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Sr.Teacher II Grade RPSC - MATHEMATICS - Trigonometric Function 2

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MATHEMATICS Study Material for Examination for the post of Sr.Teacher Grade II RPSC

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Examination Questions

TRIGONOMETRY

Sr.Teacher II Grade RPSC - MATHEMATICS - Trigonometric Function 2

Previous year Examination Questions

01. If the function \( f(x) = \sin x + \cos ax \) is periodic then (H-TET 2008)
(1) \( a \) is natural number (2) \( a \) is an integer (3) \( a \) is rational number (4) \( a \) is irrational number

02. The period of the function \( f(x) = |\sin x| + |\cos x| \) is (H-TET 2008)
(1) \( \pi \) (2) \( \pi/2 \) (3) \( 2\pi \) (4) None of these

03. The period of the function \( f(x) = \sin^4 3x + \cos^4 3x \) is (H-TET 2008)
(1) \( \pi/2 \) (2) \( \pi/3 \) (3) \( \pi/6 \) (4) None of these

04. The value of \( \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \) is (H-TET 2008)
(1) 1 (2) 1/4 (3) 1/8 (4) 1/16

05. The maximum value of \( \sin (x + \pi/6) + \cos (x + \pi/6) \) for \( 0 < x < \pi/2 \) is attained at \( x \). (H-TET 2008)
(1) \( \pi/12 \) (2) \( \pi/6 \) (3) \( \pi/3 \) (4) \( \pi/2 \)

06. The most general solution of \( \tan \theta = -1, \cos \theta = \frac{1}{\sqrt{2}} \) is (H-TET 2008)
(1) \( n\pi + \frac{7\pi}{4} \) (2) \( n\pi + (-1)^n \frac{7\pi}{4} \) (3) \( 2n\pi + \frac{7\pi}{4} \) (4) None of these

07. The principal value of \( \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \) is (H-TET 2008)
(1) \( -\frac{2\pi}{3} \) (2) \( -\frac{\pi}{3} \) (3) \( \frac{4\pi}{3} \) (4) \( \frac{5\pi}{3} \)

08. If \( \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2} \) the value of \( x^{100} + y^{100} + z^{100} - \frac{9}{x^{100} + y^{100} + z^{100}} \) is (H-TET 2008)
(1) 0 (2) 1 (3) 2 (4) 4

09. If \( S = \tan^{-1}\left(\frac{1}{2.1^2}\right) + \tan^{-1}\left(\frac{1}{2.2^2}\right) + \tan^{-1}\left(\frac{1}{2.3^2}\right) + \dots \infty \) then \( S \) is equal to (S. TET 2009)
(1) \( \pi \) (2) \( \pi/2 \) (3) \( \pi/3 \) (4) \( \pi/4 \)

10. \( \cos^{-1}\left\{\sin\left(-\frac{5\pi}{6}\right)\right\} \) (S. TET 2009)
(1) \( -5\pi/6 \) (2) \( 5\pi/6 \) (3) \( 2\pi/3 \) (4) None of these

11. If \( \sin x, \cos x, \tan x \) are in G.P. then \( \cot^5 x - \cot^2 x = \) (S. TET 2009)
(1) 0 (2) -1 (3) 1 (4) depends on x

12. The maximum value of \( 5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 3 \) is (S. TET 2009)
(1) 5 (2) 10 (3) 11 (4) -1

13. The value of \( \sin^{-1} \left( \sin \frac{13\pi}{2} \right) \) is equal to (S. TET 2009)
(1) \( \sin^{-1}\left(\frac{3}{2}\right) \) (2) \( \pi - \sin^{-1}\left(\frac{3}{2}\right) \) (3) \( \pi + \sin^{-1}\left(\frac{3}{2}\right) \) (4) None of these

14. \( \sin 163^\circ \cos 347^\circ + \sin 73^\circ \sin 167^\circ = \) (S. TET 2009)
(1) 0 (2) 1/2 (3) 1 (4) None of these

15. \( \frac{4xy}{(x+y)^2} = \sec^2 \theta \) is true if and only if (S. TET 2009)
(1) \( x + y \neq 0 \) (2) \( x = y, x \neq 0 \) (3) \( x = y \) (4) \( x \neq 0, y \neq 0 \)

16. The value of \( 2(\sec^2 30^\circ + \cos^2 45^\circ) + (2\cos 60^\circ + \sin 90^\circ + \tan 45^\circ) \) is (II GRADE RPSC DEC. 2010)
(1) 10 (2) \( \frac{10(3+2)}{3} \) (3) \( \frac{10}{3} \) (4) \( \frac{33}{4} \)

17. An observer is standing 72 m away from a building. From there, the angles of elevation of the top and bottom of a flagstaff on the building are 60° and 45° respectively. The height of the flagstaff is (II GRADE RPSC DEC. 2010)
(1) 124.7 m (2) 52.7 m (3) 98.3 m (4) 73.2 m

18. If \( \sin(A - B) = 1/2, \cos(A + B) = 1/2 \), \( 0 < A + B \le 90^\circ \), \( A > B \) then the value of A is: (II GRADE RPSC DEC. 2011)
(1) 15° (2) 45° (3) 30° (4) 60°

19. If \( \sec\theta + \tan\theta = x \) then \( \tan\theta \) is equal to: (II GRADE RPSC DEC. 2011)
(1) \( \frac{x^2+1}{x} \) (2) \( \frac{x^2-1}{x} \) (3) \( \frac{x^2-1}{2x} \) (4) \( \frac{x^2+1}{2x} \)

20. \( \sin 3A = \cos(A - 26^\circ) \) where 3A is an acute angle then the value of A is: (II GRADE RPSC DEC. 2011)
(1) 29° (2) 39° (3) 19° (4) 64°

21. If A, B and C are interior angles of \( \Delta ABC \) then the value of \( \sin\left(\frac{B+C}{2}\right) \) is: (II GRADE RPSC DEC. 2011)
(1) \( \sin A/2 \) (2) \( \cos A/2 \) (3) \( \tan A/2 \) (4) \( \cot A/2 \)

22. The value of \( \sin^2 15^\circ + \sin^2 25^\circ + \sin^2 65^\circ + \sin^2 75^\circ \) is: (II GRADE RPSC DEC. 2011)
(1) 1 (2) 0 (3) 3 (4) 2

23. If \( \cos^2 A + \cos^2 C = \sin^2 B \) then triangle ABC is (H.TET JN 2013)
(1) Equilateral (2) Right angled (3) Isosceles (4) None

24. The value of \( \sin(\cot^{-1} x) \) is (H.TET JN 2013)
(1) \( \sqrt{1+x^2} \) (2) \( x \) (3) \( \frac{1}{\sqrt{1+x^2}} \) (4) \( \frac{1}{x} \)

25. The value of \( \sin^{-1}\left[\sin\left(-\frac{\pi}{3}\right)\right] \) is (H.TET 2013)
(1) 1 (2) -1 (3) 0 (4) 1/2

26. If \( \cos^{-1}\left(\frac{4}{5}\right) + \sin^{-1}\left(\frac{13}{x}\right) = \cos^{-1}\left(-\frac{4}{5}\right) \) then x = (H.TET 2013)
(1) 0 (2) 1 (3) -1 (4) None of these

27. \( \sin 47^\circ + \sin 61^\circ - \sin 11^\circ - \sin 25^\circ = \) (H.TET 2013)
(1) \( \cos 36^\circ \) (2) \( \sin 7^\circ \) (3) \( \cos 36^\circ \) (4) \( \cos 7^\circ \)

28. If \( x = \tan 15^\circ, y = \operatorname{cosec} 75^\circ \) and \( z = 4 \sin 18^\circ \), then (H.TET 2013)
(1) \( x < z < y \) (2) \( y < z < x \) (3) \( x < y < z \) (4) \( z < x < y \)

29. The period of the function \( \sin\left(\sin \frac{x}{3}\right) \) is (H.TET 2013)
(1) \( 3\pi \) (2) \( 8\pi \) (3) \( 4\pi \) (4) \( 2\pi \)

30. One radian is equal to: (II GRADE RPSC 2014)
(1) \( \left(\frac{2}{\pi}\right)^\circ \) (2) \( \left(\frac{2}{\pi}\right)^\circ \) of a right angle (3) \( \left(\frac{1}{4\pi}\right)^\circ \) of four right angles (4) \( \left(\frac{180}{\pi}\right)^\circ \)

31. If \( \alpha + \beta = \pi/2 \) and \( \beta + \gamma = \alpha \) then \( \tan \alpha = \) (II GRADE RPSC 2014)
(1) \( 2 (\tan \beta + \tan \gamma) \) (2) \( \tan \beta + \tan \gamma \) (3) \( \tan \beta + 2\tan \gamma \) (4) \( 2\tan \beta + \tan \gamma \)

32. If \( \sec \theta + \tan \theta = m \), then \( \sin \theta = \) (II GRADE RPSC 2014)
(1) \( \frac{m^2-1}{m^2+1} \) (2) \( \frac{m^2+1}{m^2-1} \) (3) \( \frac{m^2+1}{m} \) (4) \( \frac{m^2-1}{m} \)

33. The general value of \( \theta \) satisfying the equations \( \sin 2\theta = -1 \) and \( \tan 3\theta = -1 \) is: (II GRADE RPSC 2014)
(1) \( n\pi + \frac{\pi}{6} \) (2) \( n\pi + (-1)^n \frac{\pi}{6} \) (3) \( n\pi + \frac{2\pi}{6} \) (4) \( n\pi - \frac{2\pi}{6} \)

34. If \( \tan \theta - \cot \theta = a \) and \( \sin \theta + \cos \theta = b \), then \( (b^2 - 1)^2 (a^2 + 4) = \) (II GRADE RPSC 2014)
(1) 2 (2) 4 (3) -4 (4) \( \pm 4 \)

35. The angles of a triangle are in the ratio 4 : 1 : 1, the ratio of its greatest side to its perimeter is: (II GRADE RPSC 2014)
(1) \( \frac{1}{2+\sqrt{3}} \) (2) \( \frac{\sqrt{3}}{2-\sqrt{3}} \) (3) \( \frac{\sqrt{3}}{2+\sqrt{3}} \) (4) \( \frac{\sqrt{3}}{2+\sqrt{3}} \)

36. If \( a_1, a_2, a_3, \dots, a_n \) are in AP with common difference \( d \), then \( \sum_{i=1}^n \tan^{-1} \left( \frac{d}{1+a_i a_{i+1}} \right) = \) (II GRADE RPSC 2014)
(1) \( \tan^{-1} \left( \frac{nd}{1+a_1 a_n} \right) \) (2) \( \tan^{-1} \left( \frac{d}{1+a_1 a_n} \right) \) (3) \( \tan^{-1} \left( \frac{nd}{1-a_1 a_n} \right) \) (4) \( \tan^{-1} \left( \frac{nd}{1+a_1 a_n} \right) \)

37. If \( \sin \theta + \cos \theta = 1 \), then the value of \( \sin 2\theta \) is equal to (PGT 2014)
(1) 1 (2) 1/2 (3) 0 (4) -1

38. The value of \( \cos^{-1} \left( \cos \frac{3\pi}{2} \right) \) is equal to (PGT 2014)
(1) \( \pi/2 \) (2) \( 2\pi/3 \) (3) \( 5\pi/2 \) (4) \( 7\pi/2 \)

39. The value of \( \cos (\sin^{-1} x) \) is (PGT 2014)
(1) \( \frac{\sqrt{1-x^2}}{x} \) (2) \( \frac{\sqrt{1+x^2}}{x} \) (3) \( \frac{1}{x} \) (4) \( \sqrt{1-x^2} \)

40. \( \sin \left[ \cot^{-1} \left( \cot \frac{17\pi}{3} \right) \right] = \) (H-TAT 2014)
(1) \( -\sqrt{3}/2 \) (2) \( \sqrt{3}/2 \) (3) \( 1/2 \) (4) 1

41. If \( \cos^{-1}\left(\frac{4}{5}\right) + \sin^{-1}\left(\frac{13}{x}\right) = \cos^{-1}\left(-\frac{4}{5}\right) \), then \( x = \) (H-TAT 2014)
(1) 0 (2) 1 (3) -1 (4) 2

42. Solution set of the equation \( \sin^{-1} x = 2 \tan^{-1} x \) is (H-TAT 2014)
(1) \( \{ -1, 1, 0, 1/2 \} \) (2) \( \{ -1, 1, 0 \} \) (3) \( \{ 1, 2 \} \) (4) \( \{ -1, 2 \} \)

43. \( \tan^{-1} \frac{a}{b} - \tan^{-1} \frac{a-b}{a+b} = \) (H-TAT 2014)
(1) \( \pi/3 \) (2) \( \pi/4 \) (3) \( \pi/6 \) (4) 0

44. If \( (\cos \theta + \sin \theta) = \sqrt{2} \cos \theta \) then \( \cos \theta - \sin \theta = \) (H-TAT 2014)
(1) \( \sqrt{2} \sin \theta \) (2) \( \frac{1}{\cos 2\theta} \) (3) \( -\frac{1}{\sin 2\theta} \) (4) None of these

45. If in a triangle ABC \( a^2 \cos^2\left(\frac{C}{2}\right) + c^2 \cos^2\left(\frac{A}{2}\right) = \frac{3b^2}{2} \) then the sides a, b and c: (TAT 2014)
(1) are in AP (2) are in GP (3) are in HP (4) None of these

46. The value of \( \cos 20^\circ \cos 40^\circ \cos 80^\circ \) is (Navodaya Dec. 2016)
(1) 1/2 (2) 1/8 (3) 1/16 (4) 1/32

47. If \( A + B = 90^\circ \) then the value of \( \frac{\tan A \tan B + \tan A \cot B}{\sin A \sec B} - \frac{\sin^2 B}{\cos^2 A} \) is: (Navodaya Dec. 2016)
(1) \( \sec A \) (2) \( \tan A \) (3) \( \cot A \) (4) \( \sec B \)

48. If \( \tan 7\theta \cdot \tan 2\theta = 1 \), then the value of \( \tan 3\theta \) is (Navodaya Dec. 2016)
(1) 1 (2) \( -\sqrt{3} \) (3) \( \sqrt{3} \) (4) \( 1/\sqrt{3} \)

49. If \( \sin(A + B) = 1 \) and \( \cos(A - B) = 1 \), where \( 0 < A + B \le 90^\circ \), then \( 2A - B \) is equal to (Navodaya Dec. 2016)
(1) 0° (2) 30° (3) 60° (4) 45°

50. If \( 3\sin\theta - 5\cos\theta = 3 \), then the value of \( 5\sin\theta + 3\cos\theta \) is (Navodaya Dec. 2016)
(1) \( \pm 8 \) (2) \( \pm 5 \) (3) \( \pm 3 \) (4) \( \pm 2 \)

51. The complete general solutions of the equation \( \sec^2 x = \sqrt{2}(1 - \tan x) \) are (H. TET 2016)
(1) \( x = n\pi \pm \frac{\pi}{6} \) (2) \( x = n\pi \pm \frac{\pi}{8} \) (3) \( x = n\pi \pm \frac{\pi}{4} \) (4) \( x = n\pi \)

52. The maximum value of \( \frac{1}{4} \sin \left(\theta + \frac{\pi}{4}\right) + 2 \cos \left(\theta - \frac{\pi}{4}\right); \theta \in R \) is : (H. TET 2016)
(1) 4 (2) 5 (3) 3 (4) 2

53. The value of \( \cos \left[ \tan^{-1} \left\{ \tan \left( \frac{15\pi}{4} \right) \right\} \right] \) is (H. TET 2016)
(1) 0 (2) \( -1/\sqrt{2} \) (3) 1 (4) \( 1/\sqrt{2} \)

54. \( \cos 9^\circ - \sin 9^\circ \) is equal to (H. TET 2016)
(1) \( \frac{\sqrt{5}-\sqrt{5}}{2} \) (2) \( \sqrt{5}-\sqrt{5} \) (3) \( \sqrt{5}+\sqrt{5} \) (4) \( \frac{\sqrt{5}+\sqrt{5}}{2} \)

55. The period of the function \( f(x) = \frac{\sin x + \sin 3x + \sin 5x + \sin 7x}{\cos x + \cos 3x + \cos 5x + \cos 7x} \) is (H. TET 2016)
(1) \( \pi/6 \) (2) \( \pi/3 \) (3) \( \pi/4 \) (4) \( \pi/2 \)

56. \( \sin \frac{31\pi}{3} \) is equal to (H. TET FEB. 2016)
(1) 1/2 (2) \( \sqrt{3} \) (3) \( \sqrt{3}/2 \) (4) 0

57. Value of 40°20' in Radian is (H. TET FEB. 2016)
(1) \( \frac{121\pi}{540} \) (2) \( \frac{540\pi}{121} \) (3) \( \frac{121}{540\pi} \) (4) \( \frac{540}{121\pi} \)

58. \( \cos 2x = \) (H. TET FEB. 2016)
(1) \( \frac{1-\tan^2 x}{1+\tan^2 x} \) (2) \( \frac{1+\tan^2 x}{1-\tan^2 x} \) (3) \( \frac{2\tan x}{1+\tan^2 x} \) (4) \( \frac{1-\tan^2 x}{2\tan x} \)

59. \( \frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \) (H. TET FEB. 2016)
(1) \( \cot x \) (2) \( \tan x \) (3) \( \sin x \) (4) \( \cot 7x \)

60. The greatest value of \( \sin \theta \cos \theta \) is : (K.V. 2017)
(1) 1/4 (2) 1 (3) 2 (4) 1/2

61. The number of solutions of the equation \( \tan^{-1} 2x + \tan^{-1} 3x = \pi/4 \) is: (K.V. 2017)
(1) 0 (2) 1 (3) 2 (4) 3

62. The value of \( \tan \left\{ \frac{1}{2} \cos^{-1}\left(\dots\right) + \frac{\pi}{4} \right\} \) is: (K.V. 2017)
(1) 7/17 (2) 5/12 (3) 17/12 (4) 17/7

63. If \( \tan^{-1} x > \cot^{-1} x \), the possible values of x are (K.V. 2017)
(1) any value (2) \( x > 1 \) (3) \( x < 1 \) (4) \( x = 1 \)

64. The general value of \( \theta \) satisfying \( \sin 3\theta + \cos 3\theta = 0 \) is (K.V. 2017)
(1) \( n\pi + (-1)^n \pi/4 \) (2) \( 2n\pi \pm \pi/4 \) (3) \( n\pi - \pi/4 \) (4) \( 2n\pi \pm \pi/6 \)

65. In a \( \Delta ABC \), if \( \frac{\cos B}{\sin A} = \frac{1}{2\sin C} \), then the triangle is (K.V. 2017)
(1) scalene triangle (2) acute angle triangle (3) equilateral triangle (4) an isosceles triangle

66. If \( \tan \theta + \cot \theta = \frac{4}{\sqrt{3}} \) where \( 0 < \theta < \pi/2 \), then \( \sin \theta + \cos \theta \) is equal to (RPSC II Grade 30 July, 2017)
(1) 1 (2) \( \frac{\sqrt{3}-1}{2} \) (3) \( \frac{\sqrt{3}+1}{2} \) (4) 2

67. If \( x + \frac{1}{x} = 2 \), then principal value of \( \sin^{-1} x + \cos^{-1} x \) is (RPSC II Grade 30 July, 2017)
(1) 0 (2) \( \pi \) (3) \( \pi/2 \) (4) \( \pi/4 \)

68. For a triangle ABC, with sides a, b, c the ratio of \( \sin(A - B) \) to \( \sin(A + B) \) is (RPSC II Grade 30 July, 2017)
(1) \( \frac{b^2+c^2}{a^2} \) (2) \( \frac{b^2-c^2}{bc} \) (3) \( \frac{a^2-b^2}{c^2} \) (4) \( \frac{a^2-b^2}{ab} \)

69. \( (1 - \sin A + \cos A)^2 \) equal to (RPSC II Grade 30 July, 2017)
(1) \( 2(1 - \cos A)(1 + \sin A) \) (2) \( 2(1 + \sin A)(1 + \cos A) \) (3) \( 2(1 - \cos A)(1 - \sin A) \) (4) None of these

70. If \( \cos^2 \theta - 3\cos \theta + 2 = \sin^2 \theta \) where \( 0 < \theta < \pi/2 \), then which of the following statements is/are correct? (RPSC II Grade 30 July, 2017)
(a) There are two values of \( \theta \) satisfying the above equation
(b) only \( \theta = 60^\circ \) is satisfied by the above equation
Select the correct answer.
(1) only 'a' (2) only 'b' (3) both 'a' & 'b' (4) neither 'a' nor 'b'

71. For a triangle ABC with sides a, b, c the product of \( \tan(A/2) \) and \( \tan(B/2) \) is (RPSC II Grade 30 July, 2017)
(1) \( \frac{b+c-a}{a+b+c} \) (2) \( \frac{c+a-b}{a+b+c} \) (3) \( \frac{a+b-c}{a+b+c} \) (4) \( \frac{c^2}{(a+b+c)^2} \)

72. Difference between the roots of the equation \( 8 \cos x = \operatorname{cosec} x \), \( (0 \le x \le 2\pi) \) is (RPSC II Grade 30 July, 2017)
(1) \( \pi/6 \) (2) \( \pi/4 \) (3) \( \pi/3 \) (4) \( \pi/2 \)

73. If \( \sin \theta = -4/5 \) and \( \theta \) lies in the third quadrant, then \( \cos(\theta/2) \) is equal to : (RPSC II Grade 2018)
(1) 1/5 (2) -1/5 (3) \( 2/\sqrt{5} \) (4) \( -2/\sqrt{5} \)

74. If \( \cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi \) then : (RPSC II Grade 2018)
(1) \( x^2 + y^2 = z^2 \) (2) \( x^2 + y^2 + z^2 = 0 \) (3) \( x^2 + y^2 + z^2 = 1 - 2xyz \) (4) None of these

75. If \( \tan \alpha = \frac{m}{m+1} \), and \( \tan \beta = \frac{1}{2m+1} \), then \( \alpha + \beta \) is equal to (RPSC II Grade 2018)
(1) \( \pi/4 \) (2) \( \pi/3 \) (3) \( \pi/6 \) (4) None of these

76. Which of the following number is rational ? (RPSC II Grade 2018)
(1) \( \sin 15^\circ \) (2) \( \cos 15^\circ \) (3) \( \sin 15^\circ \cos 15^\circ \) (4) \( \sin 15^\circ \cos 75^\circ \)

77. A kite is flying at an inclination of 60° with the horizontal plane. If the length of the thread is 120 m, then the height of the kite from the horizontal plane is : (RPSC II Grade 2018)
(1) \( 60\sqrt{3} \) m (2) 60 m (3) \( 60/\sqrt{3} \) m (4) 120 m

78. Each side of a square ABCD subtends an angle of 60° at the top of a tower of height h, standing at the center of the square. If a be the length of the side of square, then (K.V. Dec. 2018)
(1) \( 3a^2 = 2h^2 \) (2) \( 2a^2 = 3h^2 \) (3) \( 2h^2 = a^2 \) (4) \( h^2 = 2a^2 \)

79. In a triangle ABC, the lengths of two larger sides BC and AC are 10 and 9 respectively. If the angles are in A.P., then the length of third side can be: (K.V. Dec. 2018)
(1) \( 5+\sqrt{6} \) (2) \( 6+\sqrt{5} \) (3) \( 3\sqrt{3} \) (4) 5

80. If \( \sin \theta + \cos \theta = 1 \), then the value of \( \sin 2\theta \) is: (K.V. Dec. 2018)
(1) 1 (2) 1/2 (3) 0 (4) -1

81. In a \( \Delta ABC \), a, c, A are given and \( b_1, b_2 \) are two values of the third side b such that \( b_2 = 2b_1 \), then \( \sin A = \) (H-TAT 2019)
(1) \( \sqrt{\frac{9a^2+8c^2}{a^2}} \) (2) \( \sqrt{\frac{9a^2+8c^2}{c^2}} \) (3) \( \sqrt{\frac{9a^2-8c^2}{a^2}} \) (4) \( \sqrt{\frac{9a^2-8c^2}{c^2}} \)

82. A tree is broken by wind, its upper part touches the ground at a point 10 metre from the foot of the tree and makes at angle of 45° with the ground. The entire length of the tree is (H-TAT 2019)
(1) \( 10(1 + \sqrt{2}) \) meters (2) 20 meters (3) 30 meters (4) \( 10(1 + \sqrt{2}) \) meters

83. If \( \sin^{-1} x = \pi/5 \) for some \( x \in (-1, 1) \), then the value of \( \cos^{-1} x \) is: (H-TAT 2019)
(1) \( 9\pi/10 \) (2) \( 7\pi/10 \) (3) \( 3\pi/10 \) (4) \( \pi/10 \)

84. Number of solutions of the equation \( \tan x + \sec x = 2 \cos x \), lying in the interval \( [0, 2\pi] \) is: (H-TAT 2019)
(1) One (2) Two (3) Three (4) Four

85. What is the angle of elevation of sun when the length of the shadow of a pole is \( \sqrt{3} \) times the height of the pole? (RPSC II Grade Sansktri Dept. 2019)
(1) 45° (2) 105° (3) 60° (4) 30°

86. The most general value of \( \theta \) which satisfies both of the equations \( \sin \theta = -1/2 \) and \( \tan \theta = 1/\sqrt{3} \) is (RPSC II Grade Sansktri Dept. 2019)
(1) \( n\pi + \frac{7\pi}{6} \) (2) \( n\pi - \frac{7\pi}{6} \) (3) \( 2n\pi + \frac{7\pi}{6} \) (4) \( 2n\pi - \frac{7\pi}{6} \)

87. Solution of equation \( \tan 5\theta = \cot 2\theta \) (RPSC II Grade Sansktri Dept. 2019)
(1) \( \theta = \frac{\pi}{7} \left(n + \frac{1}{2}\right) \) (2) \( \theta = \frac{\pi}{7} \left(n - \frac{1}{2}\right) \) (3) \( \theta = \frac{\pi}{3} \left(n + \frac{1}{2}\right) \) (4) \( \theta = \frac{\pi}{3} \left(n - \frac{1}{2}\right) \)

88. \( \sin^{-1}\left(\frac{12}{13}\right) - \cos^{-1}\left(\frac{5}{13}\right) \) equals (RPSC II Grade Sansktri Dept. 2019)
(1) 0 (2) \( \sin^{-1}\left(\frac{16}{65}\right) \) (3) \( \sin^{-1}\left(\frac{56}{65}\right) \) (4) 1

89. The diameter of a wheel is 28 cm; through what distance does its centre move during one revolution of the wheel along the ground? (RPSC II Grade Sansktri Dept. 2019)
(1) 44 cm (2) \( (28/\pi) \) cm (3) 88 cm (4) 176 cm

90. \( \{(\cos 45^\circ) (\cos 60^\circ) - (\sin 45^\circ) (\sin 60^\circ)\} \) equals (RPSC II Grade Sansktri Dept. 2019)
(1) \( \frac{\sqrt{3}+1}{2\sqrt{2}} \) (2) \( \frac{1-\sqrt{3}}{2\sqrt{2}} \) (3) \( \frac{-(\sqrt{3}-1)}{2\sqrt{2}} \) (4) \( \frac{\sqrt{3}-1}{2\sqrt{2}} \)

91. Which of the following is correct? (Letters have their usual meaning in plane trigonometry) (RPSC II Grade Sansktri Dept. 2019)
(1) \( r = \frac{S}{s-a} \) (2) All the three wrong (3) \( a = 2R \sin A \) (4) \( R = \frac{abc}{2S} \)

92. For \( \sin[\cot^{-1} (x + 1)] = \cos [\tan^{-1} (x)] \), the value of x is (NVS (PGT) JUNE, 2019)
(1) 1 (2) 0 (3) -1/2 (4) 1/2

93. If \( \tan(A - B) = \frac{1}{\sqrt{3}} \) and \( 2\tan(A + B) = 3 \) then the value of A and B will be (NVS (PGT) JUNE, 2019)
(1) 30°, 30° (2) 30°, 60° (3) 40°, 20° (4) 45°, 15°

94. If \( \sin \alpha + \sin \beta + \sin \gamma = 0 \) and \( \cos \alpha + \cos \beta + \cos \gamma = 0 \), the value will be (NVS (PGT) JUNE, 2019)
(1) -3/2 (2) 0 (3) -1/2 (4) -1

95. A person walking on a straight road observes at two points 1 km apart, the angles of elevation of a pole in front of him are 30° and 75°. The height of the pole is (NVS (PGT) JUNE, 2019)
(1) \( 250(\sqrt{2}-1) \)m (2) \( 250(\sqrt{3}+1) \)m (3) \( 250(\sqrt{2}+1) \)m (4) \( 250(\sqrt{3}-1) \)m

96. If \( \sin A + \cos A = 1 \), then the value of \( \sin 2A \) is : (MP TET 2019)
(1) -1 (2) 0 (3) 2 (4) 1

97. A flag staff 5m high stands on a building 25 cm high. At an observer at a height of 30 m, the flag staff and the building subtend equal angles. The distance of the observer from the top of the flag staff is : (MP TET 2019)
(1) \( \frac{5\sqrt{2}}{3} \) (2) \( \frac{\sqrt{2}}{3} \) (3) \( \frac{2}{3} \) (4) \( \sqrt{\frac{2}{3}} \)

98. If \( \sin x + \cos x = 0 \), then \( \sin x = \) (MP TET 2019)
(1) \( \frac{\sqrt{5}-1}{2} \) (2) \( \frac{-\sqrt{5}-1}{2} \) (3) \( \frac{-\sqrt{5}+1}{2} \) (4) \( \frac{\sqrt{5}+1}{2} \)

99. If \( \sin^{-1}\left(\frac{5}{13}\right) + \sin^{-1}\left(\frac{13}{x}\right) = \sin^{-1}(1) \) then x = (MP TET 2019)
(1) 56/65 (2) 16/65 (3) 20/65 (4) 64/65

100. In triangle ABC, \( 2ac \sin\left(\frac{A - B + C}{2}\right) \) is equal to (Chandigarh (TGT) 2019)
(1) \( a^2 + b^2 + c^2 \) (2) \( a^2 + b^2 - c^2 \) (3) \( b^2 - c^2 - a^2 \) (4) \( c^2 - b^2 - a^2 \)

101. Find the solution of \( \tan^{-1}(2x) + \tan^{-1}(3x) = \pi/4 \) (NVS PGT 2019)
(1) 2 (2) -1 (3) 1 (4) 1/6

102. The value of \( \tan^{-1} (1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{2}\right) \) is: (NVS PGT 2019)
(1) \( 3\pi/4 \) (2) \( 3\pi/2 \) (3) \( \pi/4 \) (4) \( \pi/2 \)

103. Find the value of \( \sin(n + 1)x \sin(n + 2)x + \cos(n + 1)x \cos(n + 2)x \) (NVS PGT 2019)
(1) \( \cos x \) (2) \( \cos(x/2) \) (3) \( \cos 2x \) (4) \( \cos 3x \)

104. If \( \sin[\sin^{-1}(1/5) + \cos^{-1}(x)] = 1 \), then the value of x is: (NVS PGT 2019)
(1) 1/2 (2) 1/4 (3) 1 (4) 1/5

105. If \( \cos^{-1} x + \cos^{-1} y + \cos^{-1} z = \pi \), then : (H-TET 2019)
(1) \( x^2 + y^2 + z^2 + 2xyz = 1 \) (2) \( (\sin^{-1} x + \sin^{-1}y, \sin^{-1} z) = \cos^{-1}x + \cos^{-1}y + \cos^{-1}z \) (3) \( xy + yz + zx = x +y + z -1 \) (4) \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \ge 6 \)

106. If \( 0 \le x \le 3\pi \), then the number of distinct values of x, which satisfy the equation \( \sec x + \tan x = 3 \) is: (H-TET 2019)
(1) 1 (2) 2 (3) 3 (4) 4

107. Value of \( \tan \left[ \frac{\pi}{4} - \frac{1}{2} \cos^{-1}\left(\frac{2}{7}\right) \right] \) is (H-TET 2019)
(1) \( \frac{3}{\sqrt{5}} \) (2) \( 2/3 \) (3) \( 1/\sqrt{5} \) (4) \( 4/5 \)

108. Range of \( f(x) = \frac{1}{\sqrt{2}} \sin \left( \frac{x^2+1}{x^2+2} \right) \) is (H-TET 2019)
(1) \( [0, \pi/2] \) (2) \( (0, \pi/6) \) (3) \( [\pi/6, \pi/2) \) (4) None of these

109. If \( \cos(\theta+\phi) = m \cos (\theta-\phi) \), then \( \cot \left( \frac{m-1}{m+1} \tan \phi \right) \) is equal to (H-TET 2019)
(1) \( \tan \theta \) (2) \( -\tan \theta \) (3) \( 2 \tan \theta \) (4) None of these

110. If \( \sin \theta + \cos \theta = \sqrt{2} \sin(90^\circ - \theta) \) then value of \( \cot \theta \) is (ACF Feb. 2021)
(1) \( \sqrt{2} \) (2) 2 (3) \( \sqrt{2}+1 \) (4) \(\sqrt{2}-1\)

111. If \( \tan \theta + \sin \theta = m \) and \( \tan \theta - \sin \theta = n \) then value of \( (m^2 - n^2) \) is (ACF Feb. 2021)
(1) \( \frac{1}{2}\sqrt{mn} \) (2) \( 2\sqrt{mn} \) (3) \( 4\sqrt{mn} \) (4) \( mn \)

112. If \( \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2+\sqrt{3}}{2-\sqrt{3}} \), then the value of \( \theta \) in circular system will be (ACF Feb. 2021)
(1) \( \pi/3 \) (2) \( \pi/6 \) (3) \( \pi/4 \) (4) \( \pi/12 \)

113. From a tower 120 metres high, the angles of depression of two objects, which are in horizontal line through the base of the tower, are 45° and 30° and they are on the same side of the tower. The distance (in metres) between the objects is (ACF Feb. 2021)
(1) \( 120\sqrt{3} \) (2) \( 120(\sqrt{3}+1) \) (3) \( 120(\sqrt{3}-1) \) (4) \( 120(\sqrt{3}-1) \)

114. \( \frac{\cot \theta + \tan \theta}{\cot \theta - \tan \theta} - \frac{\tan \theta + \tan 3\theta}{\tan 3\theta - \tan \theta} \) is equal to (ACF Feb. 2021)
(1) 1 (2) -1 (3) 2 (4) 0

115. If \( \sin(A - B) = 1/2, \cos(A + B) = 1/2 \), \( 0^\circ < A + B \le 90^\circ, A > B \), then A and B (ACF Feb. 2021)
(1) 35° and 25° (2) 45° and 15° (3) 45° and 25° (4) 25° and 15°

116. A circus artist is climbing a tight rope of which upper end is tied to the top most point of a pillar of height 14 metre and the other end is tied to a peg on the ground. In this position the rope makes an angle 30° with the ground, then the length of the rope is (ACF Feb. 2021)
(1) 31 metres (2) 35 metres (3) 38 metres (4) 28 metres

117. If A, B, C are internal angles of a triangle, then \( \sin 2A + \sin 2B - \sin 2C \) equals to (ACF Feb. 2021)
(1) \( 4 \sin A \cos B \cos C \) (2) \( 4 \cos A \sin B \sin C \) (3) \( 4 \cos A \cos B \cos C \) (4) \( \sin A \sin B \sin C \)

118. If \( X = \cot \theta + \cos \theta \) and \( Y = \cot \theta - \cos \theta \), then \( \frac{X^2-Y^2}{XY} \) is [DSSB 2021]
(1) 0 (2) 2 (3) 1 (4) 4

119. \( \frac{\sec 8A (\tan 10A + \tan 6A)}{4(\tan 10A - \tan 6A)} \) find the value: [DSSB 2021]
(1) \( \sin 2A \) (2) \( \sin 4A \) (3) \( \cos 4A \) (4) \( \tan 4A \)

120. Value of \( \cot \left(7\frac{1}{2}^\circ\right) \) is [DSSB 2021]
(1) \( \sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{6} \) (2) \( \sqrt{3}+\sqrt{2}+\sqrt{3}+\sqrt{6} \) (3) \( \sqrt{2}+\sqrt{2}+\sqrt{3}+\sqrt{6} \) (4) \( \sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{6} \)

121. \( 3\sin 10^\circ \) is equal to [DSSB 2021]
(1) \( \sin 20^\circ + \sin 40^\circ \) (2) \( \cos 50^\circ + \cos 70^\circ \) (3) \( \cos 50^\circ - \cos 70^\circ \) (4) \( \sin 70^\circ + \sin 50^\circ \)

122. If \( \cos^{-1} \frac{x}{a} + \cos^{-1} \frac{y}{b} = \alpha \), then \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - k = \sin^2 \alpha \), then k is equal to [DSSB 2021]
(1) \( \frac{2xy}{ab} \cos \alpha \) (2) \( \frac{xy}{ab} \) (3) \( -\frac{2xy}{ab} \) (4) \( \frac{2xy}{ab} \)

123. If \( \operatorname{cosec} \alpha + \cot \alpha = 2 + \sqrt{5} \), then \( \cos \alpha \) value is [DSSB 2021]
(1) 2/5 (2) 1/3 (3) 3 (4) 5/2

124. \( \sin 6^\circ \cdot \sin 66^\circ \) is [DSSB 2021]
(1) \( -\frac{3\sqrt{5}}{4} \) (2) \( \frac{\sqrt{5}-1}{8} \) (3) \( \frac{\sqrt{5}-1}{4} \) (4) \( -\frac{3\sqrt{5}}{8} \)

125. If \( \sin 3A = \cos(A - 30^\circ) \) where 3A and \( (A - 30) \) are acute angles, find \( (\sin 2A + \cos 2A) \) [DSSB 2021]
(1) \( \frac{\sqrt{3}-1}{2} \) (2) \( \frac{\sqrt{3}+1}{3} \) (3) \( \frac{\sqrt{3}-1}{3} \) (4) \( \frac{\sqrt{3}+1}{2} \)

126. Find \( \frac{3\cot 2A + 3\cot^2 A \operatorname{cosec} 2A}{6\operatorname{cosec}^2 2A(\cos A - \sin A)} \) [DSSB 2021]
(1) 1 (2) 2 (3) 3 (4) 4

127. The minimum value of \( 4 \cos \theta + 3 \) is [UP TGT 21]
(1) -3 (2) -1 (3) 0 (4) 1

128. In a \( \Delta ABC \), \( b = 5 \) cm, \( a = 2 \) cm and \( \sin A = 3/7 \). How many such triangles are possible [UP TGT 21]
(1) 0 (2) 1 (3) 2 (4) 3

129. What is the principal value of \( \sin^{-1}\left(\sin \frac{2\pi}{3}\right) \)? [UP TGT .21]
(1) \( \pi/4 \) (2) \( \pi/2 \) (3) \( \pi/3 \) (4) \( 2\pi/3 \)

130. In the equation \( \cos^{-1}\left(\frac{1-a^2}{1+a^2}\right) - \cos^{-1}\left(\frac{1-b^2}{1+b^2}\right) = 2 \tan^{-1} x \) value of x is [UP TGT .21]
(1) \( \frac{a+b}{1+ab} \) (2) \( \frac{a+b}{1-ab} \) (3) \( \frac{a-b}{1-ab} \) (4) None of these

131. In \( \Delta ABC \), \( a = 2b \) and \( |A - B| = \pi/3 \), then \( \angle C \) is [UP TGT 21]
(1) \( \pi/2 \) (2) \( \pi/3 \) (3) \( \pi/6 \) (4) \( \pi/4 \)

132. The sides of a triangle are 15 cm, 20 cm and 25 cm respectively, then the radius of in-circle is [UP TGT 21]
(1) 10 cm (2) 12.5 cm (3) 5 cm (4) 7.5 cm

133. In a triangle ABC if \( \cos A = \frac{\sin B}{2\sin C} \) then triangle is [UP TGT 21]
(1) Isosceles (2) Equilateral (3) Right angled (4) None of the above

134. \( \sin^{-1}\left(\frac{3}{5}\right) - \cos^{-1}\left(\frac{12}{13}\right) \) equals to [UP TGT 08.08.21]
(1) \( \sin^{-1}\left(\frac{56}{65}\right) \) (2) \( \sin^{-1}\left(\frac{16}{65}\right) \) (3) 1 (4) 0

135. \( \sqrt{2 + \sqrt{2 + \sqrt{2 + 2\cos 8\theta}}} \) is equal to [UP TGT 21]
(1) \( 2 \sin \theta \) (2) \( 2 \cos \theta \) (3) \( \sin 2\theta \) (4) \( \cos 2\theta \)

136. Sum of max and min values of \( 4(\sin^2\theta + \cos^4\theta) \) is [UP TGT 21]
(1) 3 (2) 4 (3) 5 (4) 7

Answer Key (Trigonometry) Sr. Teacher Grade II

Qus. 123456789101112131415
Ans 322314214332222
Qus. 161718192021222324252627282930
Ans 122312423124342
Qus. 313233343536373839404142434445
Ans 314234314212211
Qus. 464748495051525354555657585960
Ans 22442141321114
Qus. 616263646566676869707172737475
Ans 342343334232231
Qus. 767778798081828384858687888990
Ans 31413233
Qus. 919293949596979899100101102103104105
Ans 2341221112114
Qus. 106107108109110111112113114115116117118119120
Ans 433
Qus. 121122123124125126127128129130131132133134135
Ans 31444213223122
Qus. 136
Ans 4

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