CBSE Class 10 Some Applications of Trigonometry Sure Shot Questions Set A

HEIGHTS AND DISTANCES

With the help of trigonometric ratios, we can determine the height of an object or distance between two objects. Some terms used to determine these are as follows.

Line of Sight

• The line of sight is a straight line along which an observer observes an object i.e., the line joining the eye of the observer and the point in the object is called the line of sight.
• If \( O \) is the eye of an observer and \( A, B \) are objects, then \( OA \) and \( OB \) is called the line of sight.

Angle of Elevation

• If the object is above the horizontal level, the angle between the line of sight and the horizontal is called the angle of elevation.

Angle of Depression

• If the object is below the horizontal level, the angle between the line of sight and the horizontal is called the angle of depression.

Practice Time

OBJECTIVE TYPE QUESTIONS

Multiple Choice Questions 

Question. A ladder makes an angle of \( 60^\circ \) with the ground when placed against a wall. If the foot of the ladder is \( 2 \) m away from the wall, then the length of the ladder (in meters) is
(a) \( \frac{4}{\sqrt{3}} \)
(b) \( 4\sqrt{3} \)
(c) \( 2\sqrt{2} \)
(d) \( 4 \)
Answer: (d)

Question. From a point on the ground, which is \( 15 \) m away from the foot of a vertical tower, the angle of elevation of the top of the tower, is found to be \( 60^\circ \). The height of the tower (in metres) is
(a) \( 5\sqrt{3} \)
(b) \( 15\sqrt{3} \)
(c) \( 15 \)
(d) \( 7.5 \)
Answer: (b)

Question. A lamp post \( 5\sqrt{3} \) m high casts a shadow \( 5 \) m long on the ground. The Sun’s elevation at this moment is
(a) \( 30^\circ \)
(b) \( 45^\circ \)
(c) \( 60^\circ \)
(d) \( 90^\circ \)
Answer: (c)

Question. If the height of a vertical pole is \( \sqrt{3} \) times the length of its shadow on the ground, then the angle of elevation of the Sun at that time is
(a) \( 30^\circ \)
(b) \( 60^\circ \)
(c) \( 45^\circ \)
(d) \( 75^\circ \)
Answer: (b)

Question. As some time of the day, the length of the shadow of a tower is equal to its height. Then the Sun’s altitude at that time is
(a) \( 30^\circ \)
(b) \( 60^\circ \)
(c) \( 90^\circ \)
(d) \( 45^\circ \)
Answer: (d)

Question. The length of the shadow of a tower standing on level ground is found to be \( 2x \) metres longer when the Sun’s elevation is \( 30^\circ \) than when it was \( 45^\circ \). The height of the tower (in metres) is
(a) \( (\sqrt{3} + 1)x \)
(b) \( (\sqrt{3} - 1)x \)
(c) \( 2\sqrt{3}x \)
(d) \( 3\sqrt{2}x \)
Answer: (a)

Question. If two towers of height \( h_1 \) and \( h_2 \) subtend angles of \( 60^\circ \) and \( 30^\circ \) respectively at the mid-point of the line joining their feet, then \( h_1 : h_2 \) is
(a) \( 3 : 1 \)
(b) \( \sqrt{3} : 1 \)
(c) \( 1 : \sqrt{3} \)
(d) \( 1 : 3 \)
Answer: (a)

Question. The angle of elevation of a cloud from a point \( h \) metres above a lake is \( \theta \). The angle of depression of its reflection in the lake is \( 45^\circ \). The height of the cloud (in metres) is
(a) \( h\left( \frac{1 - \tan \theta}{1 + \tan \theta} \right) \)
(b) \( h\left( \frac{1 - \cot \theta}{1 + \cot \theta} \right) \)
(c) \( h\left( \frac{1 + \tan \theta}{1 - \tan \theta} \right) \)
(d) \( h\left( \frac{1 + \cot \theta}{1 - \cot \theta} \right) \)
Answer: (c)

Question. The length of 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) \( 45^\circ \)
(b) \( 30^\circ \)
(c) \( 60^\circ \)
(d) \( 90^\circ \)
Answer: (b)

Question. The angle of elevation of the top of a tower from a point on the ground, which is \( 30 \) m away from the foot of the tower is \( 45^\circ \). The height of the tower (in metres) is
(a) \( 15 \)
(b) \( 30 \)
(c) \( 30\sqrt{3} \)
(d) \( 10\sqrt{3} \)
Answer: (b)

Question. The angle of elevation of the top of a pillar from a point on the ground is \( 15^\circ \). On walking \( 100 \) m towards the pillar, the angle of elevation becomes \( 30^\circ \). Find the height of the pillar.
(a) \( 25 \) m
(b) \( 50 \) m
(c) \( 50\sqrt{2} \) m
(d) \( 25\sqrt{2} \) m
Answer: (b)

Question. The tops of two poles of height \( 18 \) m and \( 10 \) m are connected by a wire of length \( l \). If the wire makes an angle of \( 30^\circ \) with the horizontal, then \( l \) is equal to
(a) \( 26 \) m
(b) \( 16 \) m
(c) \( 12 \) m
(d) \( 10 \) m
Answer: (b)

Question. A person walking \( 20 \) m towards a chimney in a horizontal line through its base observes that its angle of elevation changes from \( 30^\circ \) to \( 45^\circ \). The height of chimney is
(a) \( \frac{20}{\sqrt{3} + 1} \) m
(b) \( \frac{20}{\sqrt{3} - 1} \) m
(c) \( 20(\sqrt{3} - 1) \) m
(d) None of these
Answer: (b)

Question. If the angle of elevation of a cloud from a point \( h \) metres above a lake is \( \alpha \) and the angle of depression of its reflection in the lake is \( \beta \), then the height of the cloud is
(a) \( \frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha} \)
(b) \( \frac{h(\tan \beta - \tan \alpha)}{\tan \beta + \tan \alpha} \)
(c) \( \frac{h}{\tan \beta - \tan \alpha} \)
(d) \( \frac{\tan \beta + \tan \alpha}{\tan \beta - \tan \alpha} \)
Answer: (a)

Question. The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of \( 100 \) m from its base is \( 45^\circ \). If the angle of elevation of the top of the complete pillar at the same point is to be \( 60^\circ \), then the height of the incomplete pillar is to be increased by
(a) \( 100(\sqrt{3} + 1) \) m
(b) \( 100 \) m
(c) \( 100\sqrt{3} \) m
(d) \( 100(\sqrt{3} - 1) \) m
Answer: (d)

Question. A wall \( 8 \) m long casts a shadow \( 5 \) m long. At the same time, a tower casts a shadow \( 50 \) m long, then the height of tower is
(a) \( 20 \) m
(b) \( 80 \) m
(c) \( 40 \) m
(d) \( 200 \) m
Answer: (b)

Question. From the foot of a pole, the angle of elevation of the top of a tower is \( 60^\circ \) and from the top of the pole, the angle of elevation is \( 30^\circ \). If the height of the pole is \( 25 \) m, then the height of the tower is
(a) \( 35 \) m
(b) \( 42.5 \) m
(c) \( 37.5 \) m
(d) \( 27.5 \) m
Answer: (c)

Question. The length of a string between a kite and a point on the ground is \( 85 \) m. If the string makes an angle \( \theta \) with level ground such that \( \tan \theta = \frac{15}{8} \), then how high is the kite?
(a) \( 75 \) m
(b) \( 78.05 \) m
(c) \( 226 \) m
(d) None of these
Answer: (a)

Question. Two men standing on opposite sides of a flagstaff measure the angles of elevation of the top of the flagstaff is \( 30^\circ \) and \( 60^\circ \). If the height of the flagstaff is \( 20 \) m, then approximate distance between the men is (Use \( \sqrt{3} = 1.732 \))
(a) \( 46.19 \) m
(b) \( 40 \) m
(c) \( 50 \) m
(d) \( 30 \) m
Answer: (a)

Question. There are two temples one on each bank of a river just opposite to each other. One temple is \( 40 \) m high. As observed from the top of this temple, the angle of depression of the top and foot of the other temple are \( 30^\circ \) and \( 60^\circ \) respectively. The width of river is
(a) \( \frac{40\sqrt{3}}{3} \) m
(b) \( \frac{40}{3} \) m
(c) \( \frac{120}{\sqrt{3}} \) m
(d) \( \frac{80}{\sqrt{3}} \) m
Answer: (a)

Question. Suppose a straight vertical tree is broken at some point due to storm and the broken part is inclined at a certain distant from the foot of the tree. If the top of broken part of a tree touches the ground at a point whose distance from foot of the tree is equal to height of remaining part, then its angle of inclination is
(a) \( 30^\circ \)
(b) \( 60^\circ \)
(c) \( 45^\circ \)
(d) None of these
Answer: (c)

Question. The ratio of the height of a tree and its shadow is \( 1 : \frac{1}{\sqrt{3}} \). The angle of the Sun’s elevation is
(a) \( 30^\circ \)
(b) \( 45^\circ \)
(c) \( 60^\circ \)
(d) \( 90^\circ \)
Answer: (c)

Question. A steel pole is \( 30 \) m high. To keep the pole upright, one end of a steel wire is tied to the top of the pole while the other end has been fixed on the ground. If the steel wire makes an angle of \( 45^\circ \) with the horizontal through the base point of the pole, then find the length of the steel wire.
(a) \( 30\sqrt{2} \) m
(b) \( 30\sqrt{3} \) m
(c) \( 15 \) m
(d) \( 15\sqrt{2} \) m
Answer: (a)

Question. A portion of a \( 45 \) m long tree is broken by tornado and the top struck up the ground making an angle of \( 30^\circ \) with the ground level. The height of the point where the tree is broken, is equal to
(a) \( 30 \) m
(b) \( 15 \) m
(c) \( 10 \) m
(d) \( 20 \) m
Answer: (b)

Question. A man standing on the deck of a ship, which is \( 10 \) m above the water level observes the angle of elevation of the top of a hill as \( 60^\circ \) and the angle of depression of the base of the hill as \( 30^\circ \). The distance of the hill from the ship is
(a) \( 40 \) m
(b) \( 10\sqrt{3} \) m
(c) \( 10 \) m
(d) \( 20\sqrt{3} \) m
Answer: (b)

Question. The length of shadow of a building, when the Sun’s altitude is \( 60^\circ \), is \( 20 \) m less than that it was when it was \( 45^\circ \). The height of the building is (Use \( \sqrt{3} = 1.732 \))
(a) \( 54.48 \) m
(b) \( 47.32 \) m
(c) \( 64.32 \) m
(d) \( 57.48 \) m
Answer: (b)

Question. A peacock sitting on the top of a tree observes a serpent on the ground making an angle of depression \( 30^\circ \). If the peacock with a speed of \( 300 \) m per minute catches the serpent in \( 12 \) seconds, then the height of the tree is
(a) \( 30 \) m
(b) \( 30\sqrt{3} \) m
(c) \( \frac{30}{\sqrt{3}} \) m
(d) \( 15 \) m
Answer: (a)

Question. A bridge across a river makes an angle of \( 45^\circ \) with the river bank. If the length of the bridge across the river is \( 50 \) m, then what is the width of the river?
(a) \( 20\sqrt{2} \) m
(b) \( 50\sqrt{2} \) m
(c) \( 25\sqrt{2} \) m
(d) \( 10\sqrt{2} \) m
Answer: (c)

Question. The angle of elevation is always
(a) obtuse angles
(b) acute angles
(c) right angles
(d) reflex angles
Answer: (b)

Question. If the height of a tree is \( 6 \) m, which is broken by wind in such a way that its top touches the ground and makes an angle \( 30^\circ \) with the ground. At what height from the bottom of the tree is broken by the wind?
(a) \( 2 \) m
(b) \( 4 \) m
(c) \( 8 \) m
(d) \( 10 \) m
Answer: (a)

Question. A window is \( 6 \) m above the ground. A ladder is placed against the wall such that its top reaches the window. If angle made by the foot of ladder to the ground is \( 30^\circ \), then length of the ladder is
(a) \( 8 \) m
(b) \( 10 \) m
(c) \( 12 \) m
(d) \( 14 \) m
Answer: (c)

Question. If the height of the window is \( 8 \) m above the ground. A ladder is placed against the wall such that its top reaches the window. If angle of elevation is observed to be \( 45^\circ \), then horizontal distance between the foot of ladder and wall is
(a) \( 2 \) m
(b) \( 4 \) m
(c) \( 6 \) m
(d) \( 8 \) m
Answer: (d)