TS 10th Class Maths Bits Chapter 12 Applications of Trigonometry

Solving these TS 10th Class Maths Bits with Answers Chapter 12 Applications of Trigonometry Bits for 10th Class will help students to build their problem-solving skills.

Applications of Trigonometry Bits for 10th Class

Question 1.
If a pole 6 m high casts a shadow 2 \(\sqrt{3}\) m long on the ground, then the sun’s angle of elevation is
A) 60°
B) 45°
C) 30°
D) 90°
Answer:
A) 60°

Question 2.
If the angle of elevation of a tower from a distance of 100 m from its foot is 60°then the height of the tower is …………. m.
A) 100\(\sqrt{3}\)
B) \(\frac{100}{\sqrt{3}}\)
C) 50 \(\sqrt{3}\)
D) \(\frac{50}{\sqrt{3}}\)
Answer:
A) 100\(\sqrt{3}\)

Question 3.
The height of a tower is 10 m. The length of its shadow when Sun’s altitude is 45° is …………….. m.
A) 10
B) 20
C) 10\(\sqrt{3}\)
D) 50
Answer:
A) 10

Question 4.
The length of the shadow of a tower on the plane ground is \(\sqrt{3}\) times the height of the tower, the angle of elevation of sun is
A) 30°
B) 45°
C) 60°
D) 90°
Answer:
A) 30°

Question 5.
The ratio of the length of a rod and it’s shadow is 1 : \(\sqrt{3}\) then the angle of elevation of the Sun is
A) 45°
B) 30°
C) 75°
D) 90°
Answer:
B) 30°

Question 6.
If two towers of height X and Y subtend angles of 30° and 60°respectively at the centre of the line joining their feet, then X : Y is equal to
A) 1 : 3
B) 1 : \(\sqrt{3}\)
C) 3 : 1
D) \(\sqrt{3}\) : 1
Answer:
A) 1 : 3

Question 7.
A wall of 8 m long casts a shadow 5 m long at the same time a tower casts a shadow 50m long, then the height of tower is
A) 20 m
B) 40 m
C) 80 m
D) 200 m
Answer:
C) 80 m

Question 8.
If the Sun’s angle of elevation is 60°, then a pole of height 6 m will cast a shadow of length ………….. m.
A) \(\sqrt{3}\)
B) 5\(\sqrt{3}\)
C) 6\(\sqrt{3}\)
D) 2\(\sqrt{3}\)
Answer:
D) 2\(\sqrt{3}\)

Question 9.
A pole of 12 m high casts a shadow 4\(\sqrt{3}\)m on the ground, then the Sun’s angle of elevation is
A) 60°
B) 120°
C) 45°
D) 30°
Answer:
A) 60°

Question 10.
If the height and length of the shadow of a man are the same then the angle of elevation of the Sun is
A) 60°
B) 45°
C) 90°
D) 120°
Answer:
B) 45°

Question 11.
What is the angle of elevation of the top of a temple of height 10 m at a point whose distance from the base of the tower is 10\(\sqrt{3}\) m ?
A) 30°
B) 60°
C) 45°
D) 90°
Answer:
A) 30°

Question 12.
The length of the shadow of 5m height tree whose angle of elevation of the Sun is 30° is?
A) 5 m
B) \(\sqrt{3}\) m
C) 5\(\sqrt{3}\) m
D) 10 m
Answer:
C) 5\(\sqrt{3}\) m

Question 13.
From the top of a 10m height tree the angle of depression of a point on the ground is 30° then the distance of the point from the foot of the tree is
A) 10 m
B) 10\(\sqrt{3}\) m
C) \(\frac{10}{\sqrt{3}}\) m
D) 5\(\sqrt{3}\) m
Answer:
B) 10\(\sqrt{3}\) m

Question 14.
Ladder ‘x’ meters long is laid against a well making an angle ‘0’ with the ground. If we want to directly find the distance between the foot of ladder and foot of the wall, which trigonometrical ratio should be considered ?
A) sin θ
B) cos θ
C) tan θ
D) cot θ
Answer:
B) cos θ

Question 15.
Top of a building was observed at an angle of elevation ‘α’ from a point, which is at distance ‘d’ meters from the foot of the building. Which trigonometrical ratio should be considered for finding height of buildings.
A) tan α
B) sin α
C) cos α
D) sec α
Answer:
A) tan α

Question 16.
In the given figure, the value of angle θ is

A) 30°
B) 60°
C) 45°
D) 90°
Answer:
A) 30°

Question 17.
The given figure shows the observation of point ‘C’ from point A. The angle of depression from A is

A) 30°
B) 45°
C) 90°
D) 75°
Answer:
A) 30°

Question 18.
If the length of the shadow of a tower is \(\frac{1}{\sqrt{3}}\) times the height of the tower, then the angle of elevation of the sun is ……………..
A) 30°
B) 45°
C) 60°
D) 75°
Answer:
C) 60°

Question 19.
A tower is 50 m high. Its shadow is x m shorter when the sun’s altitude is 45° then when it is 30°, then x = ………… cm
A) 105
B) 20
C) 10
D) 100
Answer:
D) 100

Question 20.
The length of the string of a kite flying at 100 m above the ground with the elevation of 60° is ………….
A) \(\frac{200}{\sqrt{3}}\)
B) \(\frac{20}{\sqrt{3}}\)
C) \(\frac{291}{\sqrt{3}}\)
D) none
Answer:
A) \(\frac{200}{\sqrt{3}}\)

Question 21.
A player sitting on the top of a tower of height 40 m observes the angle of depression of a ball lying on the ground is 60 The distance between the foot of the tower and ball is …………… m.
A) 20
B) \(\frac{80}{\sqrt{61}}\)
C) \(\frac{40}{\sqrt{3}}\)
D) \(\frac{40}{\sqrt{6}}\)
Answer:
C) \(\frac{40}{\sqrt{3}}\)

Question 22.
If the ratio of height of a tower and the length of its shadow on the ground is \(\sqrt{3}\) :1, then the angle of elevation of the sun is ……………….
A) 80°
B) 60°
C) 70°
D) 100°
Answer:
B) 60°

Question 23.
The angle of depression of the top of a tower at a point 100 m from the house is 45°, then the height of the tower is …………. m.
A) 18.1
B) 16.3
C) 36.6
D) 26.7
Answer:
C) 36.6

Question 24.
An object is placed above the observer’s horizontal, we call the angle between the line of sight and observer’s horizontal is ……………..
A) angle of elevation
B) angle of depression
C) point
D) none
Answer:
A) angle of elevation

Question 25.
Angle of elevation of the top of a building from a point on the ground is 30. Then the angle of depression of this point from the top of the building is …………………
A) 65°
B) 60°
C) 70°
D) 30°
Answer:
D) 30°

Question 26.
What change will be observed in the angle of elevation as we move away from the object ?
A) increase
B) decrease
C) can’t be determined
D) none
Answer:
A) increase

Question 27.
An object is placed below the observer’s horizontal, then what is the angle between line of sight and observer’s horizontal ?
A) angle of elevation
B) angle of depression
C) can’t be determined
D) none
Answer:
B) angle of depression

Question 28.
What change will be observed in the angle of elevation as we approach the foot of the tower ?
A) 0
B) 60°
C) Data not correct
D) none
Answer:
D) none

Question 29.
In the figure given below, the imaginary line through the object and eye of the observer is called …………………

A) line of sight
B) angle of depression
C) angle of elevation
D) none
Answer:
A) line of sight

Question 30.
In the figure given below, a man on the top of cliff observers a boat coming towards him. Then 6 represents the angle of …………….

A) depression
B) elevation
C) equal
D) none
Answer:
A) depression

Question 31.
In the figure given below, if AB = 10 m and AC = 20 m, then θ = ………………..

A) 60°
B) 30°
C) 70°
D) none
Answer:
B) 30°

Question 32.
A pole 6 m high casts a shadow 2\(\sqrt{3}\) m long on the ground, then the sun’s elevation is …………….
A) 70°
B) 20°
C) 80°
D) 60°
Answer:
D) 60°

Question 33.
In the figure given below, if AB = CD = 10\(\sqrt{3}\) m then BC = ………………

A) 90
B) 60
C) 40
D) None
Answer:
C) 40

Question 34.
In the figure given below, if AB = 10\(\sqrt{3}\) m, then CD = …………….. (take \(\sqrt{3}\) = 1.732)

A) 7.32
B) 8.14
C) 3.1
D) 1.92
Answer:
A) 7.32

Question 35.
In the figure given below, if AD = 7\(\sqrt{3}\) m, then BC = ………………. m.

A) 13
B) 19
C) 28
D) None
Answer:
C) 28

Question 36.
The length of the shadow of a tree is 7 m high, when the sun’s elevation is …………………..
A) 45°
B) 60°
C) 70°
D) 90°
Answer:
A) 45°

Question 37.
If two tangents inclined at an angle of 60 are drawn to a circle of radius 3 cm, then length of tangent is equal to …………. m.
A) 4\(\sqrt{3}\)
B) 2\(\sqrt{91}\)
C) \(\sqrt{3}\)
D) 3\(\sqrt{3}\)
Answer:
D) 3\(\sqrt{3}\)

Question 38.
The angle formed by the line of sight with horizontal, when the point being viewed is above the horizontal level is called
A) angle of elevation
B) angle of depression
C) point
D) none
Answer:
A) angle of elevation

Question 39.
cot² B – Cosec² B = ………………
A) 0
B) – 1
C) 1
D) 2
Answer:
B) – 1

Question 40.
\(\frac{\tan \theta}{\sec \theta}\) = ……………….
A) – cos θ
B) sin θ
C) – tan θ
D) none
Answer:
B) sin θ

Question 41.
A boy observed the top of an electrical pole to be at angle of elevation of 60° when the observation point is 8 m away from the foot of the pole then the height of the pole is ……………… m.
A) 18\(\sqrt{3}\)
B) 14
C) 7\(\sqrt{3}\)
D) 8\(\sqrt{3}\)
Answer:
D) 8\(\sqrt{3}\)

Question 42.
Suppose you are shooting an arrow from the top of a building at a height of 6 m to a target on the ground at an angle of depression of 60 what is the distance between you and the object ?
A) 9
B) 7\(\sqrt{3}\)
C) 12\(\sqrt{3}\)
D) None
Answer:
D) None

Question 43.
Sin \(\frac{\pi^{\mathrm{c}}}{2}\) = ……………….
A) 4
B) 3
C) 1
D) -1
Answer:
C) 1

Question 44.
Domain of sin θ = ………………..
A) R
B) R – {30°}
C) N
D) None
Answer:
D) None

Question 45.
tan \(\frac{\pi^{\mathrm{c}}}{4}\) = ……………….
A) 2
B) 3
C) -1
D) 1
Answer:

Question 46.
cot 15° = ………………
A) 2 + \(\sqrt{3}\)
B) 2 – \(\sqrt{3}\)
C) \(\sqrt{2}\)
D) \(\sqrt{3}\) – 1
Answer:
A) 2 + \(\sqrt{3}\)

Question 47.
A + B = 180° then cos A + cos B = ………………
A) 4
B) 1
C) 0
D) none
Answer:
C) 0

Question 48.
sin 15° = ……………….
A) \(\frac{\sqrt{3}}{9 \sqrt{2}}\)
B) \(\frac{\sqrt{3}-1}{2 \sqrt{2}}\)
C) \(\frac{\sqrt{3}+1}{2}\)
D) none
Answer:
B) \(\frac{\sqrt{3}-1}{2 \sqrt{2}}\)

Question 49.
tan A = \(\frac{\mathrm{n}}{\mathrm{n}+1}\), tan B = \(\frac{\mathrm{n}}{2\mathrm{n}+1}\), A + B = …………..
A) 4
B) 3
C) -1
D) 1
Answer:
D) 1

Question 50.
The angle of elevation of tower at a point 40 m apart from it is cot^-1 \(\left(\frac{3}{5}\right)\). Obtain the height of the tower.
A) \(\frac{200}{3}\) m
B) \(\frac{100}{3}\) m
C) \(\frac{210}{17}\) m
D) none
Answer:
A) \(\frac{200}{3}\) m

Question 51.
A ladder 20 m long is placed against a vertical wall of height 10 m, then the distance between the foot of the ladder and the wall is …………………. m.
A) 7\(\sqrt{3}\)
B) 20\(\sqrt{3}\)
C) 30\(\sqrt{3}\)
D) none
Answer:
C) 30\(\sqrt{3}\)

Question 52.
sin 18° = ………………
A) \(\frac{\sqrt{5}}{4}\)
B) \(\frac{\sqrt{5}-1}{4}\)
C) \(\frac{1+\sqrt{3}}{2}\)
D) \(\frac{\sqrt{3}-1}{4}\)
Answer:
B) \(\frac{\sqrt{5}-1}{4}\)

Question 53.
In the below figure x = …………………. cm.

A) 10
B) 12
C) 13
D) 19
Answer:
A) 10

Question 54.
cot (90 – A) = ………………
A) 3 tan A
B) sin A
C) cot A
D) tan A
Answer:
D) tan A

Question 55.
cos⁴ A – sin⁴ A = …………….
A) sin² A
B) cos² A
C) cos 2A
D) cos 3A
Answer:
C) cos 2A

Question 56.
If cosec θ + cot θ = k then cos θ ……………..
A) \(\frac{k^2-1}{k^2+1}\)
B) \(\frac{k^2}{k^2-1}\)
C) \(\frac{k^2+1}{k}\)
D) none
Answer:
D) none

Question 57.
x = (sec θ + tan θ), y = (sec θ – tan θ) then xy ………………
A) -1
B) 0
C) 1
D) -2
Answer:
C) 1

Question 58.
tan 15° = ……………..
A) \(\frac{\sqrt{3}}{\sqrt{3}+1}\)
B) \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\)
C) \(\frac{\sqrt{3}-1}{2}\)
D) none
Answer:
B) \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\)

Question 59.
cosec θ = ……………….
A) \(\sqrt{1+\cot ^2 \theta}\)
B) \(\sqrt{\cot ^2 \theta-1}\)
C) \(\sqrt{1+\sin \theta}\)
D) \(\sqrt{\cot \theta-1}\)
Answer:
A) \(\sqrt{1+\cot ^2 \theta}\)

Question 60.
x = a sin θ, y = a cos θ then x² + y² = ………………
A) \(\frac{a}{3}\)
B) \(\frac{a}{2}\)
C) a
D) a²
Answer:
D) a²

Question 61.
Example of a Pythagorean Triplet is ………………
A) 5, 12, 13
B) 5, 10, 11
C) 8, 9, 11
D) none
Answer:
A) 5, 12, 13

Question 62.
sec² A = …………….
A) 1 – tan² A
B) 1 + tan² A
C) cot² A
D) none
Answer:
B) 1 + tan² A

Question 63.
\(\frac{1}{\cos \theta}\) – cos θ = ………………
A) tan θ . sin θ
B) sec θ . cos θ
C) tan θ . cot θ
D) none
Answer:
A) tan θ . sin θ

Question 64.
sin θ = cos θ, θ ∈ Q₁ then θ = …………..
A) \(\frac{\pi^c}{2}\)
B) \(\frac{\pi^c}{3}\)
C) \(\frac{2 \pi^c}{3}\)
D) \(\frac{\pi^c}{4}\)
Answer:
D) \(\frac{\pi^c}{4}\)

Question 65.
72° = …………………
A) \(\frac{\pi^c}{2}\)
B) \(\frac{\pi^c}{3}\)
C) \(\frac{2 \pi^c}{5}\)
D) \(\frac{\pi^c}{5}\)
Answer:
C) \(\frac{2 \pi^c}{5}\)

Question 66.
sin² 105° + cos² 105° = ……………….
A) 1
B) 0
C) 9
D) 10
Answer:
A) 1

Question 67.
sin 45° (cos 45°) = ………………..
A) 1
B) \(\frac{1}{2}\)
C) 3
D) none
Answer:
B) \(\frac{1}{2}\)

Question 68.
cos 40° = 0.76 then sin 50² = ………………..
A) 0.76
B) 7.6
C) 76.6
D) none
Answer:
A) 0.76

Question 69.
At a point 15 m away from the base of a 15 m high pole, the angle of elevation of the top is …………………
A) 30°
B) 45°
C) 60c
D) 90°
Answer:
B) 45°

Question 70.
When the length of the shadow of a person is equal to his height, then the elevation of source of light is …………
A) 15°
B) 30°
C) 45°
D) 60°
Answer:
C) 45°

Question 71.
The angle of elevation of top of a tree is 30. On moving 20 m nearer, the angle of elevation is 60. The height of the tree is
A) 15\(\sqrt{3}\) m
B) 2\(\sqrt{3}\) m
C) 10\(\sqrt{3}\) m
D) 5\(\sqrt{3}\) m
Answer:
C) 10\(\sqrt{3}\) m

Question 72.
The ratio of length of a pole and its shadow is 1 :\(\sqrt{3}\).The angle of elevation is
A) 90°
B) 60°
C) 45°
D) 30°
Answer:
D) 30°

Question 73.
The upper part of a treee is broken by wind and makes an angle of 30° with the ground and at a distance of 21 m from the foot of the tree. Find the total height of the tree.
A) 30\(\sqrt{3}\) m
B) 21 m
C) 30 m
D) 21\(\sqrt{3}\) m
Answer:
D) 21\(\sqrt{3}\) m

Question 74.
From a bridge 25 m high, the angle of depression of a boat is 45°. Find the horizontal distance of the boat from the bridge.
A) 25\(\sqrt{3}\) m
B) 25 m
C) \(\frac{25}{\sqrt{3}}\)
D) 45 m
Answer:
B) 25 m

Question 75.
A tower makes an angle of elevation equal to the angle of depression from the top of a cliff 25 m height. Find the height of the tower.
A) 25 m
B) 75 m
C) 5m
D) 50 m
Answer:
D) 50 m

Question 76.
When the angle of elevation of a pole is 45°, the length of the pole and its shadow are
A) equal
B) length > shadow
C) shadow > length
D) none of the above
Answer:
A) equal

Question 77.
In a rectangle, if the angle between a diagonal and a side is 30, and the length of the diagonal is 6 cm, the area of the rectangle is
A) 18 cm²
B) 9 cm²
C) 18\(\sqrt{3}\) cm²
D) 9\(\sqrt{3}\) cm²
Answer:
D) 9\(\sqrt{3}\) cm²

Question 78.
Two posts are 15 m and 25 m high and the line joining their tops make an angle of 45° with the horizontal. The distance between the two posts is
A) 15 m
B) 25 m
C) 18 m
D) 10 m
Answer:
D) 10 m

Question 79.
An electric pole 20 m high stands up right! on the ground with the help of steel wire to its top and affixed on the ground. If the steel wire makes 60° with the horizontal ground, find the length of steel wire.
A) 60\(\sqrt{3}\) m
B) 20 m
C) 60 m
D) \(\frac{20}{\sqrt{3}}\) m
Answer:
D) \(\frac{20}{\sqrt{3}}\) m

Question 80.
A building casts a shadow of length 50\(\sqrt{3}\) m when the sun is 30° about the horizontal. The height of the building is
A) 30 m
B) 40 m
C) 50 m
D) 60 m
Answer:
C) 50 m

Question 81.
When the angle of elevation of a light! changes from 30° to 45°, the shadow of pole becomes 100\(\sqrt{3}\) m less. The height of the pole is
A) 30 m
B) 120 m
C) 75 m
D) 100 m
Answer:
D) 100 m

Question 82.
From the top of a building 50 m from horizontal, the angle of depression made by a car is 30°. How far is the car from the building ?
A) \(\frac{50}{\sqrt{3}}\)
B) 50\(\sqrt{3}\) m
C) 150 m
D) 30\(\sqrt{3}\) m
Answer:
B) 50\(\sqrt{3}\) m

Question 83.
From the top of a building with height 30°(\(\sqrt{3}\) + 1) m two cars make angles of depression of 45° and 30° due east. What is the distance between two cars ?
A) 30 m
B) 60 m
C) 45 m
D) 75 m
Answer:
B) 60 m

Question 84.
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he retires 40 m from the bank, he finds the angle to be 30°. The breadth of the river is
A) 10 m
B) 15 m
C) 20 m
D) 25 m
Answer:
C) 20 m

Question 85.
A ladder of 10 m length touches a wall at a height of 5 m. The angle made by it with the horizontal is
A) 30°
B) 45°
C) 60°
D) 90°
Answer:
A) 30°

Question 86.
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance between the top of the tree and the ground is 10 m. Find the height of the tree.
A) 10 m
B) 30\(\sqrt{3}\) m
C) 10\(\sqrt{3}\) m
D) 30 m
Answer:
C) 10\(\sqrt{3}\) m

Question 87.
The angle of elevation of a cloud from a point 200 m above the lake is 30° and the angle of depression of its reflection in the lake is 60°. The height of the cloud above the lake is
A) 100 m
B) 200 m
C) 300 m
D) 400 m
Answer:
D) 400 m

Question 88.
An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60. After a flight of 10 seconds, its angle of elevation is observed to be 30 from the same point on the ground. Find the speed of the aeroplane.
A) 415.7 km/h
B) 215.3 km/h
C) 700 km/h
D) none of the above
Answer:
A) 415.7 km/h

Question 89.
If AB = 4m, and AC = 8m, then the angle of elevation of A as observed from C is
A) 30°
B) 45°
C) 60°
D) 90°
Answer:
A) 30°

Question 90.
If a pole of height 6 m casts a shadow 2\(\sqrt{3}\) m long on the ground, then the sun’s elevation is
A) 30°
B) 60°
C) 45°
D) 90°
Answer:
B) 60°

Question 91.
Find the elevation of the sun at the moment when the length of the shadow of a tower is just equal to its height.
A) 30°
B) 45°
C) 60°
D) 90°
Answer:
B) 45°

Question 92.
If the shadow of a tree is \(\frac{1}{\sqrt{3}}\) times the height of the tree, then the angle of elevation of the sun is
A) 30°
B) 45°
C) 60°
D) 90°
Answer:
C) 60°