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

Question. If \( \tan A + \sin A = m \) and \( \tan A - \sin A = n \), then \( \frac{(m^2 - n^2)^2}{mn} = \)
(a) 4
(b) 3
(c) 16
(d) 9
Answer: (c) 16

Question. If \( \text{cosec } \theta - \sin \theta = m \) and \( \sec \theta - \cos \theta = n \) then \( (m^2n)^{2/3} + (mn^2)^{2/3} = \)
(a) –1
(b) 1
(c) 0
(d) None of these
Answer: (b) 1

Question. If \( 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \), then \( \tan \theta = \)
(a) \( \pm \frac{1}{\sqrt{3}} \)
(b) \( \pm \frac{1}{2} \)
(c) \( \pm \sqrt{3} \)
(d) \( \pm \frac{1}{\sqrt{2}} \)
Answer: (a) \( \pm \frac{1}{\sqrt{3}} \)

Question. The value of \( \frac{\sin^3 A + \cos^3 A}{\sin A + \cos A} + \frac{\cos^3 A - \sin^3 A}{\cos A - \sin A} \) is
(a) 0
(b) 1
(c) 2
(d) None of these
Answer: (c) 2

Question. \( \frac{\cot \theta - \text{cosec } \theta + 1}{\cot \theta + \text{cosec } \theta - 1} \) is equal to
(a) 1
(b) \( \cot \theta + \text{cosec } \theta \)
(c) \( \text{cosec } \theta - \cot \theta \)
(d) None of these
Answer: (c) \( \text{cosec } \theta - \cot \theta \)

Question. One side of a parallelogram is 12 cm and its area is 60 cm². If the angle between the adjacent sides is 30°, then its other side is
(a) 10 cm
(b) 8 cm
(c) 6 cm
(d) 4 cm
Answer: (a) 10 cm

Question. A flagstaff stands vertically on a pillar, the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from the pillar is
(a) 1 : 3
(b) \( 3 : 1 \)
(c) \( 1 : \sqrt{3} \)
(d) \( \sqrt{3} : 2 \)
Answer: (c) \( 1 : \sqrt{3} \)

Question. The distance between two multistoried buildings is 60 m. The angle of depression of the top of the first building as seen from the top of the second building which is 150 m high is 30°. The heigth of the first building is
(a) \( (150 + 20\sqrt{3}) \) m
(b) \( (150 - 20\sqrt{3}) \) m
(c) \( (150 + 10\sqrt{3}) \) m
(d) \( (15 - 10\sqrt{3}) \) m
Answer: (b) \( (150 - 20\sqrt{3}) \) m

Question. An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°. If after 10 s the elevation be 30°, the uniform speed of the aeroplane is
(a) \( 240\sqrt{3} \) km/hrs
(b) \( 240/\sqrt{3} \) km/hrs
(c) \( 120/\sqrt{3} \) km/hrs
(d) \( 120/\sqrt{3} \) km/hrs
Answer: (b) \( 240/\sqrt{3} \) km/hrs

Question. A balloon leaves the earth at point A and rises at a uniform velocity. At the end of \( 1 \frac{1}{2} \) min, an observer situated at a distance of 200 m from A finds the angular elevation of the balloon to be 60°. The speed of the balloon is
(a) 5.87 m/s
(b) 4.87 m/s
(c) 3.87 m/s
(d) 6.87 m/s
Answer: (c) 3.87 m/s

Question. At the foot of a mountain, the elevation of its summit is 45°. After ascending one kilometer the mountain upon and incline of 30°, the elevation changes to 60°. The height of the mountain is
(a) 1.366 km
(b) 1.266 km
(c) 1.166 km
(d) 1.466 km
Answer: (a) 1.366 km

Question. If \( x = r \cos \alpha \cos \beta, y = r \cos \alpha \sin \beta \text{ and } z = r \sin \alpha \) then \( x^2 + y^2 + z^2 \) is equal to
(a) \( r^2 \)
(b) \( r^4 \)
(c) 1
(d) None of these
Answer: (a) \( r^2 \)

Question. From the top of a light house, the angles of depression of two stations on opposite sides of it at distance ‘a’ apart are \( \alpha \text{ and } \beta \). The height of the light house is
(a) \( \frac{a}{\cot \alpha \cot \beta} \)
(b) \( \frac{a}{\cot \alpha + \cot \beta} \)
(c) \( \frac{a \cot \alpha \cot \beta}{\cot \alpha + \cot \beta} \)
(d) \( \frac{a \tan \alpha \cot \beta}{\cot \alpha + \cot \beta} \)
Answer: (b) \( \frac{a}{\cot \alpha + \cot \beta} \)

Question. If the angle of elevation of an object from a point 100 m above a lake is found to be 30° and the angle of depression of its image in lake is 45°, then the height of the object above the lake is
(a) \( 100(2 - \sqrt{3}) \) m
(b) \( 100(2 + \sqrt{3}) \) m
(c) \( 100(\sqrt{3} - 1) \) m
(d) \( 1000(\sqrt{3} + 1) \) m
Answer: (c) \( 100(\sqrt{3} - 1) \) m

Question. A person standing on the bank of a river observes that the angles 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) 40 m
(b) 60 m
(c) 20 m
(d) 30 m
Answer: (c) 20 m

Question. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60° and when he retires 40 m away from the tree, the angle of elevation becomes 30°. The breadth of the river is
(a) 40 m
(b) 20 m
(c) 30
(d) 60 m
Answer: (b) 20 m