Model Test Paper

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GRADE 10 TRIGONOMETRY
Class 10 - Mathematics
Time Allowed: 2 hours Maximum Marks: 127

Section A
–
1. If √3 tan 2θ − 3 = 0 then θ = ? [1]

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a) 30o b) 60o

c) 15o d) 45o

h
2 ∘

2. (
1−tan 30
) is equal to: [1]

rut
2 ∘
1+tan 30

a) sin 60o b) tan 60o

c) cos 30o d) cos 60o

Sh
60
3. If (sinα + cosecα)2 + (cosα + secα)2 = k + tan2α + cot2α, then k = ________. [1]

a) 7 b) 3
08
y

c) 5 d) 9
db

4. If 2 cos 3θ = 1 then θ = ? [1]

20
a) 30° b) 10°
are

c) 15° d) 20°
40

5. If 3x = cosec θ and
3
= cot θ then 3 (x 2
−
1
) =? [1]
x x
2

a) 1
b) 1
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9 81

c) 1
d) 1

3
27

6. If x = a cos θ and y = b sin θ , then b2x2 + a2y2 = [1]

Pr

a) a2 + b2 b) ab

c) a4b4 d) a2b2

7. If sec A + tan A = m and sec A - tan A = n, then the value of mn is [1]

a) 0 b) 1

c) -1 d) 2
8. If sin A = 1

2
, then the value of cot A is [1]
–
a) √3 b) √3

c) 1
d) 1
√3

9. If x = r sin θ cos ϕ , y = r sin θ sin ϕ and z = r cos θ , then [1]

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 a) x2 + y2 + z2 = r2 b) x2 - y2 + z2 = r2

c) z2 + y2 - x2 = r2 d) x2 + y2 - z2 = r2

10. If cos A = 1

2
, then tan A is equal to [1]

a) 1
b) 3

2
√3

–
c) 3 d) √3

11. sec4A - sec2A is equal to [1]

a) tan2 A - tan4A b) tan4A - tan2A

c) tan2A + tan3A d) tan4A + tan2A

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[1]
∘

12. (
2 tan 30

2 ∘
) is equal to:
1+tan 30

a) cos 60o b) sin 60o

h
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c) sin 30° d) tan 60o

[1]
2 2

If θ is an acute angle such that sec2θ = 3, then the value of

tan θ− cosec θ
13. 2 2
is
tan θ+ cosec θ
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a) b)
1 3

60
7 7

c) 2

7
d) 4

14. For θ = 30o, the value of (2 sin θ cos θ ) is: [1]

08
y

a) √3
b) 1
db

c) d)
√3 3
20
2 2

15. If tan2 θ = 3, where θ is an acute angle, then the value of θ is: [1]
are
40

a) 0o b) 60o

c) 45o d) 30o
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16. (sec A + tan A) (1 – sin A) [1]

a) cos A b) sec A
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c) sin A d) cosec A
3

17. If 3 cos θ = 5 sin θ , then the value of

5 sin θ−2 sec

3
θ+2 cos θ
is [1]
5 sin θ+2 sec θ−2 cos θ

a) b)
542 271

2937 979

c) 322

2657
d) 316

2937
−−−−−
18. √
1−sin A
=? [1]
1+sin A

a) sec A - tan A b) sec A + tan A

c) - sec A tan A d) sec A tan A

19. If sin θ = 1 , then the value of

1
sin(
θ
) is: [1]
2 2

a) 1
b) 0
√2

c) 1
d) 1

2
2√2

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20. 5 cot2 A - 5 cosec2 A = [1]

a) 0 b) 5

c) 1 d) -5
21. A monkey is climbing a 10 m long rope which is tightly stretched and tied from the top of a vertical pole to the [1]
ground. If the angle made by the rope with the ground level is 45o, then the height of pole is

a) 15 m b) 25 m
–
c) 20 m d) 5√2 m

22. A boy is flying a kite, the string of the kite makes an angle of 30o with the ground. If the height of the kite is 18 [1]

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m, then the length of the string is

a) 18 m b) 36 m
– –
c) 18√3m d)

h
36√3m

The angle of elevation of a cliff from a fixed point A is 45o. After going up a distance of 600 meters towards the [1]

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23.

top of the cliff at an inclination of 30o, it is found that the angle of elevation is 60o. Find the height of the cliff.
–
(Use √3 = 1.732)
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60
a) 819.6 m b) 817.8 m

c) 818.5 m 08 d) 820.5 m
24. If the angle of elevation of a tower from a distance of 100 meters from its foot is 60°, then the height of the [1]
y

tower is
db

–
a) m b) 50 √3 m
200
20
√3

–
c) 100√3 m d) 100
m
√3
are
40

25. It is found that on walking x meters towards a chimney in a horizontal line through its base, the elevation of its [1]
top changes from 30° to 60°. The height of the chimney is
– –
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a) 2√3x b)
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3√2x

c) 2
x d) √3
x
√3 2
Pr

26. A vertical pole 10 m long casts a shadow of length 5 m on the ground. At the same time, a tower casts a shadow [1]
of length 12.5 m on the ground. The height of the tower is:

a) 22 m b) 25 m

c) 24 m d) 20 m
27. Two men are on opposite sides of a tower. They observe the angles of elevation of the top of the tower as [1]
60
∘
and 45 respectively. If the height of the tower is 60m, then the distance between them is
∘

– –
a) 20(3 − √3)m b) 20( √3 − 3)m

– –
c) 10(√3 − 3)m d) 20( √3 + 3)m

28. The tops of two poles of height 16 m and 10 m are connected by a wire. If the wire makes an angle of 30o with [1]

the horizontal, then the length of the wire is

–
a) 12 m b) 10√3 m

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 c) 16 m d) 10 m
29. The angles of elevation of the top of a tower from two points on the ground at distances 8 m and 18 m from the [1]
base of the tower and in the same straight line with it are complementary. The height of the tower is

a) 12 m b) 18 m

c) 8 m d) 16 m
30. The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of [1]
the tower becomes x meters less. The value of x is
–
a) 100 m b) 100 (√3 - 1) m
–
c) 100√3 m d) m
100

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√3

31. The angle of depression of a car, standing on the ground, from the top of a 75 m tower, is 30°. The distance of [1]
the car from the base of the tower (in metres) is

h
– –
a) 25√3 b) 75√3

rut
–
c) 150 d) 50√3

32. A tree casts a shadow 7 m long on the ground, when the angle of elevation of the Sun is 45o. The height of the
Sh [1]
tree is:

60
–
a) 3.5 m b) 7√3 m

–
c) 7 m d)
7
√3 m
08
y
3

33. From the top of a cliff 20 m high, the angle of elevation of the top of a tower is found to be equal to the angle of [1]
db

depression of the foot of the tower. The height of the tower is

20
a) 20 m b) 40 m

c) 80 m d) 60 m
are
40

34. In a right △ ABC, AC is the hypotenuse of length 10cm. If ∠A = 30°, then the area of the triangle is [1]
–
a) 25√3cm 2
b) 25cm
2
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ep

– –
c) 25

3
√3cm
2
d) 25

2
√3cm
2

35. From the top of a hill, the angles of depression of two consecutive km stones due east are found to be 30° and [1]
Pr

45°. The height of the hill is

– –
a) (√3 − 1) Km b) 1

2
( √3 − 1) Km
– –
c) (√3 + 1) Km d) 1

2
( √3 + 1) Km
36. From the top of a cliff 25 m high the angle of elevation of a tower is found to be equal to the angle of depression [1]
of the foot of the tower. The height of the tower is

a) 75 m b) 25 m

c) 100 m d) 50 m
37. If the angle of elevation of a cloud from a point h metres above a lake is x and the angle of depression of its [1]
reflection in the lake is y, then the distance of the cloud from the point of observation is

a) 2h cos x

tan x+tan y
b) 2h cot x

tan y+tan x

c) 2h sec x

tan y+tan x
d) 2h sec x

tan y−tan x

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Contact 9940200860 Prepared by Shruthi
 –
38. The angle of elevation of the top of a 15 m high tower at a point 15√3 m away from the base of the tower is: [1]

a) 90o b) 60o

c) 30o d) 45o

39. The angle of elevation of an aeroplane from a point on the ground is 45°. After a flight of 10 sec, the elevation [1]

changes to 30o. If the aeroplane is flying at a height of 3 km, then find the speed of the aeroplane. (Use √3 =
–

1.732)

a) 782.65 km/hr b) 785.46 km/hr

c) 790.56 km/hr d) 780.56 km/hr

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40. The tops of two towers of heights x and y, standing on a level ground subtend angles of 30° and 60° respectively [1]
at the centre of the line joining their feet. Then, x : y is

a) 1 : 3 b) 2 : 1

h
c) 1 : 2 d) 3 :1

rut
Section B
41. Prove that: 1

(sec θ−tan θ)
−
1

cos θ
=
1

cos θ
−
1

(sec θ+tan θ)
[2]
3
Sh
42. Prove that: 2 cos θ−cos θ
= cot θ [2]

60
3
sin θ−2 sin θ

43. If ∠A and ∠B are acute angles such that cos A = cos B then show that ∠A = .
∠B [2]
44. If sin θ = a
, 0 < θ < 90 ,
∘
find the values of cos θ and tan θ [2]
√a2 +b2
08
y

45. If a cosθ + b sinθ = m and a sinθ - b cosθ = n, prove that a2 + b2 = m2 + n2 [2]

db

−−−−−
46. Prove that: √ 1+sin A
= sec A + tan A [2]
20
1−sin A

47. Evaluate (sin230° + 4 cot245° - sec260°)(cosec245° sec230°). [2]

48. If A = B = 60°, verify that cos (A - B) = cosA cosB + sin A sinB [2]
are
40

49. Verify: sin 60° cos 30° - cos 60° sin 30° = sin 30° [2]
50. Prove that: sin θ−cos θ+1
=
1
[2]
cos θ+sin θ−1 sec θ−tan θ

51. The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 9.5 m away from the [2]
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ep

wall. Find the length of the ladder.

52. A person standing on the bank of a river, observes that the angle of elevation of the top of the tree standing on [2]
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the opposite bank is 60o. When he retreats 20 m from the bank, he finds the angle of elevation to be 30o. Find
the height of the tree and the breadth of the river.

53. Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation of 30o. [2]

Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to be 45o.
Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak.
–
54. Find the angle of elevation when the shadow of a pole 'h' m high is √3h m long. [2]
55. The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° [2]
and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance
between the two towers and also the height of the other tower.

56. A 7 m long flagstaff is fixed on the top of a tower standing on the horizontal plane. From point on the ground, [2]
the angles of elevation of the top and bottom of the flagstaff are 60o and 45o respectively. Find the height of the

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Contact 9940200860 Prepared by Shruthi
 tower correct to one place of decimal.
–
57. The ratio of the height of a tower and the length of its shadow on the grow is √3 : 1 . What is the angle of [2]
elevation of the sun?
–
58. An observer, 1.7 m tall, is 20√3 m away from a tower. The angle of elevation from the eye of observer to the [2]
top of tower is 30°. Find the height of tower.
Section C
59. In ΔABC , right angled at B, if, tan A = 1
Find the value of. [3]
√3

i. sin A cos C + cos A sin C

ii. cos A cos C - sin A sin C
60. If secθ + tanθ = p, obtain the values of secθ , tanθ and sinθ in terms of p. [3]

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61. Given that 16 cot A = 12; find the value of . [3]
sin A+cos A

sin A−cos A

62. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A. [3]
If (tan θ + sin θ ) = m and (tan θ - sin θ ) = n, prove that (m2 - n2)2 = 16mn [3]

h
63.
64. Given 15 cot A = 8, compute cos A and tan A. [3]

rut
65. If tan θ = , find the value of sin θ + cos θ . [3]
24

[3]
√3
66. If sin θ = 2
, find the value of all T-ratios of θ .
Sh cos θ+sin θ
67. Prove that . [3]
tan θ cot θ
− =

60
1−tan θ 1−cot θ cos θ−sin θ

68. In the given △PQR, right-angled at Q, QR = 9 cm and PR - PQ = 1 cm. Determine the value of sin R + cos R. [3]
08
y
db
20
are
40

69. The angle of elevation of an aeroplane from a point on the ground is 60 . After a flight of15 seconds, the angle
∘
[3]
–
of elevation changes to 30 . If the aeroplane is flying at a constant height of 1500
∘
√3 m, find the speed of the
plane in km/hr.
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70. The angle of elevation of an aeroplane from a point A on the ground is 60o. After a flight of 30 seconds, the [3]

angle of elevation changes to 30o. If the plane is flying at a constant height of 3600√3 metres, find the speed of
–
Pr

the aeroplane.
71. From the top of hill, the angles of depression of two consecutive kilometer stones due East are found to be [3]
30
o
and 45 . Find the height of the hill.
o

72. A man standing on the deck of a ship, which is 10 m above the water level, observes that the angle of elevation [3]
of the top of a hill is 60o and the angle of depression of the base of the hill is 30o. Find the height of the hill.
73. A kite is flying at a height of 66 m above the ground. The string attached to the kite is temporarily tied to a point [3]
on the ground. The inclination of the string with the ground is 60o. Find the length of the string, assuming that
there is no slack in the string.
74. A path separates two walls. A ladder leaning against one wall rests at a point on the path. It reaches a height of [3]
90 m on the wall and makes an angle of 60° with the ground. If while resting at the same point on the path, it
were made to lean against the other wall, it would have made an angle of 45° with the ground. Find the height it
would have reached on the second wall.

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Contact 9940200860 Prepared by Shruthi
75. From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are [3]
30o and 45o. If the bridge is at a height of 8 m from the banks, then find the width of the river.

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Sh
60
08
y
db
20
are
40
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Pr

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