ⎛ π ⎞ cot ⎜ ⎟ 2 ⎝ 2n ⎠ (d) a ⎛ π ⎞ cot ⎜ ⎟ 4 ⎝ 2n ⎠ [ AIEEE 2003 ] ( 11 ) The upper 3 ⎛ 3 ⎞ th portion of a vertical pole subtends an angle tan - 1 ⎜ ⎟ at a point in 4 ⎝ 5 ⎠ the horizontal plane through its foot and at a distance 40 m from the foot. The height of the vertical pole is ( a ) 20 m ( b ) 40 m ( c ) 60 m ( d ) 80 m [ AIEEE 2003 ] ( 12 ) The value of cos α + cos ( α + 120° ) + cos ( α - 120° ) is (a) 3 2 2 2 2 (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) < lal < ( d ) all real values of a 2 2 2 2 [ AIEEE 2003 ] ⎛ θ - φ ⎞ ( 14 ) If sin θ + sin φ = a and cos θ + cos φ = b, then the value of tan ⎜ ⎟ ⎝ 2 ⎠ 1 1 is (a) a2 + b2 4 - a2 - b2 a2 + b2 4 + a2 + b2 (b) 4 - a2 - b2 a2 + b2 4 + a2 + b2 a2 + b2 (c) (d) [ AIEEE 2002 ] 15 - TRIGONOMETRY ( Answers at the end of all questions ) 2π , then the value of x is 3 3 Page 3 ( 15 ) If tan - ( x ) + 2 cot - ( x ) = 1 1 (a) 2 (b) 3 (c) (d) 3 - 1 3 + 1 [ AIEEE 2002 ] ⎛ 1 ⎜ ⎜ 2 ⎝ n + n + 1 ⎞ ⎟ ⎟ ⎠ ( 16 ) The value of tan π 2 1 ⎛ 1 ⎞ -1 ⎛ 1 ⎞ -1 ⎛ 1 ⎞ -1 ⎜ ⎟ + tan ⎜ ⎟ + tan ⎜ ⎟ + … + tan ⎝ 3 ⎠ ⎝ 7 ⎠ ⎝ 13 ⎠ is (a) (b) π 4 (c) 2π 3 (d) 0 [ AIEEE 2002 ] ( 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° respectively. If the tower and the building stand at the same level, then the height of the tower is (a) a 3 (b) a 3 3 - 1 (c) a(3 + 2 3) (d) a( 3 - 1) [ AIEEE 2002 ] ( 18 ) If cos ( α - β ) = 1 and cos ( α + β ) = ordered pairs ( α, β ) = (a) 0 (b) 1 (c) 2 (d) 4 1 , e - π ≤ α, β ≤ π, then the number of [ IIT 2005 ] ( 19 ) Which of the following is correct for triangle ABC having sides a, b, c opposite to the angles A, B, C respectively A ⎛ B - C⎞ ( a ) a sin ⎜ ⎟ = ( b - c ) cos 2 ⎠ 2 ⎝ A ⎛ B + C⎞ ( c ) ( b + c ) sin ⎜ ⎟ = a cos 2 ⎠ 2 ⎝ A ⎛ B + C⎞ ( b ) a sin ⎜ ⎟ = ( b + c ) cos 2 2 ⎠ ⎝ A ⎛ B - C⎞ ( d ) sin ⎜ ⎟ = a cos 2 ⎠ 2 ⎝ [ IIT 2005 ] ( 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 (c) 7 + 4 3 3 ( b ) 12 + 8 3 7 (d) 4 + 3 2 [ IIT 2005 ] 15 - TRIGONOMETRY ( Answers at the end of all questions ) ( 21 ) If θ and φ are acute angles such that sin θ = lies in ⎤ π π ⎤ (a) ⎥ , ⎥ ⎦ 3 2 ⎦ ⎤ π 2π ⎡ (b) ⎥ , 3 ⎢ ⎦ 2 ⎣ 1 Page 4 1 , then θ and φ 3 1 2 and cos θ = ⎤ 2π 5π ⎡ (c) ⎥ , 3 ⎢ ⎦ 3 ⎣ ⎤ 5π ⎡ (d) ⎥ , π⎢ ⎦ 6 ⎣ 1 [ IIT 2004 ] ( 22 ) For which value of x, sin [ cot - ( x + 1 ) ] = cos ( tan - x ) ? (a) 1 2 (b) 0 (c) 1 (d) - 1 2 [ IIT 2004 ] ( 23 ) If a, b, c are the sides of a triangle such that a : b : c = 1 : then A : B : C is (a) 3 : 2 : 1 (b) 3 : 1 : 2 tan 2 α x2 + x 3 : 2, (c) 1 : 3 : 2 (d) 1 : 2 : 3 [ IIT 2004 ] ( 24 ) Value of equal to (a) 2 x2 + x + , x > 0, α ∈ ⎜ 0, ⎝ ⎛ π ⎞ ⎟ 2 ⎠ is always greater than or (b) 5 2 ( c ) 2 tan α ( d ) sec α [ IIT 2003 ] ( 25 ) 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 (a) 1:1 + 3 (b) 2:3 (c) π 6 3 :2 + 3 (d) 1:2 + 3 [ IIT 2003 ] ( 26 ) The natural domain of ⎡ 1 1⎤ (a) ⎢- , ⎥ ⎣ 4 2 ⎦ sin - 1 ( 2x ) + for all x ∈ R, is ⎡ 1 1⎤ (d) ⎢- , ⎥ ⎣ 2 4 ⎦ 3 ⎡ 1 1⎤ (b) ⎢- , ⎥ ⎣ 4 4 ⎦ ⎡ 1 1⎤ (c) ⎢- , ⎥ ⎣ 2 2 ⎦ [ IIT 2003 ] ( 27 ) The length of a longest interval in which the function 3 sin x - 4 sin x is increasing is (a) π 3 (b) π 2 (c) 3π 2 (d) π [ IIT 2002 ] ( 28 ) 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 ( Answers at the end of all questions ) ( 29 ) Page 5 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 (b) 8 ( c ) 10 ( d ) 12 [ IIT 2002 ] ( 30 ) Let 0 < α < sin (α - π be a fixed angle. If P = ( cos θ, sin θ ) 2 θ ) ], then Q is obtained from P by and Q = [ cos ( α - θ ), ( 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 tan 2 [ IIT 2002 ] ( 31 ) 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 point X on the circumference of the circle, then 2r equals PQ ⋅ RS (a) (b) PQ + RS 2 (c) 2 PQ ⋅ RS PQ + RS (d) PQ 2 + RS 2 2 [ IIT 2001 ] ( 32 ) A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of 30°. After some time, the angle of depression becomes 60°. The distance in ( metres ) traveled by the car during this time is ( a ) 100 3 (b) 200 3 3 (c) 100 3 3 ( d ) 200 3 [ IIT 2001 ] ( 33 ) If α + β = π 2 and β + γ = α , then tan α equals ( b ) tan β + tan γ ( d ) 2tan β + tan γ ( a ) 2 ( tan β + tan γ ) ( c ) tan β + 2tan γ [ IIT 2001 ] ( 34 ) ⎛ ⎞ ⎛ ⎞ π x6 x4 x3 x2 + - ... ⎟ + cos - 1 ⎜ x 2 + - ... ⎟ = for 0 < l x l < sin - 1 ⎜ x ⎜ ⎟ ⎜ ⎟ 4 2 2 4 2 ⎝ ⎠ ⎝ ⎠ then x equals If (a) 1 2 2, (b) 1 (c) - 1 2 (d) - 1 [ IIT 2001 ] 15 - TRIGONOMETRY ( Answers at the end of all questions ) Page 6 ( 35 ) The maximum value of π 0 ≤ α1, α2, ….. αn ≤ 2 (a) 1 2 2 n ( cos α1 ) and ⋅ ( cos α2 ) ….. ( cos αn ), under the restrictions ( cos α1 ) ⋅ ( cos α2 ) ….. ( cos αn ) = 1 is 1 2n (b) 1 2n (c) (d) 1 [ IIT 2001 ] sin x cos x sin x cos x cos x cos x sin x ( 36 ) The number of distinct real roots of π π ≤ x ≤ 4 4 cos x cos x = 0 in the interval is (c) 1 (d) 3 [ IIT 2001 ] (a) 0 (b) 2 ( 37 ) If f ( θ ) = sin θ ( sin θ + sin 3θ ), then f ( θ ) ( a ) ≥ 0 only when θ ≥ 0 ( c ) ≥ 0 for all real θ ( b ) ≤ 0 for all real θ ( d ) ≤ 0 only when θ ≤ 0 [ IIT 2000 ] ( 38 ) In a triangle ABC, 2ac sin (a) a +b -c 2 2 2 2 1 (A 2 2 2 - B + C) = (c) b - c - a 2 2 2 (b) c + a - b (d) c - a - b 2 2 2 [ IIT 2000 ] ( 39 ) In a triangle ABC, if ∠ C = (a) a + b (b) b + c π , r = inradius and R = circum-radius, then 2 ( r + R ) = 2 (c) c + a (d) a + b + c [ IIT 2000 ] ( 40 ) A pole stands vertically inside a triangular park Δ ABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in Δ ABC, the foot of the pole is at the ( a ) centroid ( b ) circumcentre ( c ) incentre ( d ) orthocentre [ IIT 2000 ] 15 - TRIGONOMETRY ( Answers at the end of all questions ) π . 2 Page 7 ( 41 ) ⎛ P ⎞ If tan ⎜ ⎟ ⎝ 2 ⎠ 2 equation ax + bx + c = 0 ( a ≠ 0 ), then In a triangle PQR, ∠ R = and ⎛ Q ⎞ tan ⎜ ⎟ ⎝ 2 ⎠ are the roots of the (a) a+b =c (b) b+c =a (c) c+a =b (d) b = c [ IIT 1999 ] π ( 42 ) The number of real solutions of ( a ) zero ( b ) one tan - 1 x ( x + 1 ) + sin - 1 x2 + x + 1 = 2 is ( c ) two ( d ) infinite [ IIT 1999 ] ( 43 ) The number of values of x where the function f ( x ) = cos x + cos ( maximum is (a) 0 (b) 1 (c) 2 ( d ) infinite 2x ) attains its [ IIT 1998 ] ( 44 ) If, for a positive integer n, θ ⎞ ⎛ fn ( θ ) = ⎜ tan ⎟ ( 1 + sec θ ) ( 1 + sec 2θ ) ... ( 1 + sec 2 n θ ) , then 2 ⎠ ⎝ (a) (c) ⎛ π ⎞ f2 ⎜ ⎟ = 1 ⎝ 16 ⎠ ⎛ π ⎞ f4 ⎜ ⎟ = 1 ⎝ 64 ⎠ (b) (d) ⎛ π ⎞ f3 ⎜ ⎟ = 1 ⎝ 32 ⎠ ⎛ π ⎞ f5 ⎜ ⎟ = 1 ⎝ 128 ⎠ [ IIT 1999 ] ( 45 ) If in a triangle PQR, sin P, sin Q, sin R are in A. P., then ( b ) the altitudes are in H. P. ( d ) the medians are in A. P. [ IIT 1998 ] ( a ) the altitudes are in A. P. ( c ) the medians are in G. P. ( 46 ) The number of values of 2 3 sin x - 7 sin x + 2 = 0 is (a) 0 (b ) 5 (c) 6 x 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 ( Answers at the end of all questions ) n Page 8 ( 48 ) Let n be an odd integer. If sin nθ = b1 respectively are ( a ) 1, 3 ( b ) 0, n ( c ) - 1, n r=0 ∑ br sinr θ, for every value of θ, then b0 and ( d ) 0, n 2 - 3n + 3 [ IIT 1998 ] ( 49 ) The parameter, on which the value of the determinant 1 cos ( p - d ) x sin ( p - d ) x a cos px sin px a2 cos ( p + d ) x sin ( p + d ) x does not depend upon is (a) a (b) p (c) d (d) x [ IIT 1997 ] ( 50 ) The graph of the function cos x cos ( x + 2 ) - cos ( x + 1 ) is ⎛ 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 through ( 0, - sin 1 ) with slope 2 ( c ) a straight line passing through ( 0, 0 ) 2 ( d ) a parabola with vertex ( 1, - sin 1 ) 2 [ IIT 1997 ] ( 51 ) If A0 A1 A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius, then the product of the lengths of the line segments A0 A1, A0 A2 and A0 A4 is (a) 3 4 (b) 3 3 (c) 3 (d) 3 2 3 [ IIT 1998 ] ( 52 ) sec θ = 2 4 xy ( x + y )2 is true if and only if ( d ) x ≠ 0, y ≠ 0 (a) x + y ≠ 0 ( b ) x = y, x ≠ 0 (c) x = y [ IIT 1996 ] ( 53 ) 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 ( Answers at the end of all questions ) π 3 Page 9 ( 54 ) In a triangle ABC, ∠ B = 1 : 3, then 1 6 and ∠ C = π . If D divides BC internally in the ratio 4 sin ∠ BAD sin ∠ CAD (b) 1 3 equals 1 3 2 3 (a) (c) (d) [ IIT 1995 ] ( 55 ) Number of solutions of the equation [ 0, 2π ], is (a) 0 (b) 1 (c) 2 tan x + sec x = 2 cos x, lying in the interval (d) 3 [ IIT 1993 ] ( 56 ) If x = then ∞ n=0 ∑ cos 2n φ , y = ∞ n=0 ∑ sin 2n φ , z = ∞ n=0 ∑ cos 2n φ sin 2n φ , for 0 < φ < π , 2 ( a ) xyz = xz + y ( c ) xyz = x + y + z ( b ) xyz = xy + z ( d ) xyz = yz + x [ IIT 1993 ] ( 57 ) If f ( x ) = cos [ function, then ⎛ π ⎞ (a) f ⎜ ⎟ = -1 ⎝ 2 ⎠ π2 ] x + cos [ - π2 ] x , where [ x ] stands for the greatest integer (b) f(π) = 1 (c) f(- π) = 0 ⎛ π ⎞ (d) f ⎜ ⎟ = 2 ⎝ 4 ⎠ [ IIT 1991 ] ( 58 ) The 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 ⎛ π π ⎞ (c) ⎜ - , ⎟ ⎝ 2 2 ⎠ 2 ( a ) ( 0, 2π ) ( 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 ( Answers at the end of all questions ) x x Page 10 ( 60 ) The number of real solutions of the equation sin ( e ) = 5 (a) 0 (b) 1 (c) 2 ( d ) infinitely many + 5– x is [ IIT 1990 ] ( 61 ) The general solution of sin x - 3 sin 2x + sin 3x = cos x - cos 2x + cos 3x is π 8 π n nπ + (c) (-1) 2 8 (a) nπ + (b) nπ π + 2 8 1 ( d ) 2 n π + cos - 3 2 [ IIT 1989 ] ( 62 ) The value of the expression (a) 2 (b) 4 (c) 3 cosec 20° - sec 20° is equal to (d) 4 sin 20 o sin 40 o π 2 2 sin 20 o sin 40 o [ IIT 1988 ] ( 63 ) The values of θ lying between θ = 0 and θ = 1 + sin 2 θ sin 2 θ sin 2 θ 7π 24 cos 2 θ 1 + cos 2 θ cos 2 θ 5π 24 4 sin 4θ 4 sin 4θ 1 + 4 sin 4θ 11 π 24 and satisfying the equation = 0 are (a) (b) (c) (d) π 24 [ IIT 1988 ] ( 64 ) In a triangle, the lengths of the two larger sides are 10 and 9 respectively. If the angles are in A. P., then the lengths of the third side can be (a) 5 6 (b) 3 3 (c) 5 (d) 5 + 6 [ IIT 1987 ] ( 65 ) The smallest positive root of the equation tan x = x lies π ⎞ 3π ⎞ ⎞ ⎛ ⎛ π ⎛ ( a ) ⎜ 0, (b) ⎜ , π ⎟ ( c ) ⎜ π, (d) ⎟ ⎟ 2 ⎠ 2 2 ⎠ ⎝ ⎝ ⎠ ⎝ in 3π ⎞ ⎛ ⎜ π, ⎟ 2 ⎠ ⎝ [ 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 ( Answers at the end of all questions ) 2π ⎞ ⎛ ⎜ sin ⎟ 3 ⎠ ⎝ 4π 3 Page 11 ( 67 ) The principal value of sin – 2π 3 2π 3 1 is 5π 3 (a) - (b) (c) (d) ( e ) none of these [ IIT 1986 ] ( 68 ) The expression ⎡ ⎤ ⎡ ⎤ ⎛ 3π ⎞ ⎛ π ⎞ 3 ⎢ sin 4 ⎜ + α ⎟ + sin 6 ( 5 π - α ) ⎥ is equal to - α ⎟ + sin 4 ( 3π + α ) ⎥ - 2 ⎢ sin 6 ⎜ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎣ ⎦ ⎣ ⎦ (a) 0 (b) 1 (c)3 ( d ) sin 4α + cos 4α ( e ) none of these [ IIT 1986 ] ( 69 ) There exists a triangle ABC satisfying the conditions ( a ) b sin A = a, ( c ) b sin A > a, ( e ) b sin A < a, π ( b ) b sin A > a, 2 π A < ( d ) b sin A < a, 2 π A > , b = a 2 A < π 2 π A < , b > a 2 A > [ IIT 1986 ] 7π ⎞ 5π ⎞ ⎛ 3π ⎞ ⎛ π⎞⎛ ⎛ ( 70 ) ⎜ 1 + cos ⎟ ⎜ 1 + cos ⎟ ⎜ 1 + cos ⎟ ⎜ 1 + cos ⎟ 8 ⎠ 8 ⎠⎝ 8 ⎠⎝ 8⎠⎝ ⎝ (a) 1 2 is equal to ( b ) cos π 8 (c) 1 8 (d) 1+ 2 2 2 [ IIT 1984 ] ( 71 ) From the top of a light-house 60 m high with its base at the sea-level, the angle of depression of a boat is 15°. The distance of the boat from the foot of the lighthouse is ⎛ ⎜ ⎜ ⎝ 3 - 1 ⎞ ⎟ 60 metres 3 + 1⎟ ⎠ 3 + 1⎞ ⎟ 60 metres 3 - 1 ⎟ ⎠ (a) (b) ⎛ ⎜ ⎜ ⎝ 3 + 1⎞ ⎟ 3 - 1 ⎟ ⎠ 2 metres ⎛ (c) ⎜ ⎜ ⎝ ( d ) None of these [ IIT 1983 ] ⎡ ⎛ 4 ⎞ -1 ⎛ 2 ⎞ ⎤ ( 72 ) The value of tan ⎢ cos - 1 ⎜ ⎟ + tan ⎜ ⎟⎥ ⎝ 5 ⎠ ⎝ 3 ⎠⎦ ⎣ is (a) 6 17 (b) 7 16 (c) 16 7 ( d ) None of these [ IIT 1983 ] 15 - TRIGONOMETRY ( Answers at the end of all questions ) ( 73 ) If f ( x ) = cos ( ln x ), then f ( x ) f ( y ) 1 2 ⎡ ⎢ ⎢ ⎣ ⎤ ⎛ x ⎞ f⎜ ⎜ y ⎟ + f ( xy ) ⎥ ⎟ ⎥ ⎝ ⎠ ⎦ Page 12 has the value (a) -1 (b) 1 2 (c) -2 ( d ) none of these [ IIT 1983 ] ( 74 ) The general solution of the trigonometric equation sin x + cos x = 1 is given by ( a ) x = 2 n π, n = 0, ± 1, ± 2, … ( b ) x = 2 n π + (c) x + nπ + (-1) n π , n = 0, ± 1, ± 2, … 2 π 4 - π , n = 0, ± 1, ± 2, … 4 ( d ) none of these [ IIT 1981 ] ( 75 ) If A = sin θ + cos θ, then for all real values of θ (a) 1 ≤ A ≤ 2 (c) 13 ≤ A ≤ 1 16 2 4 (b) 3 ≤ A ≤ 1 4 13 3 (d) ≤ A ≤ 16 4 [ IIT 1980 ] 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 real solution has [ IIT 1980 ] ( 77 ) If tan θ = -4 5 4 5 -4 , then sin θ is 3 4 5 -4 5 (a) (c) but not but not (b) -4 5 or 4 5 ( d ) none of these [ IIT 1979 ] ( 78 ) If α + β + γ = 2 π, then γ β α + tan + tan 2 2 2 α β β ( b ) tan tan + tan tan 2 2 2 γ β α ( c ) tan + tan + tan 2 2 2 ( d ) none of these ( a ) tan α β γ tan tan 2 2 2 γ γ α + tan tan = 1 2 2 2 α β γ = - tan tan tan 2 2 2 = tan [ IIT 1979 ] 15 - TRIGONOMETRY ( Answers at the end of all questions ) Page 13 Answers 1 b 21 b 41 a 61 b 2 b 22 d 42 c 62 b 3 c 23 d 43 a 63 a,c 4 b 24 c 5 a 25 c 6 d 26 a 45 d 66 d 7 c 27 a 46 c 67 e 47 c 8 a 28 d 48 b 68 b 9 a 29 b 49 b 69 a,d 10 c 30 d 50 a 70 c 11 b 31 a 51 c 71 c 12 a 32 b 52 b 72 d 13 a 33 c 53 c 73 d 14 b 34 b 54 a 74 c 15 c 35 a 55 d 75 b 16 b 36 c 56 b 76 a 17 c 37 c 57 a,c 77 b 18 d 38 b 58 b 78 a 19 a 39 a 59 c 79 20 a 40 b 60 0 80 44 a,b,c,d 64 a,c 65 a