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CHAPTER PRACTICE PROBLEMS

Subject: Mathematics Topic: Trigonometry

1. If tanγ = secα secβ + tanα tanβ, then cos2γ is necessarily

(a) ≥ 0 (b) ≤ 0 (c) < 0 (d) > 0

2. Minimum value of f (x ) = sin x + cos x + tan x + cot x + sec x + cosec x is

(a) 6 (b) 10 (c) 2 (d) none of these

2 1
3. If cosec x = , cot x = − , then cos x + cos 2 x + cos 3 x + ...... + cos 100 x is
3 3
equal to
1 1 3
(a) (b) − (c) (d) none of these
2 2 2

nπ x x
4. If x= , satisfies the equation sin − cos = 1 − sin x and the inequality
2 2 2
x π 3π
− ≤ , then
2 2 4
(a) n = −1, 0, 3, (b) n = 1, 2, 4, 5 (c) n = 0, 2, 4 (d) n = −1, 1, 3, 5

10
πr
5. The value of  cos
r =0
3
3
is equal to

9 7 9 1
(a) − (b) − (c) − (d) −
2 2 8 8

7  π x
6. If sin x + cos x = where x ∈ 0,  , then tan is equal to
2  4  2
3− 7 7 −2 4− 7
(a) (b) (c) (d) none of these
3 3 4

7. If e sin x − e − sin x − a = 0 has atleast one real solution, then

 e2 − 1   e 2 − 1
(a) | a |∈  , ∞  (b) | a |∈ 0,  (c) | a |∈ (e, ∞ ) (d) | a |∈ [0, e]
 e   e 

8. The general solution of the equation sin100 x − cos100 x = 1 is

π π π
(a) 2nπ + , n ∈ I (b) nπ + , n ∈ I (c) 2nπ − , n ∈ I (d) none of these
2 2 2

π π n
9. Let n be a positive real such that sin + cos = , then
2n 2n 2
(a) n ∈ [3, 5] (b) n ∈ (1, 4] (c) n ∈ [1, 4) (d) n ∈ (4, 8 )

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 x
10. The number of solutions of the equation = sin x is
100
(a) 0 (b) 33 (c) 63 (d) none of these

11. The number of pairs (x, y) satisfying the equations sin x + sin y = sin( x + y ) and
| x | + | y |= 1 is
(a) 2 (b) 4 (c) 6 (d) infinite

π 8π 
12. The solution of the trigonometric equation cos 2  cos x −  = 1, must be
3 3 
1
(a) cos −1 (3k + 8) (b) cos −1 (3k − 8) (c) 2nπ (d) none of these
3

13. The number of real solutions of sin e x . cos e x = 2 x −2 + 2 − x −2 is

(a) zero (b) one (c) two (d) infinite

14. If sin x = cos y , 6 sin y = tan z and 2 sin z = 3 cos x; u, v , w denote respectively
sin2 x, sin 2 y , sin2 z then the value of the triplet (u, v , w ) is
1 1 3  1 3 1
(a) (1, 0, 0) (b) (0, 1, 0) (c)  , ,  (d)  , , 
2 2 4 2 4 2

15. If [sin x ] + [ 2 cos x ] = −3, x ∈ [0, 2π]([.] denotes the greatest integer function), then x
belongs to
 5π   5π   5π   5π 
(a)  π,  (b) π,  (c)  ,2π  (d)  ,2π
 4   4  4  4 

16. Values of x and y satisfying the equation

sin 7 y =| x 3 − x 2 − 9 x + 9 | + | x 3 − x 2 − 4 x + 4 | + sec 2 2y + cos 4 y are
π
(a) x = 1, y = nπ, n ∈ I (b) x = 1, y = 2nπ + , n ∈ I
2
(c) x = 1, y = 2nπ, n ∈ I (d) none of the above

2
17. The equation ( cos P − 1) x 2 + ( cos P ) x + sin P = 0, where x is a variable, has real
roots. Then the interval of P may be
 −π π 
(a) ( 0, 2π ) (b) ( −π , 0 ) (c)  ,  (d) ( 0, π )
 2 2

 π
18. The set of all values of x in the interval 0, 2  for which

2 sin 2 x − 3sin x + 1 ≥ 0 contains

 π  π  2π   2π 5π 
(a) 0,  (b) 0,  (c)  , π  (d)  , 
 6  3  3   3 6 

19. The number of distinct solutions of sec θ + tan θ = 3, 0 ≤ θ ≤ 3π

(a) 3 (b) 5 (c) 4 (d) None of these

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20. Let α , β be any two positive values of x for which 2 cos x, cos x and 1 − 3cos 2 x are
in G.P. then the maximum value of α − β is
π π π
(a) (b) (c) (d) None of these
3 4 2

21. The number of solutions of cos x = sin x, 0 ≤ x ≤ 4π

(a) 8 (b) 4 (c) 2 (d) None of these

 πx  2
22. The number of solutions of the equation sin   = x − 2 3x + 4
2 3
(a) Is zero (b) Is only one
(c) Is only two (d) Is greater than 2

23. The equation 4 sin 2 x + 4sin x + a 2 − 3 = 0 possesses a solution it a belongs to the

interval
(a) ( −1,3) (b) ( −3,1) (c) ( −2, 2 ) (d) R − ( −2, 2 )

24. If sin x + cos x = sin x + cos x , then x belongs to the quadrant

(a) I or III (b) II or IV (c) I or II (d) III or IV

25. The smallest positive root of the equation tan x − x = 0 lies in

 π π   3π   3π 
(a)  0,  (b)  , π  (c)  π ,  (d)  , 2π 
 2 2   2   2 

 π
26. The number of distinct solutions of sin 5θ ⋅ cos 3θ = sin 9θ ⋅ cos 7θ in 0,  is
 2
(a) 4 (b) 5 (c) 8 (d) 9

27. The equation p cos x − q sin x = r admits of a solution for x only if

(a) r < Max { p, q} (b) − p 2 + q 2 ≤ r ≤ p2 + q2
(c) r 2 = p 2 + q 2 (d) None of these

28. The equation k cos x − 3sin x = k + 1 is solvable only if k belongs to the interval
(a) [ 4, + ∞ ] (b) [ −4, 4] (c) ( −∞, 4] (d) None of these

29. The number of solutions of cos θ + 3 sinθ = 5, 0 ≤ θ ≤ 5π , is

(a) 4 (b) 0 (c) 5 (d) None of these

30. The number of values of x in [ 0, 5π ] satisfying the equation

3cos 2 x − 10 cos x + 7 = 0 is
(a) 5 (b) 6 (c) 8 (d) 10

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31. The smallest positive integral value of P for which the equation
cos ( P sin x ) = sin ( P cos x ) in x has a solution in [ 0, 2π ] is
(a) 2 (b) 1 (c) 3 (d) None of these

32. Maximum value of the expression 2 sin x + 4 cos x + 3 is

(a) 2 5 + 3 (b) 2 5 − 3 (c) 5 + 3 (d) None of these

33. If sin A = sin B and cos A = cos B , then

(a) A = B + nπ , n ∈ I (b) A = B − nπ , n ∈ I
(c) A = 2nπ + B, n ∈ I (d) A = nπ − B, n ∈ I

34. If cos 4 x + a cos 2 x + 1 = 0 has at least one solution, then

(a) a ∈ [ 2, ∞ ) (b) a ∈ [ −2, 2]
(c) a ∈ ( −∞, − 2] (d) a ∈ R − ( −2, 2 )

35. Total number of solutions of the equation sin 2 x + cos 2 x = sin x.cos x in [0, 2 π ] is
(a) 2 (b) 4 (c) 6 (d) None of these

36. Total number of integral values of n such that sin x ( sin x + cos x ) = n has at least one
real solution is
(a) 2 (b) 1 (c) 3 (d) Zero

π 
37. If tan (π cos x ) = cot (π sin x ) , then cos  − x  is equal to
4 
1 1 1
(a) ± (b) ±1 (c) ± (d) ±
2 2 2 2

38. If the equation x 2 + 4 + 3sin ( ax + b ) − 2 x = 0 has atleast one real solution where a, b
∈ [0, 2 π ], then one possible value of (a + b) can be equal to
7π 5π 9π
(a) (b) (c) (d) None of these
2 2 2

39. If sin 2 x − 2sin x − 1 = 0 has exactly four different solutions in x ∈ [ 0, nπ ] then

minimum value of n can be, (n ∈ N)
(a) 2 (b) 3 (c) 4 (d) 5

40. x1 and x2 are two positive values of x for which 2 cos x, cos x and 3sin 2 x − 2 are ( )
in geometric sequence. One possible value of x1 − x2 can be equal to
2π π 2 2
(a) (b) (c) 2 cos −1   (d) cos −1  
3 3 3 3

41. If x, y ∈ [ 0, 2π ] , then total number of ordered pairs (x, y) satisfying the equation
sin x.cos y = 1 , is equal to
(a) 1 (b) 3 (c) 5 (d) 7

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 1
42. Total number of solutions of cot x = cot x + , x ∈ [ 0, 3π ] is equal to
sin x
(a) 1 (b) 2 (c) 3 (d) Zero

 π x2 
43. Total number of ordered pairs (x, y) satisfying x + y = 2, sin   = 1 ; is equal to
 3 
(a) 2 (b) 3 (c) 4 (d) 6

sin x − sin x
44. e +e + 4a = 0 will exactly four different solutions in [0, 2 π ], if
 e 1  −1 − e 2 
(a) a ∈ R (b) a ∈  − , −  (c) a ∈  , ∞  (d) None of these
 4 4  4e 

45. Complete set of values of x in (0, π ) satisfying 1 + log 2 sin x + log 2 sin 3 x ≥ 0 , is
 2π 3π  π 2π 
(a)  , (b)  , 
 3 4  3 3 
 π   2π  π 2π 
(c)  0,  ∪  , π (d)  , 
 2  3  2 3 

2
46. If tan ( A − B ) = 1 and sec ( A + B ) = − , then smallest positive values of A and B
3
respectively are
25π 19π 19π 25π 31π 13π 13π 31π
(a) , (b) , (c) , (d) ,
24 24 24 24 24 24 24 24

5 1  π
47. If tan α = and tan β = , then  0 < α , β < 
6 11  2
π π π
(a) α + β = (b) α + β = (c) α + β = (d) None of these
6 4 3

π
48. If tan x tan y = a and x + y = , then tan x and tan y satisfy the equation
6
(a) x 2 − 3 (1 − a ) x + a = 0 (b) 3 x 2 − (1 − a ) x + a 3 = 0
(c) x 2 + 3 (1 − a ) x − a = 0 (d) 3 x 2 + (1 + a ) x − a 3 = 0

49. The number of integral values of k for which the equation 7 cos x + 5sin x = 2k + 1
has a solution is
(a) 4 (b) 8 (c) 10 (d) 12

7 π α
50. If sin α + cos α = , 0 < α < , then tan is
2 6 2
1 1
(a) 7 −2 (b)
3
(7 −2 )(c) 2 − 7 (d)
3
(2− 7 )

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51. If α + β = 90°, then the maximum value of sin α .sin β is
1 3 3
(a) 1 (b) (c) (d)
2 2 4

52. The value of tan 40° + 2 tan10° is

(a) cot 50° (b) cot 40° (c) cot10° (d) None of these

ANSWERS
1. B 2. D 3. B 4. B
5. C 6. B 7. B 8. B
9. D 10. C 11. C 12. D
13. A 14. A 15. A 16. B
17. D 18. A 19. D 20. D
21. B 22. B 23. C 24. A
25. C 26. D 27. B 28. C
29. B 30. C 31. A 32. A
33. C 34. C 35. D 36. A
37. C 38. A 39. C 40. C
41. B 42. B 43. C 44. D
45. A 46. A 47. B 48. B
49. B 50. B 51. B 52. B

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