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

Question. Man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be 60°. Find the time taken by the boat to reach the shore.
(a) 6 min.
(b) 7 min.
(c) 8 min.
(d) 9 min.
Answer: (d) 9 min.

Question. The value of \( \frac{\tan^3 \theta}{1 + \tan^2 \theta} + \frac{\cot^3 \theta}{1 + \cot^2 \theta} = \)
(a) \( \frac{1 - \sin^2 \theta \cos^2 \theta}{2 \sin \theta \cos \theta} \)
(b) \( \frac{1 + 2 \sin^2 \theta \cos^2 \theta}{\sin \theta \cos \theta} \)
(c) \( \frac{1 - 2 \sin^2 \theta \cos^2 \theta}{\sin \theta \cos \theta} \)
(d) \( \frac{2 \sin^2 \theta \cos^2 \theta}{1 - \sin \theta \cos \theta} \)
Answer: (c) \( \frac{1 - 2 \sin^2 \theta \cos^2 \theta}{\sin \theta \cos \theta} \)

Question. The value of \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \dots \tan 89^\circ \) is
(a) 0
(b) 1
(c) – 1
(d) 2
Answer: (b) 1

Question. If \( x \tan 45^\circ \sin 30^\circ = \cos 30^\circ \tan 30^\circ \), then \( x \) is equal to
(a) \( \sqrt{3} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{\sqrt{2}} \)
(d) 1
Answer: (d) 1

Question. If \( x \) and \( y \) are complementary angles, then
(a) \( \sin x = \sin y \)
(b) \( \tan x = \tan y \)
(c) \( \cos x = \cos y \)
(d) \( \sec x = \text{cosec } y \)
Answer: (d) \( \sec x = \text{cosec } y \)

Question. If A, B and C are interior angles of a \( \Delta ABC \), then \( \cos \left( \frac{B + C}{2} \right) \) is equal to
(a) \( \sin \frac{A}{2} \)
(b) \( - \sin \frac{A}{2} \)
(c) \( \cos \frac{A}{2} \)
(d) \( - \cos \frac{A}{2} \)
Answer: (a) \( \sin \frac{A}{2} \)

Question. If \( \cot \theta + \cos \theta = p \) and \( \cot \theta - \cos \theta = q \), then the value of \( p^2 - q^2 \) is :
(a) \( 2\sqrt{pq} \)
(b) \( 4\sqrt{pq} \)
(c) \( 2pq \)
(d) \( 4pq \)
Answer: (b) \( 4\sqrt{pq} \)

Question. If \( \sec A + \tan A = x \), then \( \sec A = \)
(a) \( \frac{x^2 - 1}{x} \)
(b) \( \frac{x^2 - 1}{2x} \)
(c) \( \frac{x^2 + 1}{x} \)
(d) \( \frac{x^2 + 1}{2x} \)
Answer: (d) \( \frac{x^2 + 1}{2x} \)

Question. The value of \( 5 \tan^2 A - 5 \sec^2 A + 1 \) is equal to
(a) 6
(b) –5
(c) 1
(d) – 4
Answer: (d) – 4

Question. If \( x = a \cos \theta \) and \( y = b \sin \theta \), then \( b^2x^2 + a^2y^2 = \)
(a) \( ab \)
(b) \( b^2 + a^2 \)
(c) \( a^2b^2 \)
(d) \( a^4b^4 \)
Answer: (c) \( a^2b^2 \)

Question. Given that \( \sin A = \frac{\sqrt{3}}{2} \) and \( \cos B = \frac{\sqrt{3}}{2} \), then \( \tan(A + B) = \)
(a) \( \frac{1}{\sqrt{3}} \)
(b) 1
(c) \( \sqrt{3} \)
(d) Not defined
Answer: (d) Not defined

Question. If \( \cos \theta - \sin \theta = \sqrt{2} \sin \theta \), then \( \cos \theta + \sin \theta \)
(a) \( \sqrt{2} \sin \theta \)
(b) \( \sqrt{2} \cos \theta \)
(c) \( 2 \cos \theta \)
(d) \( 2 \sin \theta \)
Answer: (b) \( \sqrt{2} \cos \theta \)

Question. If \( x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta \) and \( x \sin \theta - y \cos \theta = 0 \). Then \( x^2 + y^2 = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. If \( \sin A - \cos A = 0 \), then the value of \( \sin^4 A + \cos^4 A \) is
(a) 2
(b) 1
(c) 3/4
(d) 1/2
Answer: (d) 1/2

Question. The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sunrays meet the ground at an angle of 60°. find the angle between the sunrays and the ground at the time of long shadow.
(a) 60°
(b) 90°
(c) 45°
(d) 30°
Answer: (d) 30°

Question. A boy standing on the ground and flying a kite with 75 m of string at an elevation of 45°. Another boy is standing on the roof of 25 m high building and is flying his kite at an elevation of 30°. Both the boys are on the opposite side of the two kites. Find the length of the string that the second boy must have, so that the kites meet.
(a) 43.05 m
(b) 34.05 m
(c) 45.05 m
(d) 56.05 m
Answer: (d) 56.05 m

Question. If \( \frac{\sin A}{\sin B} = p \) and \( \frac{\cos A}{\cos B} = q \), then \( \tan B \) is equal to
(a) \( \pm \sqrt{\frac{p^2 - q^2}{q^2(1 - p^2)}} \)
(b) \( \pm \sqrt{\frac{q^2 - 1}{1 - p^2}} \)
(c) \( \pm \frac{p}{q} \sqrt{\frac{p^2 - 1}{1 - q^2}} \)
(d) None of these
Answer: (b) \( \pm \sqrt{\frac{q^2 - 1}{1 - p^2}} \)

Question. If \( T_n = \sin^n \theta + \cos^n \theta \), then \( 2T_6 - 3T_4 + 1 \) is equal to
(a) 0
(b) \( \sin \theta \)
(c) \( \cos \theta \)
(d) \( 2 \sin \theta \cos \theta \)
Answer: (a) 0

Question. If \( T_n = \sin^n \theta + \cos^n \theta \), then \( 6T_{10} - 15T_8 + 10T_6 - 1 \) is equal to
(a) 0
(b) 1
(c) \( \sin^2 \theta \)
(d) \( \sin^3 \theta \)
Answer: (a) 0

Question. If \( \sin \theta \) and \( \cos \theta \) are the roots of \( ax^2 + bx + c = 0 (ac \neq 0) \), then
(a) \( a^2 + b^2 - 2ac = 0 \)
(b) \( a^2 - b^2 + 2ac = 0 \)
(c) \( (a + c)^2 = b^2 + c^2 \)
(d) None of these
Answer: (b) \( a^2 - b^2 + 2ac = 0 \)

Question. A flagstaff 5 m high stands on a building 25 m high. At an observer at a height of 30 m, the flagstaff and the building subtended equal angles. The distance of the observer from the top of the flagstaff is :-
(a) \( \frac{5\sqrt{3}}{2} \)
(b) \( 5\sqrt{\frac{3}{2}} \)
(c) \( 5\sqrt{\frac{2}{3}} \)
(d) None
Answer: (c) \( 5\sqrt{\frac{2}{3}} \)

Question. In an equilateral triangle the inradius r and circumradius R are connected by
(a) \( r = \frac{R}{3} \)
(b) \( r = \frac{R}{2} \)
(c) \( r = 4R \)
(d) None
Answer: (b) \( r = \frac{R}{2} \)

Question. The perimeter of a triangle ABC is 6 times the A.M. of the sines of its angles. If \( a = 1 \), the angle A (acute) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{6} \)
Answer: (d) \( \frac{\pi}{6} \)

Question. The angle of elevation of the top of two vertical towers as seen from the middle point of the line joining the feet of the towers are 60° and 30° respectively. The ratio of heights of the towers is :-
(a) 2 : 1
(b) \( \sqrt{3} : 1 \)
(c) 3 : 2
(d) 3 : 1
Answer: (d) 3 : 1

Question. A tower of height h standing at the centre of a square with sides of length a makes the same angle \( \alpha \) at each of the four corners. Then \( \frac{a^2}{h^2 \cot^2 \alpha} \) is :-
(a) 1
(b) 3/2
(c) 2
(d) 4
Answer: (c) 2

Question. A river flows due North, and a tower stands on its left bank. From a point A upstream and on the same bank as the tower, the elevation of the tower is 60° and from a point B just opposite A on the other bank the elevation is 45°. If the tower is 360 m high, the breadth of the river is :-
(a) \( 120\sqrt{6} \) m
(b) \( \frac{240}{\sqrt{3}} \) m
(c) \( 240\sqrt{3} \) m
(d) \( 240\sqrt{6} \) m
Answer: (b) \( \frac{240}{\sqrt{3}} \) m

Question. A tower subtends an angle of 30° at a point on the same level as the foot of tower. At a second point h m high above the first, the depression of the foot of tower is 60°. The horizontal distance of the tower from the point is:-
(a) \( \frac{h}{\sqrt{3}} \)
(b) \( \frac{h \cot 60^\circ}{\sqrt{3}} \)
(c) \( \frac{h \cot 60^\circ}{3} \)
(d) \( h \cot 30^\circ \)
Answer: (a) \( \frac{h}{\sqrt{3}} \)

Question. The angle of elevation of a cloud at a height h above the level of water in a lake is \( \alpha \) and the angle of depression of its image in the lake is \( \beta \). The height of the cloud above the surface of the lake is equal to
(a) \( \frac{h(\tan \beta + \tan \alpha)}{(\tan \beta - \tan \alpha)} \)
(b) \( \frac{h \cos(\alpha + \beta)}{\sin(\beta - \alpha)} \)
(c) \( \frac{h(\cot \alpha + \cot \beta)}{\cos \alpha - \cos \beta} \)
(d) h
Answer: (a) \( \frac{h(\tan \beta + \tan \alpha)}{(\tan \beta - \tan \alpha)} \)

Question. A balloon of radius r subtends an angle \( \alpha \) at the eye of an observer and the elevation of the centre of the balloon from the eye is \( \beta \), the height h of the centre of the balloon is given by :-
(a) \( \frac{r \sin \beta}{\sin \alpha} \)
(b) \( r \sin \beta \sin \alpha \)
(c) \( \frac{r \sin \beta}{\sin(\alpha / 2)} \)
(d) \( \frac{r \sin \alpha}{\sin(\beta / 2)} \)
Answer: (c) \( \frac{r \sin \beta}{\sin(\alpha / 2)} \)

Question. A man observes that when he moves up a distance c metres on a slope, the angle of depression of a point on the horizontal plane from the base of the slope is 30°, and when he moves up further a distance c metres, the angle of depression of that point is 45°. The angle of inclination of the slope with the horizontal is :-
(a) 60°
(b) 45°
(c) 75°
(d) 30°
Answer: (d) 30°

Question. The angle of elevation of the top of a tower from a point A due south of the tower is \( \alpha \) and from a point B due east of the tower is \( \beta \). If AB = d, then the height of the tower is :-
(a) \( \frac{d}{\sqrt{\tan^2 \alpha - \tan^2 \beta}} \)
(b) \( \frac{d}{\sqrt{\tan^2 \alpha + \tan^2 \beta}} \)
(c) \( \frac{d}{\sqrt{\cot^2 \alpha + \cot^2 \beta}} \)
(d) \( \frac{d}{\sqrt{\cot^2 \alpha - \cot^2 \beta}} \)
Answer: (c) \( \frac{d}{\sqrt{\cot^2 \alpha + \cot^2 \beta}} \)

Question. AB is vertical tower. The point A is on the ground and C is the middle point of AB. The part CB subtend an angle \( \alpha \) at a point P on the ground. If AP = nAB, then \( \tan \alpha = \)
(a) \( n(n^2 + 1) \)
(b) \( \frac{n}{2n^2 - 1} \)
(c) \( \frac{n}{2n^2 + 1} \)
(d) \( \frac{n}{2n^2 + 1} \)
Answer: (c) \( \frac{n}{2n^2 + 1} \)

Question. The top of a hill observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. The height of hill is :-
(a) \( \frac{h \cot p}{\cot q - \cot p} \)
(b) \( \frac{h \cot p}{\cot p - \cot q} \)
(c) \( \frac{h \tan p}{\tan p - \tan q} \)
(d) \( \frac{h \tan p}{\tan q - \tan p} \)
Answer: (d) \( \frac{h \tan p}{\tan q - \tan p} \)

Question. A man standing on a level plane observes the elevation of the top of a pole to be \( \alpha \). He then walks a distance equal to double the height of the pole and finds that the elevation is now 2\( \alpha \). Then \( \alpha = \)
(a) \( \pi / 6 \)
(b) \( \pi / 4 \)
(c) \( \pi / 12 \)
(d) \( \pi / 8 \)
Answer: (c) \( \pi / 12 \)

Question. The angle of elevation of a cloud from a point x m above a take is \( \theta \) and the angle of depression of its reflection in the take is 45°. The height of the cloud is
(a) \( x \tan(45^\circ - \theta) \)
(b) \( x \tan(45^\circ + \theta) \)
(c) \( \frac{1}{x} \cot(45^\circ - \theta) \)
(d) \( \frac{1}{x} \cot(45^\circ + \theta) \)
Answer: (b) \( x \tan(45^\circ + \theta) \)

Question. Two poles of height a and b stand at the centres of two circular plots which touch each other externally at a point and the two poles subtend angles of 30° and 60° respectively at this point, then distance between the centres of these plots is :-
(a) \( a + b \)
(b) \( \frac{3a + b}{\sqrt{3}} \)
(c) \( \frac{a + 3b}{\sqrt{3}} \)
(d) \( a\sqrt{3} + b \)
Answer: (b) \( \frac{3a + b}{\sqrt{3}} \)

Question. From a point on the horizontal plane, the elevation of the top of a hill is 45°. After walking 500 m towards its summit up a slope inclined at an angle of 15° to the horizon the elevation is 75°, the height of the hill is :-
(a) \( 500\sqrt{6} \) m
(b) \( 500\sqrt{3} \) m
(c) \( 250\sqrt{6} \) m
(d) \( 250\sqrt{3} \) m
Answer: (d) \( 250\sqrt{3} \) m

Question. If \( x = a \text{ cosec}^n \theta \) and \( y = b \cot^n \theta \), then by eliminating \( \theta \)
(a) \( (x/a)^{2/n} + (y/b)^{2/n} = 1 \)
(b) \( (x/a)^{2/n} - (y/b)^{2/n} = 1 \)
(c) \( (x/a)^2 - (y/b)^2 = 1 \)
(d) \( (x/a)^{1/n} - (y/b)^{1/n} = 1 \)
Answer: (b) \( (x/a)^{2/n} - (y/b)^{2/n} = 1 \)

Question. Find the value of \( \frac{1}{(1 + \tan^2 \theta)} + \frac{1}{(1 + \cot^2 \theta)} \)
(a) 1/2
(b) 2
(c) 1
(d) 1/4
Answer: (c) 1

Question. If \( \tan \theta = p/q \), then \( \frac{p \sin \theta - q \cos \theta}{p \sin \theta + q \cos \theta} = \)
(a) \( (p^2 + q^2)/(p^2 - q^2) \)
(b) \( (p^2 - q^2)/(p^2 + q^2) \)
(c) \( (p^2 + q^2)/(p^2 - q^2) \)
(d) None of these
Answer: (b) \( (p^2 - q^2)/(p^2 + q^2) \)