MATHEMATICS:

1. In which quadrant 740° lies:

a) 1st b) 2nd c) 3rd d) 4th
2. Find another positive angle whose initial and final positions are same as that of −135°.
a) 210˚ b) 225˚ c) 240˚ d) All of these
3. If θ lies in second quadrant, in which quadrant –θ will lie?
a) 1st b) 2nd c) 3rd d) Both b and c
4. If θ lies in second quadrant, in which quadrant 2θ will lie?
a) 1st b) 2nd c) 3rd d) Both b and c
5. Convert angle 40˚37’30’’ into radian measurement.
65𝜋 75𝜋 65𝜋 75𝜋
a) b) c) d)
288 288 298 298
6. Find the length of an arc of a circle of diameter 20 cm which subtends an angle of 45˚ at the centre of the circle.
a) 5π b) 5π/2 c) 3π/2 d) 3π
7. Find the radius of the circle in which a central angle of 45˚ makes an arc of length 187 cm.
a) 310 cm b) 210 cm c) 238 cm d) 283 cm
8. If 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 = √2𝑐𝑜𝑠𝜃 then 𝑐𝑜𝑠𝜃 − 𝑠𝑖𝑛𝜃 =
a) √2𝑐𝑜𝑠𝜃 b) √2𝑠𝑖𝑛𝜃 c) 𝑐𝑜𝑠𝜃 d) 𝑠𝑖𝑛𝜃
2𝜋
9. Convert radians into degree measure.
15
a) 42˚ b) 45˚ c) 24˚ d) 35˚
𝜋
10. Convert radians into degree measure.
8
a) 22˚30’ c) 20˚10’ c) 28˚50’ d) 12˚30’
4𝜋
11. The sign of 𝑡𝑎𝑛 is:
3
a) Positive b) Negative
7𝜋
12. The sign of 𝑐𝑜𝑠 (− ) is:
3
a) Positive b) Negative
3 𝑠𝑒𝑐𝑥−𝑡𝑎𝑛𝑥
13. If 𝑠𝑖𝑛𝑥 = then find the value of .
5 𝑠𝑒𝑐𝑥+𝑡𝑎𝑛𝑥
a) ½ b) 1/8 c) ¼ d) 3/4
5
14. If 𝑡𝑎𝑛𝜃 = − and θ lies in the second quadrant then the value of 𝑠𝑖𝑛𝜃 =
12
a) 5/13 b) 4/13 c) -5/13 d) -13/5
15. Find the value of 𝑠𝑖𝑛30°𝑐𝑜𝑠0° + 𝑠𝑖𝑛45°𝑐𝑜𝑠45° + 𝑠𝑖𝑛60°𝑐𝑜𝑠30°.
a) ¼ b) 5/4 c) 7/4 f) 3/4
𝜋 𝜋 𝜋 𝜋
16. Find the value of 𝑡𝑎𝑛2 + 2𝑐𝑜𝑠 2 + 3𝑠𝑒𝑐 2 + 4𝑐𝑜𝑠 2 .
3 4 6 2
a) 4 b) 8 c) 16 d) 5
17. 𝑠𝑖𝑛4 𝜃 − 𝑐𝑜𝑠 4 𝜃 =
a) 1 − 2𝑐𝑜𝑠 2 𝜃 b) 2𝑠𝑖𝑛2 𝜃 − 1 c) 𝑠𝑖𝑛2 𝜃 − 𝑐𝑜𝑠 2 𝜃 d) All of these
2 1
18. 𝑐𝑜𝑡 𝜃 − 2 + 1 =
𝑠𝑖𝑛 𝜃
a) 0 b) 1 c) -1 d) 𝑐𝑜𝑠𝑒𝑐𝜃
19. (1 + 𝑡𝑎𝑛2 𝜃)(1 − 𝑠𝑖𝑛𝜃)(1 + 𝑠𝑖𝑛𝜃) =
a) 0 b) 1 c) -1 d) 𝑠𝑒𝑐𝜃
𝑐𝑜𝑠𝜃 1−𝑠𝑖𝑛𝜃
20. + =
1−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
a) 𝑠𝑖𝑛𝜃 b) 𝑐𝑜𝑠𝜃 c) 𝑠𝑒𝑐𝜃 d) 𝑡𝑎𝑛𝜃
21. tan 105 ° =
a) – (2 + √3) b) – (2 − √3) c) – (3 + √2) d) – (3 + √2)
22. 𝑠𝑖𝑛50°𝑐𝑜𝑠10° + 𝑐𝑜𝑠50°𝑠𝑖𝑛10° =
1 √3 1
a) b) c) 1 d)
2 2 √2
23. 𝑐𝑜𝑠70°𝑐𝑜𝑠10° + 𝑠𝑖𝑛70°𝑠𝑖𝑛10° =
1 √3 1
a) b) c) 1 d)
2 2 √2
24. Covert 2𝑠𝑖𝑛5𝜃. 𝑐𝑜𝑠3𝜃 as the sum.
a) 𝑠𝑖𝑛8𝜃 + 𝑠𝑖𝑛4𝜃 b) 𝑐𝑜𝑠8𝜃 + 𝑐𝑜𝑠4𝜃 c) 𝑐𝑜𝑠8𝜃 + 𝑐𝑜𝑠2𝜃 d) 𝑠𝑖𝑛8𝜃 + 𝑠𝑖𝑛2𝜃
25. 𝑐𝑜𝑠75°. 𝑐𝑜𝑠15° =
a) ½ b) ¾ c) ¼ d) 0
cos(𝜋+𝜃)cos⁡(−𝜃)
26. 𝜋 =
sin(𝜋−𝜃)𝑐𝑜𝑠( +𝜃)
2
a) 𝑡𝑎𝑛2 𝑥 b) 𝑐𝑜𝑡 2 𝑥 c) −𝑡𝑎𝑛2 𝑥 d) −𝑐𝑜𝑡 2 𝑥
cos(90°+𝜃) sec(270°+𝜃)⁡sin⁡(180°+𝜃)
27. =
cosec(−𝜃)cos⁡(270°−𝜃)tan⁡(180°+𝜃)
a) 𝑠𝑖𝑛𝜃 b) −𝑠𝑖𝑛𝜃 c) 𝑐𝑜𝑠𝜃 d) −𝑐𝑜𝑠𝜃
𝑡𝑎𝑛69°+𝑡𝑎𝑛66°
28. =
1−𝑡𝑎𝑛69°𝑡𝑎𝑛66°
a) 1 b) -1 c) 0 d) ∞
29. Convert 𝑐𝑜𝑠5𝛼 + 𝑐𝑜𝑠3𝛼 as product.
a) 2𝑐𝑜𝑠4𝛼⁡𝑐𝑜𝑠𝛼 b) 𝑐𝑜𝑠4𝛼⁡𝑐𝑜𝑠𝛼 c) 2𝑠𝑖𝑛4𝛼⁡𝑠𝑖𝑛𝛼 d) 𝑠𝑖𝑛4𝛼⁡𝑠𝑖𝑛𝛼
 𝑠𝑖𝑛𝛼+𝑠𝑖𝑛3𝛼
30. 𝑐𝑜𝑠𝛼+𝑐𝑜𝑠3𝛼 =
a) 𝑡𝑎𝑛𝛼 b) 𝑡𝑎𝑛2𝛼 c) 𝑡𝑎𝑛3𝛼 d) −𝑡𝑎𝑛𝛼
31. 𝑐𝑜𝑠3𝜃 =
a) 3𝑐𝑜𝑠 3 𝜃 − 4𝑐𝑜𝑠𝜃 b) 3𝑐𝑜𝑠 3 𝜃 − 𝑐𝑜𝑠𝜃 c) 4𝑐𝑜𝑠 3 𝜃 + 3𝑐𝑜𝑠𝜃 d) 4𝑐𝑜𝑠 3 𝜃 − 3𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝑥−𝑠𝑖𝑛3𝑥
32. 𝑠𝑖𝑛2 𝑥−𝑐𝑜𝑠2 𝑥 =
a) 𝑠𝑖𝑛𝑥 b) 2𝑠𝑖𝑛𝑥 c) 𝑐𝑜𝑠𝑥 d) 2𝑐𝑜𝑠𝑥
1
33. If 𝑠𝑖𝑛𝛼 = then find the value of 𝑠𝑖𝑛3𝛼.
3
a) 4/27 b) 23/27 c) 31/27 d) 11/27
3
34. If 𝑠𝑖𝑛𝛼 = then find the value of 𝑐𝑜𝑠2𝛼.
5
a) 7/25 b) 18/25 c) -7/25 d) -18/25
35. 𝑠𝑖𝑛51° + 𝑐𝑜𝑠81° =
a) 𝑠𝑖𝑛21° b) −𝑠𝑖𝑛21° c) 𝑐𝑜𝑠21° d) −𝑐𝑜𝑠21°
1
If 𝑠𝑖𝑛𝛼 = √3 2
and 𝑐𝑜𝑠𝛽 = ,⁡and both α and β lies in 1st quadrant then
2
36. Find the value of cos(𝛼 − 𝛽).
a) -56/65 b) -33/65 c) -16/65 d) -63/65
37. Find the value of sin(𝛼 + 𝛽).
a) -56/65 b) -33/65 c) -16/65 d) -63/65
38. Which of the is not correct?
a) 𝑠𝑖𝑛𝜃 = 23 b) 𝑐𝑜𝑠𝜃 = − 14 c) 𝑡𝑎𝑛𝜃 = 300 d) 𝑐𝑜𝑠𝑒𝑐𝜃 = 23
39. Find the sign of 𝑠𝑒𝑐2000°.
a) Positive, lies in 1st quadrant b) Positive, lies in 4th quadrant
nd
c) Negative, lies in 2 quadrant d) Negative, lies in 3rd quadrant
3𝑐𝑜𝑡(90°−𝜃) 2𝑠𝑖𝑛𝜃
40. 𝑡𝑎𝑛𝜃 − 𝑐𝑜𝑠(90°−𝜃) =
a) ½ b) ¾ c) 1 d) 0
𝑠𝑖𝑛𝛼−𝑠𝑖𝑛𝛽
41. 𝑐𝑜𝑠𝛼+𝑐𝑜𝑠𝛽 =
a) 𝑡𝑎𝑛 (𝛼−𝛽
2
) b) 𝑡𝑎𝑛 (𝛼+𝛽
2
) c) 𝑡𝑎𝑛 (𝛼+𝛽
2
𝛼−𝛽
) 𝑐𝑜𝑡 ( )
2
d) 𝑡𝑎𝑛 (𝛼−𝛽
2
𝛼+𝛽
) 𝑐𝑜𝑡 ( )
2
𝛼
42. 𝑠𝑖𝑛 is equal to:
2

a) ±√1+𝑠𝑖𝑛𝛼
2
b) ±√1−𝑐𝑜𝑠𝛼
2
c) ±√1−𝑠𝑖𝑛𝛼
2
d) ±√1+𝑐𝑜𝑠𝛼
2

43. 𝑐𝑜𝑠𝛼 − 𝑐𝑜𝑠𝛽 =

𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽 𝛼+𝛽 𝛼−𝛽
a) 2𝑠𝑖𝑛 ( ) 𝑐𝑜𝑠 ( ) b) 2𝑐𝑜𝑠 ( ) 𝑠𝑖𝑛 ( ) c) 2𝑐𝑜𝑠 ( ) 𝑐𝑜𝑠 ( ) d) −2𝑠𝑖𝑛 ( ) 𝑠𝑖𝑛 ( )
2 2 2 2 2 2 2 2
44. An allied angle to θ is:
a) 270˚+θ b) 60˚+θ c) 45˚+θ d) 30˚+θ
45. The value of cos(𝛼 − 2𝜋) is equal to:
a) – 𝑐𝑜𝑠𝛼 b) 𝑐𝑜𝑠𝛼 c) 𝑠𝑖𝑛𝛼 d) – 𝑠𝑖𝑛𝛼
46. The radian measure of the angle described by a wheel in 5 revolution is:
a) 5π b) 15π c) 10π d) 20π
47. A wheel of radius 1 foot rotates through ½ a rotation. How far will it travel?
a) 3.14 feet b) 6.2 feet c) 9.2 feet d) 1 foot
48. In which quadrant tangent function increases from -∞ to 0.
a) 1st b) 2nd c) 3rd d) 4th
49. The range of cosecθ is:
a) -1≤cosecθ≤1 b) R c) All real numbers ≥1 and ≤ -1
50. Find the principal solution of 𝑐𝑜𝑡 −1 (−√3).
a) 11π/6 b) π/6 c) 5π/6 d) 7π/6
51. 𝑠𝑖𝑛 (− 11𝜋
3
)=
√3 1
a) b) c) 1 d) 0
2 2
52. 𝑐𝑜𝑡 (− 15𝜋
4
)=
√3 1
a) b) c) 1 d) 0
2 2
53. 𝑐𝑜𝑠(−1710°) =
√3 1
a) b) c) 1 d) 0
2 2
54. Find the range of the function 𝑠𝑖𝑛3𝑥.
a) {𝑦: 𝑦 ∈ ℝ, −3 ≤ 𝑦 ≤ 0} b) {𝑦: 𝑦 ∈ ℝ, −3 ≤ 𝑦 ≤ 3} c) {𝑦: 𝑦 ∈ ℝ, −1 ≤ 𝑦 ≤ 1} d) {𝑦: 𝑦 ∈ ℝ, 0 ≤ 𝑦 ≤ 1}
55. Find the period of the function 𝑠𝑖𝑛3𝑥.
2𝜋 𝜋
a) 𝜋 b) c) 2𝜋 d)
3 2
56. Find the range of the function 3𝑐𝑜𝑠2𝑥.
2𝜋 𝜋
a) 𝜋 b) c) 2𝜋 d)
3 2
3𝜋
57. 𝑠𝑖𝑛 ( 2 + 𝜃) =
a) 𝑠𝑖𝑛𝜃 b) −𝑠𝑖𝑛𝜃 c) 𝑐𝑜𝑠𝜃 d) −𝑐𝑜𝑠𝜃
58. 𝑠𝑒𝑐 (𝜋2 + 𝜃) =
a) 𝑠𝑒𝑐𝜃 b) −𝑠𝑒𝑐𝜃 c) 𝑐𝑜𝑠𝑒𝑐𝜃 d) −𝑐𝑜𝑠𝑒𝑐𝜃
59. 𝑠𝑖𝑛1230° + cot(−315°) =
a) 1 b) ½ c) 0 d) 3/2
60. If r1, r2, r3 are the radii of escribed circle then r1r2r3 =
a) r2 b) rs c) rs2 d) r2s
2
61. In a ΔABC, if a = 2, b = 3 and 𝑠𝑖𝑛𝛼 = then find β.
3
a) 45˚ b) 30˚ c) 60˚ d) 90˚
62. In a ΔABC, if a = 2, b = 3 and c = 4 then find 𝑐𝑜𝑠𝛼.
a) 8/7 b) 7/8 c) 1/8 d) 1/7
𝛼
63. In a ΔABC, if a = 13, b = 14 and c = 15, find the value of 𝑠𝑖𝑛 .
2
1 1 7 7
a) b) c) d) −
√5 5 √65 √65
64. In any triangle ΔABC, 𝑏𝑐𝑜𝑠 2 (𝛼2) + 𝑐𝑐𝑜𝑠 2 (𝛽2) =
a) s b) 2s c) s(s-a) d) ½ s
65. Find the area of triangle ABC in which 𝑎 = 60°, 𝑏 = 4⁡𝑐𝑚 and 𝑐 = ⁡ √3⁡𝑐𝑚.
a) 3 sq. cm b) 5 sq. cm c) 8 sq. cm d) None of these
66. In any triangle ABC, if 𝑎 = √2⁡𝑐𝑚, 𝑏 = ⁡ √3 and 𝑐 = √5⁡𝑐𝑚. Find the area of triangle.
1 1
a) √3 sq. cm b) √3 sq. cm c) √6 sq. cm d) √6 sq. cm
2 2
67. In a ΔABC, if a = 18 cm, b = 24 cm and c = 30 cm then R =
a) 12 cm b) 13 cm c) 14 cm d) 15 cm
68. Find the height of an object if the angle of elevation of the sun is 30˚ and the length of the shadow of the object is 1.7 meters.
a) 0.34 m b) 1.4 cm c) 0.577 d) 0.98
69. From the top of a tower, the angle of depression to the nearest port of a ship at its waterline is 45˚. If the height of the tower is 35 m,
find the distance between the ship and the foot of the tower.
a) 70 m b) 35 m c) 17.43 m d) 55 m
70. An aeroplane is flying at a height of 8000 metres. If the angle of depression to a field marker measures 30˚, find the aerial distance.
a) 12000 m b) 14500 m c) 16000 d) 4000 m
71. In a ΔABC, if a = 5 cm, b = 10 cm and c = 13 cm then r =
a) 3.1 cm b) 2.57 cm c) 4 cm d) 3.57 cm
72. Find the area of triangle ABC, with 𝑎 = 2√2 cm, β = 60˚ and γ = 90˚.
a) √3 cm b) 2√3 cm c) √3/2 cm d) 3√3 cm
73. Find r1, if a = b = c = 2 cm.
a) √3 cm b) 2√3 cm c) √3/2 cm d) 3√3 cm
74. From the top of a cliff 80√3 m high the angle of depression of a boat is α. If the distance between the boat and foot of cliff is 80 m,
then the angle α is:
𝜋 𝜋 𝜋 𝜋
a) b) c) d)
4 6 3 2
75. Radius of the circle passing through all the vertices of a triangle is called:
a) circum-radius b) In-radius c) e-radius d) None of these
76. In any triangle ABC, √(𝑠−𝑎)(𝑠−𝑐)
𝑎𝑐
is equal to:
a) 𝑠𝑖𝑛 𝛼2 b) 𝑐𝑜𝑠 𝛼2 c) 𝑠𝑖𝑛 𝛽2 d) 𝑠𝑖𝑛 𝛾2
𝛾
77. In any triangle ABC, 𝑐𝑜𝑠 is equal to:
2

a) √𝑠(𝑠−𝑎)
𝑎𝑏
b) √𝑠(𝑠−𝑏)
𝑎𝑐
c) √𝑠(𝑠−𝑎)
𝑏𝑐
d)√𝑠(𝑠−𝑐)
𝑎𝑏⁡⁡

78. In any triangle ABC, √(𝑠−𝑎)(𝑠−𝑏)

𝑠(𝑠−𝑐)
is equal to:
𝛾 𝛾 𝛼 𝛾
a) 𝑐𝑜𝑡 b) 𝑡𝑎𝑛 c) 𝑡𝑎𝑛 d) 𝑠𝑖𝑛
2 2 2 2
79. In any triangle ABC, with usual notation, abc =
a) 2Δ b) 4Δ c) 4RΔ d) 4r
80. In an equilateral triangle, 𝑟: 𝑅: 𝑟1 =
a) 1:2:3 b) 1:3:2 c) 3:2:1 d) 2:3:1
𝛽
81. 𝑠⁡𝑡𝑎𝑛 =
2
a) r b) r1 c) r2 d) r3
82. 𝑟1 + 𝑟2 + 𝑟3 − 𝑟 =
a) s2 b) Δ c) 4R d) rs2
83. 𝑟1 𝑟2 + 𝑟2 𝑟3 + 𝑟3 𝑟1 =
a) s2 b) Δ c) 4R d) rs2
84. In a ΔABC, if a = 18 cm, b = 24 cm and c = 30 cm then the value of r3.
a) 12 cm b) 18 cm c) 36 cm d) 10 cm
85. (𝑠𝑒𝑐𝜃 + 𝑡𝑎𝑛𝜃)(𝑠𝑒𝑐𝜃 − 𝑡𝑎𝑛𝜃) =
a) secθ b) tanθ c) 1 d) 0
86. The domain of y = cos-1x is:
a) -∞<x<∞ b) -1<x<1 c) x≤-1 or x≥1
87. 𝑡𝑎𝑛−1 14 + 𝑡𝑎𝑛−1 15 = ______.
𝜋 3𝜋 9 19
a) b) c) 𝑡𝑎𝑛−1 d) 𝑡𝑎𝑛−1
4 4 19 9
88. 𝑠𝑖𝑛−1 𝑥 − 𝑐𝑜𝑠 −1𝑦 =⁡
a) 𝑐𝑜𝑠 −1 {𝑥𝑦 + √1 − 𝑥 2 √1 − 𝑦 2 } b) 𝑐𝑜𝑠 −1 {𝑥𝑦 − √1 − 𝑥 2√1 − 𝑦 2 }
c) 𝑐𝑜𝑠 −1 {𝑥𝑦 + √1 + 𝑥 2 √1 − 𝑦 2 } d) 𝑐𝑜𝑠 −1 {𝑥𝑦 + √1 − 𝑥 2√1 + 𝑦 2 }
89. 𝑠𝑖𝑛−1 𝑥 − 𝑐𝑜𝑠 −1𝑥 =
𝜋
a) 0 b) 1 c) -1 d)
2
90. Find the value of 𝑠𝑖𝑛 (𝑐𝑜𝑠 −1 √3
2
)=
1 √3 2
a) b) c) d) √3
2 2 √3
91. Find the value of 𝑠𝑒𝑐 [𝑠𝑖𝑛−1 (12)]=
1 √3 2
a) b) c) d) ∞
2 2 √3
92. Find the solution of the equation 1+cosθ = 0
a) {π+2nπ} b) {2π+2nπ} c) {π/2+2nπ} d) {π+nπ}
93. Solve the equation 4𝑐𝑜𝑠 2 𝑥 − 3 = 0
a) {𝜋2 + 2𝑛𝜋} ∪ {𝜋6 + 𝑛𝜋} b) {𝜋6 + 2𝑛𝜋} ∪ {11𝜋
3
+ 2𝑛𝜋} c) {𝜋3 + 2𝑛𝜋} ∪ {11𝜋
6
+ 2𝑛𝜋} d) {𝜋6 + 2𝑛𝜋} ∪ {11𝜋
6
+ 2𝑛𝜋}
94. Find the principal solution of 𝑐𝑜𝑡𝑥 = √3.
2𝜋 5𝜋 𝜋 7𝜋 2𝜋 7𝜋 𝜋 5𝜋
a) , b) , c) , d) ,
3 4 6 6 3 6 6 4
√3
95. Find the principal solution of 𝑠𝑖𝑛𝑥 = −
2
𝜋 5𝜋 𝜋 2𝜋 4𝜋 5𝜋 4𝜋 2𝜋
a) , b) , c) , d) ,
6 3 3 3 3 3 3 3
96. The range of cosecθ is:
a) -1≤ cosecθ ≤1 b) R c) All real numbers ≥1 and ≤ -1
97. Solve 𝑐𝑜𝑡𝑥. 𝑐𝑜𝑠 2 𝑥 = 2𝑐𝑜𝑡𝑥
a) {𝜋4 + 2𝑛𝜋} ∪ {𝜋2 + 𝑛𝜋} b) {𝜋4 + 2𝑛𝜋} c) {𝜋2 + 𝑛𝜋} d) {𝜋4 + 2𝑛𝜋} ∪ {𝜋2 + 2𝑛𝜋}
98. Find the value of 𝑠𝑖𝑛−1 (𝑐𝑜𝑠 (13𝜋
6
)).
𝜋 𝜋 𝜋 𝜋
a) b) c) d)
6 3 4 2
−1 4
99. Find the value of 𝑠𝑖𝑛 (𝑐𝑜𝑠 ( )).
5
a) 2/3 b) ½ c) ¾ d) 4/3
100. Solution of trigonometric equation 2cos2θtanθ = tanθ in the interval [0, π]
𝜋 3𝜋 5𝜋 7𝜋 3𝜋 5𝜋 7𝜋 5𝜋 7𝜋 𝜋 3𝜋
a) 0, , , 𝜋, and b) 0, , 𝜋, and c) 0,⁡ and d) 0, , ,𝜋
4 4 4 4 4 4 4 4 4 4 4