15 - Trigonometry

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15 - TRIGONOMETRY Page 1 ( Answers at the end e nd of all questions que stions ) (1) In triangle triangle PQR, ax 2 + b x + c = 0, (a) a = b + c (2) If cos -1 Let 2 are the roots of the equation 2 0, then (c) b = c (d) b = a + c [ AIEEE 2005 ] C = (b) a + b x - cos -1 y 2 = (c) a + b+c , then 4x (b) 4 ( b ) A.P. , 2 ( c ) 4 sin (d) c + a - 4xy cos 2 + y 2 [ AIEEE 2005 ] is equal to ( d ) - 4 sin 2 ( c ) Arithmetic-Geometric Arithmetic-Geometric Progression be such that - [ AIEEE 2005 ] - < < 3 . If sin ( d ) H.P. = - + sin [ AIEEE 2005 ] 21 , then the value value of  65 is 2 (a) - 3 (b) 130 If u = sin 3 (c) 130 a 2 cos 2 6 ( a) 2 (a 2 2 + b ) b 2 sin 2 (b) 2 The sides sides of a triangle triangle are a2 sin (d) - 65 a 2 sin 2 the maximum and minimum minimum values of of u (7) Q and tan 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) P If tan . If r is the inradius and R is the circumradius circumra dius of the 2 triangle ABC, then 2 ( r + R ) equals ( a ) G.P. (5) a In triangle ABC, let ( a ) 2 sin 2 (4) 2 . (b) c = a + b (a) b + c (3) R = b2 , 2 [ AIEEE 2004 ] 65 b 2 sin 2 , then difference between is given by (c) (a + b) cos 6 and 1 2 2 ( d) ( a - b ) sin cos [ AIEEE AIEEE 2004 ] for some 0 < < 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 e nd of all questions que stions ) (8) 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 breadth of the the river is ( a ) 20 m (9) ( b ) 30 m If in a triangle a cos ( a ) in A. P. 2 ( c ) 40 m C 2 + c cos ( b ) in G. P. 2 ( d ) 60 m A 2 = ( c ) in H. P. [ AIEEE 2004 ] 3b , then the sides a, b and c are 2 ( d ) satisfy a + b = c [ AIEEE 2003 ] ( 10 ) The sum of the radii of of inscribed and circumscribed circumscribed circles, for an n sided regular  polygon of side a, is ( a ) a cot ( b ) b cot 2n (c) n a cot 2 (d) 2n a cot 4 2n [ AIEEE 2003 ] 3 ( 11 ) The upper  uppe r  tan - 1 3 at a point in 5 4 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 th portion of a vertical pole pole subtends an angle angle ( b ) 40 m ( 12 ) The value of cos (a) 3 2 2 2 1 ( a) l a l< [ AIEEE 2003 ] - 120 ) is (d) 0 -1 1 ( b) l al 2 + 120 ) + cos ( (c) 1 ( 13 ) The trigonometric trigonometric equation sin ( d ) 80 m 2 + cos ( 1 2 (b) ( c ) 60 m x = 2 sin (c) 2 1 2 [ AIEEE 2003 ] -1 a has a solution for  < lal < 1 2 ( d ) all real values of a [ AIEEE 2003 ] ( 14 ) If sin (a) + sin a2 = a and cos b2 4 - a2 - b2 a2 (c) 4 a2 b2 b2 (b) (d) + cos = b, then the value of tan θ 2 is 4 - a2 - b2 a2 4 b2 a2 a2 b2 b2 [ AIEEE 2002 ] 15 - TRIGONOMETRY Page 3 ( Answers at the end of all questions ) ( 15 ) If tan -1 2 (a) 2 , then the value of x is 3 (c) 1 -1 1 -1 + tan 7 2 (c) 4 (d) 3 + tan 3 (b) 2 (x) = (b) 3 ( 16 ) The value of tan (a) -1 ( x ) + 2 cot 3 - 1 3 1 -1 1 [ AIEEE AIEEE 2002 ] + … + tan 13 n2 (d) 0 3 1 -1 is n 1 [ 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 ( 18 ) If cos ( a (b) 3 - 3 - 1 , a(3 (c) ) = 1 and cos ( ordered pairs ( (a) 0 3 3) ( d) a ( 2 + ) = 1 e , - 3 , - 1) [ AIEEE 2002 ] , then the number of  ) = (b) 1 (c) 2 (d) 4 [ IIT 2005 ] ( 19 ) Which of the following following is correct for triangle triangle ABC having sides sides a, b, c opposite to the angles A, B, C respectively respectively ( a ) a sin B - C 2 ( c ) ( b + c ) sin = ( b - c ) cos B C 2 A ( b ) a sin 2 = a cos A 2 ( d ) sin B C = ( b + c ) cos 2 B - C 2 = a cos ( 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 ( b ) 12 + 8 (c) 7 + 4 3 (d) 4 + 7 2 3 3 [ IIT 2005 ] A 2 A 2 [ IIT 2005 ] 15 - TRIGONOMETRY Page 4 ( Answers at the end of all questions que stions ) ( 21 ) If  and are acute angles such that sin = 1 and cos 2 = 1 3 , then and lies in (a) 3 , (b) 2 2 2 , ( 22 ) For which value of x, sin [ cot (a) ( 23 ) 1 (b) 0 2 2 (c) -1 5 3 , ( x + 1 ) ] = cos ( tan (d) - (c) 1 5 (d) -1 , x)? 1 [ IIT 2004 ] 2 If a, b, c are the sides of a triangle triangle such that a : b : c = 1 : then A : B : C is (a) 3 : 2 : 1 (b) 3 : 1 : 2 x2 ( 24 ) Value of  tan 2 x x , 2 [ IIT 2004 ] (c) 1 : 3 : 2 (d) 1 : 2 : 3 x > 0, 0, 3 : 2, [ IIT 2004 ] is alway ways greater than or  x equal to (a) 2 ( 25 ) (b) 5 ( c ) 2 tan 2 ( d ) sec [ IIT 2003 ] If the angles of of a triangle are in in the ratio 4 : 1 : 1, then the ratio ratio of the largest largest side to the perimeter is equal to (a) 1:1 + (b) 2:3 3 sin - 1 ( 2x ) ( 26 ) The natural domain of  (a) 1 - 4 , 1 (b) 2 (c) 1 - 4 , 1 3 :2 + 6 (c) 4 3 for all x 1 - 2 , 1 2 (d) 1:2 + 3 [ IIT 2003 ] R, is (d) 1 - 2 , 1 [ IIT 2003 ] 4 3 ( 27 ) The length of a longest interval in which the function 3 sin x - 4 sin x is increasing is (a) ( 28 ) 3 (b) 2 (c) 2 (d) [ IIT 2002 ] Which of the following pieces of data does NOT uniquely determ ine 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 ) ( 29 ) 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 Let 0 < sin ( - ( c ) 10 ( d ) 12 [ IIT 2002 ] < be a fixed angle. If P = ( cos , sin 2 ) ], then Q is obtained from P by ) and Q = [ cos ( - ), ( a ) clockwise rotation around ar ound origin through an angle ( b ) anticlockwise anticlockw ise rotation rotat ion around origin through an angle angl e ( c ) reflection in the line through origin with w ith slope tan ta n ( d ) reflection in the line through origin w ith slope tan ( 31 ) PQ RS (b) PQ RS (c) 2 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 depression of 30 . After some time, the angle angle of depression becomes becomes 60 . The distance in ( metres ) traveled by the car during this time is ( a ) 100 ( 33 ) If  + (b) 3 = 2 and ( a ) 2 ( tan + tan ( c ) tan + 2tan ( 34 ) [ IIT 2002 ] 2 Let PQ and RS be tangents tangents at the the extremities extremities of the diameter 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 (a) ( 32 ) α If sin - 1 x2 x 2 200 (c) 3 + ) 3 = 100 3 3 , then tan ( d ) 200 [ IIT 2001 ] 3 equals ( b ) tan + tan ( d ) 2tan + tan x3 - ... 4 cos - 1 x [ IIT 2001 ] 2 x4 2 x6 - ... 4 2 for  0 lxl 2, then x equals (a) 1 2 (b) 1 (c) - 1 2 (d) - 1 [ IIT 2001 ] 15 - TRIGONOMETRY Page 6 ( Answers at the end e nd of all questions que stions ) ( 35 ) The maximum value of 0 1, ≤ 2, ….. 1 (a) n ≤ (b) n 2 2 ( cos and 2 1 1 ( cos ( cos (c) 2n ) x 4 4 (a) 0 ( 37 ) If f ( (a) (c) ≥ ≥ ( sin 0 only when 0 for all real (c) 1 2 2 2 ≥ ( 40 ) ( cos 2 ( cos n ), ) ….. ( cos under the restrictions n ) = 1 is [ IIT 2001 ] sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0 in the the interval interval (d) 3 0 (b) (d) 2 (b) b + c ≤ ≤ [ IIT 2001 ] ) 0 for all real 0 only when ≤ 0 [ IIT 2000 ] 1 (A - B + C) = 2 2 2 ( b) c + a - b ( 39 ) In a triangle tr iangle ABC, ABC , if  (a) a + b ….. (d) 1 + sin 3 ), then f ( ( 38 ) In a triangle ABC, 2ac sin (a) a +b -c ) is (b) 2 ) = sin ) 1 2n ( 36 ) The number of distinct real roots of  - 1 2 C = 2 2 2 2 (c) b - c - a 2 2 2 (d) c - a - b [ IIT 2000 ] , r = inradius and R = circum-radius, circum-radius, then 2 ( r + R ) = (c) c + a (d) a + b + c [ IIT 2000 ] 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 Page 7 ( Answers at the end e nd of all questions que stions ) ( 41 ) In a triangle equation ax 2 PQR, R = 2 . 2 and tan Q are the roots of the 2 + bx + c = 0 ( a ≠ 0 ), then (a) a+b =c (b) b+c =a (c) c+a =b ( 42 ) The number of real solutions of  tan - 1 ( a ) zero P If tan ( b ) one ( c ) two x(x (d) b = c [ IIT 1999 ] sin - 1 1) x2 x ( d ) infinite (b) 1 (c) 2 is [ IIT 1999 ] ( 43 ) The number numbe r of values of x where the function func tion f ( x ) = cos x + cos ( maximum is (a) 0 1 2x ) attains its ( d ) infinite [ IIT 1998 ] ( 44 ) If, for a positive integer n, f n ( θ ) tan (a) f 2 (c) f 4 16 64 θ (1 2 1 1 ( 45 ) If in a triangle PQR, sec θ ) ( 1 (b) (d) sec 2θ ) ... ( 1 f 3 f 5 128 1 [ IIT 1999 ] ( b ) the altitudes altitudes are are in H. P. ( d ) the medians are in A. P. ( 46 ) The number of values of 2 3 sin x - 7 sin x + 2 = 0 is (b ) 5 1 sin P, sin Q, sin R are in A. P., then ( a ) the altitudes altitudes are in A. P. ( c ) the medians are in G. P. (a) 0 32 sec 2 n θ ) , then (c) x 6 in the interval [ 0, 5 ] [ IIT 1998 ] satisfying ( d ) 10 the equation [ IIT 1998 ] ( 47 ) Which of the following fol lowing number( numbe r( 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 e nd of all questions que stions ) n ( 48 ) Let n be an odd odd integer. If sin n = b r  sinr  , for every every value of  , then b0 and r  0 b1 respectively are ( a ) 1, 3 ( c ) - 1, n ( b ) 0, n ( d ) 0, n 2 - 3n + 3 [ IIT 1998 ] ( 49 ) The parameter, on which the value of the determinant 1 a2 a cos ( p - d ) x cos px cos ( p d)x sin ( p - d ) x sin px sin ( p d)x (a) a (b) p (c) d does not depend upon is (d) x [ IIT 1997 ] 2 ( 50 ) The graph of the function cos x cos ( x + 2 ) - cos ( x + 1 ) is , - sin 2 1 ( a ) a straight strai ght line passing pas sing through the point and parallel to the X-axis 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 ) [ 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 A 0 A1, A0 A2 and A0 A4 is 3 (a) ( 52 ) sec (b) 3 4 2 4xy = (x (a) x + y ≠ 0 ( 53 ) 3 y )2 (c) 3 3 (d) [ IIT 1998 ] 2 is true if and only if  ( b ) x = y, x ≠ 0 (c) x = y The minimum minimum value of the expression expression sin real numbers satisfying ( a ) positive 3 ( b ) zero + + = + sin ( d ) x ≠ 0, y ≠ 0 + sin , where [ IIT 1996 ] , , are the is ( c ) negative (D) - 3 [ IIT 1995 ] 15 - TRIGONOMETRY Page 9 ( Answers at the end e nd of all questions que stions ) ( 54 ) In a triangle triangle ABC, sin sin 1 : 3, then 1 (a) ( 55 ) ( 56 ) B = BAD CAD equals 1 3 (c) (b) 6 and 3 C = 1 (b) 1 3 If x = cos 2n n (c) 2 , 2 3 [ IIT 1995 ] tan x + sec x = 2 cos x, lying in the interval (d) 3 sin 2n y = 0 . If D divides BC internally in the ratio (d) Number of solutions of the equation [ 0, 2 ], is (a) 0 4 n [ IIT 1993 ] , cos 2n z = 0 n sin 2n , for 0 < < 0 2 , then ( a ) xyz = xz + y ( c ) xyz = x + y + z ( 57 ) If f ( x ) = cos [ function, then ( a ) f  ( 58 ) = -1 2 2 ( b ) xyz x yz = xy + z ( d ) xyz = yz + x ] x + cos [ - (b) f( ) = 1 2 [ IIT 1993 ] ] x , where [ x ] stands for the greatest integer  (c) f(- ) = 0 ( d ) f  4 = 2 [ IIT 1991 ] 2 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 ( a ) ( 0, 2 ) (b) (- , 0) (c) - 2 , 2 ( d ) ( 0, ) [ 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 the measure of of angle C is (a) 3 (b) 2 (c) 2 3 (d) 5 6 [ IIT 1990 ] 15 - TRIGONOMETRY Page 10 ( Answers at the end e nd of all questions que stions ) x ( 60 ) The number of real solutions of the equation sin ( e ) = 5 (a) 0 (b) 1 (c) 2 x + 5  – x is ( d ) infinitely many [ IIT 1990 ] ( 61 ) The general solution of sin x - 3 sin 2x + sin 3x = cos x - cos 2x + cos 3x is (a) n + n (b) 8 n n (c) (-1) 2 2 (d) 2n 8 ( 62 ) The value of the expression (a) 2 8 (b) 4 (c) + cos ( 63 ) The values of  2 sin 20 o sin 2 1 sin 2 θ sin 2 θ (a) ( 64 ) cos 2 θ cos 2 (b) 24 4 sin 20 o (d) o is equal to sin 40 = 0 and = 2 [ IIT 1988 ] o and satisfying the equation 4 sin 4θ θ cos 2 1 7 4 sin 4θ θ 1 θ 5 are 4 sin 4θ (c) 24 = 0 11 24 (d) [ IIT 1988 ] 24 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 - ( 65 ) lying between [ IIT 1989 ] - sec 20 3 cosec 20 sin 40 3 2 -1 6 (b) 3 3 (c) 5 (d) 5 + 6 The smallest positive root of the equation equation tan x = x lies in 3 3 (a) 0, (b) , (c) , (d) , 2 2 2 2 [ IIT IIT 1987 1987 ] [ IIT 1987 ] ( 66 ) The number of all triplets ( a 1, a2, a3 ) such that 2 a1 + a2 cos 2x + a 3 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 e nd of all questions que stions ) ( 67 ) The principal value of sin 2 (a) - ( 68 ) (b) 3 2  – 1 (c) 3 sin 2 is 3 4 5 (d) 3 ( e ) none of these 3 [ IIT 1986 ] The expression expression 3 sin 4 3 (a) 0 (b) 1 - sin 4 ( 3 α (c)3 α) - 2 sin 6 ( d ) sin 4 sin 6 ( 5 α 2 + cos 4 - ( e ) none of these α) is equal to [ IIT 1986 ] ( 69 ) There exists a triangle ABC satisfying the conditions conditions ( 70 ) ( a ) b sin A = a, A < ( c ) b sin A > a, A < ( e ) b sin A < a, A > 1 (a) cos 1 1 8 cos ( b ) cos 2 2 2 2 3 8 ( b ) b sin A > a, A > ( d ) b sin A < a, A < 2 2 , b > a , b = a 1 [ IIT 1986 ] cos 1 1 8 (c) 8 5 8 cos (d) 7 8 is equal to 1 2 2 2 [ IIT IIT 1984 ] ( 71 ) From the top of a light-house light-house 60 m high with its base at the sea-level, the angle angle of  depression of a boat is 15 . The distance of the boat from the foot of the lighthouse is 3 - 1 (a) (c) 3 1 3 1 3 - 1 6 17 1 2 60 metres (b) 60 metres ( d ) None of these ( 72 ) The value of tan (a) 3 (b) cos - 1 4 5 7 16 (c) metres 3 - 1 tan - 1 16 7 2 3 [ IIT 1983 ] is ( d ) None of these [ IIT 1983 ] 15 - TRIGONOMETRY Page 12 ( Answers at the end e nd of all questions que stions ) 1 2 ( 73 ) If f ( x ) = cos ( ln x ), then f ( x ) f ( y ) - (a) -1 (b) 1 2 (c) -2 x y f  f ( xy ) has the value ( 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 , n = 0, ± 1, ± 2, … ( b ) x = 2 n n + (-1) (c) x + n ( 75 ) If A = sin (a) 1 (c) ≤ 13 16 2 + cos A ≤ 2 ≤ A ≤ 4 - 4 4 , n = 0, ± 1, ± 2, … 3 4 3 (d) 4 ( 76 ) The equation 2 cos 2 2 ( d ) none of these 1 2 ≤ A ≤ 1 ≤ A ≤ 13 16 (a) (c) ( 78 ) If  = -4 2 sin x = x x ( c ) tan , then sin but not + ( b ) tan 3 but not 5 4 5 ( a ) tan -4 + 2 2 [ IIT 1981 ] 2 [ IIT 1980 ] + x -2 , 0 < x ≤ 2 ( a ) no real solution ( b ) one real solution ( c ) more than one real solution ( 77 ) If tan , n = 0, ± 1, ± 2, … , then for for all real values values of  (b) 1 + has [ IIT 1980 ] is 4 (b) 5 -4 5 -4 5 4 or  5 ( d ) none of these [ IIT 1979 ] = 2 , then + tan tan 2 2 + tan + tan + tan 2 2 ( d ) none of these 2 + tan 2 tan 2 = tan 2 tan 2 + tan = - tan 2 2 2 tan tan tan 2 2 2 = 1 tan 2 [ IIT 1979 ] 15 - TRIGONOMETRY Page 13 ( Answers at the end e nd of all questions que stions ) Answers 1 b 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 15 c 16 b 17 c 18 d 19 a 20 a 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 35 a 36 c 37 c 38 b 39 a 40 b 41 a 42 c 43 a 59 c 60 0 61 b 62 b 63 a,c a,c 79 80 44 a,b, a,b,c, c,d d 64 a,c a,c 65 a 45 d 66 d 46 c 67 e 47 c 48 b 68 b 49 b 69 a,d a,d 50 a 70 c 51 c 71 c 52 b 72 d 53 c 73 d 54 a 74 c 55 d 75 b 56 b 76 a 57 a,c a,c 77 b 58 b 78 a