JEE Mathematics Heights and Distances MCQs

Type – 1

Question. Choose the most appropriate option (a, b, c or d).

Question. A 6–ft tall man finds that the angle of elevation of the top of a 24-ft-high pillar and the angle of depression of its base are complementary angles.
(a) \(2\sqrt{3}\text{ft}\)
(b) \(8\sqrt{3}\text{ft}\)
(c) \(6\sqrt{3}\text{ft}\)
(d) None of the options
Answer: (c) \(6\sqrt{3}\text{ft}\)

Question. A rocket of height h metres is fired vertically upwards. Its velocity at time t seconds is (2t + 3) metres/second. If the angle of elevation of the top of the rocket from a point on the ground after 1 second of firing is \(\pi/6\) and after 3 seconds it is \(\pi/3\) then the distance of the point from the rocket is
(a) \(14\sqrt{3}\) metres
(b) \(7\sqrt{3}\) metres
(c) \(2\sqrt{3}\) metres
(d) cannot be found without the value of h
Answer: (b) \(7\sqrt{3}\) metres

Question. Three vertical poles of heights \(h_1, h_2\) and \(h_3\) at the vertices A, B and C of a \(\triangle ABC\) subtend angles \(\alpha, \beta\) and \(\gamma\) respectively at the circumcentre of the triangle. If \(\cot \alpha, \cot \beta\) and \(\cot \gamma\) are in AP then \(h_1, h_2, h_3\) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. The angle of elevation of the top of a hill from each of the vertices A, B, C of a horizontal triangle is \(\alpha\). The height of the hill is
(a) \(b \tan \alpha \cdot \text{cosec } B\)
(b) \(\frac{1}{2} a \tan \alpha \cdot \text{cosec } A\)
(c) \(\frac{1}{2} c \tan \alpha \cdot \text{cosec } C\)
(d) None of the options
Answer: (b) \(\frac{1}{2} a \tan \alpha \cdot \text{cosec } A\)

Question. A piece of paper in the shape of a sector of a circle of radius 10 cm and of angle \(216^\circ\) just covers the lateral surface of a right circular cone of vertical angle \(2\theta\). Then \(\sin \theta\) is
(a) \(3/5\)
(b) \(4/5\)
(c) \(3/4\)
(d) None of the options
Answer: (a) \(3/5\)

Question. The angle of elevation of the top of a vertical pole when observed from each vertex of a regular hexagon is \(\pi/3\). If the area of the circle circumscribing the hexagon be \(A \text{ metre}^2\) then the are of the hexagon is
(a) \(\frac{3\sqrt{3}}{8} A \text{ metre}^2\)
(b) \(\frac{\sqrt{3}}{\pi} A \text{ metre}^2\)
(c) \(\frac{3\sqrt{3}}{4\pi} A \text{ metre}^2\)
(d) \(\frac{3\sqrt{3}}{2\pi} A \text{ metre}^2\)
Answer: (d) \(\frac{3\sqrt{3}}{2\pi} A \text{ metre}^2\)

Question. A vertical pole PO is standing at the centre O of a square ABCD. If AC subtends an angle \(90^\circ\) at the top, P, of the pole then the angle subtended by a side of the square at P is
(a) \(45^\circ\)
(b) \(30^\circ\)
(c) \(60^\circ\)
(d) None of the options
Answer: (c) \(60^\circ\)

Question. A vertical lamp–post of height 9 metres stands at the corner of a rectangular field. The angle of elevation of its from the farthest corner is \(30^\circ\), while from another corner it is \(45^\circ\). The area of the field is
(a) \(81\sqrt{2} \text{ metre}^2\)
(b) \(9\sqrt{2} \text{ metre}^2\)
(c) \(81\sqrt{3} \text{ metre}^2\)
(d) \(9\sqrt{3} \text{ metre}^2\)
Answer: (a) \(81\sqrt{2} \text{ metre}^2\)

Question. A vertical lamp–post, 6 m high, stand at a distance of 2 m from a wall, 4 m high. A 1.5–m–tall man starts to walk away from the wall on the other side of the wall, in line with the lamp–post. The maximum distance to which the man can walk remaining in the shadow is
(a) \(\frac{5}{2}\) m
(b) \(\frac{3}{2}\) m
(c) 4 m
(d) None of the options
Answer: (a) \(\frac{5}{2}\) m

Question. A circular of radius 3 cm is suspended horizontally from a point 4 cm vertically above the centre by 4 strings attached at equal intervals to its circumference. If the angle between two consecutive stings be \(\theta\) then \(\cos \theta\) is
(a) \(4/5\)
(b) \(16/25\)
(c) \(7/25\)
(d) None of the options
Answer: (c) \(7/25\)

Type 2

Question. Choose the correct options. One or more options may be correct.

Question. A man standing between two vertical posts finds that the angle subtended at his eyes by the tops of the post is a right angle. If the heights of the two posts are two times and four times the height of the man, and the distance between them is equal to the length of the longer post, then the ratio of the distance of the man from the shorter and the longer post is
(a) 3 : 1
(b) 2 : 3
(c) 3 : 2
(d) 1 : 3
Answer: (a) 3 : 1, (d) 1 : 3

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) \(\sqrt{3} : 1\)
(b) 1 : 3
(c) \(1 : \sqrt{3}\)
(d) \(\sqrt{3} : 2\)
Answer: (c) \(1 : \sqrt{3}\)