CBSE Class 10 Mathematics Some Application of Trigonometry MCQs Set 03

Question. A vertical pole consists of two parts, the lower part being one third of the whole. At a point in the horizontal plane through the base of the pole and distance 20 meters from it, the upper part of the pole subtends an angle whose tangent is \( \frac{1}{2} \). The possible heights of the pole are
(a) 20 m and \( 20\sqrt{3} \) m
(b) 20 m and 60 m
(c) 16 m and 48 m
(d) None of these
Answer: B

Question. If the angle of depression of an object from a 75 m high tower is \( 30^{\circ} \), then the distance of the object from the tower is
(a) \( 25\sqrt{3} \) m
(b) \( 50\sqrt{3} \) m
(c) \( 75\sqrt{3} \) m
(d) 150 m
Answer: C

Question. The height of a tree, if it casts a shadow 15 m long on the level of ground, when the angle of elevation of the sun is \( 45^{\circ} \), is
(a) 10 m
(b) 14 m
(c) 8 m
(d) 15 m
Answer: D

Question. If the height and length of the shadow of a man are the same, then the angle of elevation of the sun is,
(a) \( 45^{\circ} \)
(b) \( 60^{\circ} \)
(c) \( 90^{\circ} \)
(d) \( 120^{\circ} \)
Answer: A

Question. The ratio of the length of a rod and its shadow is \( 1 : \sqrt{3} \) then the angle of elevation of the sun is
(a) \( 90^{\circ} \)
(b) \( 45^{\circ} \)
(c) \( 30^{\circ} \)
(d) \( 75^{\circ} \)
Answer: C

Question. A tree is broken by the wind. The top struck the ground at an angle of \( 30^{\circ} \) and at distance of 10 m from its root. The whole height of the tree is (\( \sqrt{3} = 1.732 \))
(a) \( 10\sqrt{3} \) m
(b) \( 3\sqrt{10} \) m
(c) \( 20\sqrt{3} \) m
(d) \( 3\sqrt{20} \) m
Answer: A

Question. A circle artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground, then the height of pole, if the angle made by the rope with the ground level is \( 30^{\circ} \), is
(a) 5 m
(b) 10 m
(c) 15 m
(d) 20 m
Answer: B

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 the height of kite is
(a) 75 m
(b) 78.05 m
(c) 226 m
(d) None of these
Answer: A

Question. The angle of elevation of the top of a tower at point on the ground is \( 30^{\circ} \). If on walking 20 meters toward the tower, the angle of elevation become \( 60^{\circ} \), then the height of the tower is
(a) 10 meter
(b) \( \frac{10}{\sqrt{3}} \) metre
(c) \( 10\sqrt{3} \) metre
(d) None of these
Answer: C

Question. The top of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of \(30^\circ\) with the horizontal, then the length of the wire is
(a) 12 m
(b) 10 m
(c) 8 m
(d) 6 m
Answer: A

Question. An observer, 1.5 m tall is 20.5 m away from a tower 22 m high, then the angle of elevation of the top of the tower from the eye of observer is
(a) \(30^\circ\)
(b) \(45^\circ\)
(c) \(60^\circ\)
(d) \(90^\circ\)
Answer: B

Question. A tree 6 m tall cast a 4 m long shadow. At the same time, a flag pole cast a shadow 50 m long. How long is the flag pole?
(a) 75 m
(b) 100 m
(c) 150 m
(d) 50 m
Answer: A

Question. From the top of a 7 m high building the angle of elevation of the top of a cable tower is \(60^\circ\) and the angle of depression of its foot is \(45^\circ\), then the height of the tower is
(a) 14.124 m
(b) 17.124 m
(c) 19.124 m
(d) 15.124 m
Answer: C

Question. The angles of elevation of the top of a tower from the points \(P\) and \(Q\) at distance of \(a\) and \(b\) respectively from the base and in the same straight line with it, are complementary. The height of the tower is
(a) \(ab\)
(b) \(\sqrt{ab}\)
(c) \(\sqrt{\frac{a}{b}}\)
(d) \(\sqrt{\frac{b}{a}}\)
Answer: B

Question. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are \(45^\circ\) and \(60^\circ\) respectively, then the height of the tower is
(a) 14.64 m
(b) 28.64 m
(c) 38.64 m
(d) 19.64 m
Answer: A

Question. A tower stands at the centre of a circular park. If \(A\) and \(B\) are two points on the boundary of the park, such that \(AB = a \text{ m}\) subtends an angle of \(60^\circ\) at the foot of the tower and the angle of elevation of the top of the tower from \(A\) or \(B\) is \(30^\circ\). Find, then the height of the tower is
(a) \(\sqrt{3} a \text{ m}\)
(b) \(a/\sqrt{3} \text{ m}\)
(c) \(\frac{\sqrt{3}}{a} \text{ m}\)
(d) None of these
Answer: B

Question. A ladder rests against a vertical wall at an inclination \(\alpha\) to the horizontal. If its foot is pulled away from the wall through a distance \(p\) so that its upper end slides at distance \(q\) down the wall and then the ladder makes an angle \(\beta\) to the horizontal, then \(\frac{\cos \beta - \cos \alpha}{\sin \alpha - \sin \beta}\) is equal to
(a) \(p/q\)
(b) \(p/q\)
(c) \(pq\)
(d) \(1/pq\)
Answer: B

Question. A kite is flying at a height of 80 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with ground is \(60^\circ\), then the length of the string is
(a) 62.37 m
(b) 92.37 m
(c) 52.57 m
(d) 72.57 m
Answer: B

Question. A spherical balloon of radius \( r \) subtends an angle \( \theta \) at the eye of the observer. If the angle of elevation of its centre is \( \phi \), then the height of the centre of balloon is
(a) \( r \sin \phi / 2 \cos \theta \)
(b) \( r \sin \theta \text{ cosec } \phi \)
(c) \( r \sin \phi \text{ cosec } \theta / 2 \)
(d) None of these
Answer: C

FILL IN THE BLANK

Question. The .......... of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object.
Answer: angle of elevation

Question. The .......... of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we raise our head to look at the object.
Answer: angle of depression

Question. The .......... is the line drawn from the eye of an observer to the point in the object viewed by the observer.
Answer: line of sight

Question. .......... are used to find height or length of an object or distance between two distant objects.
Answer: Trigonometric ratios

Question. The top of a building from a fixed point is observed at an angle of elevation \( 60^\circ \) and the distance from the foot of the building to the point is 100 m. then the height of the building is ..........
Answer: \( 100\sqrt{3} \)

TRUE/FALSE

DIRECTION : Read the following statements and write your answer as true or false.
A straight highway leads to the foot of a tower of height 50 m. From the top of the tower, the angles of depression of two cars standing on the highway are \( 30^\circ \) and \( 60^\circ \). Now, based on the above information, mark the given statements as true or false.

Question. Distance between the cars is 57.6 m.
Answer: True

Question. First car is at a distance of 38.90 m from the tower.
Answer: False

Question. Second car is at a distance of 86.50 m from the tower.
Answer: True

Question. Car at point C is at a distance of 200 m away from the top of the tower.
Answer: False

Question. When the length of the shadow of a pole is equal to the height of the pole, then the angle of elevation of source of light is \( 90^\circ \).
Answer: False

Question. When we lower our head to look at the object, the angle formed by the line of sight with horizontal is known as angle of depression.
Answer: True

Question. If two towers of height \( h_1 \) and \( h_2 \), and \( \frac{h_1}{x} = \tan 60^\circ \); \( \frac{h_2}{x} = \tan 30^\circ \) respectively at the mid-point of the line joining their feet then, \( h_1 : h_2 = 3 : 1 \).
Answer: True

Question. If the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains unchanged.
Answer: True

Question. The angle of elevation of the top of a building from the top and bottom of a tree are \( x \) and \( y \) respectively. If the height of the tree is \( h \) metre, then the height of the building is \( \frac{h \cot x}{\cot x - \cot y} \).
Answer: True

Question. The angle for which sine and cosine have equal values is \( 90^\circ \).
Answer: False

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : If the length of shadow of a vertical pole is equal to its height, then the angle of elevation of the sun is \( 45^\circ \).
Reason : According to pythagoras theorem, \( h^2 = l^2 + b^2 \), where \( h = \text{hypotenuse} \), \( l = \text{length} \) and \( b = \text{base} \)
Answer: (b)
Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A). Both Assertion and Reason are correct, but Reason is not the correct explanation of the Assertion.