Height and Distance
2023
Q1
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of the top $\mathrm{P}$ of a tower from the feet of one person standing due South of the tower is $45^{\circ}$ and from the feet of another person standing due west of the tower is $30^{\circ}$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to
A.
10
B.
$\frac{5}{2} \sqrt{5}$
C.
$5 \sqrt{5}$
D.
5
2023
Q2
JEE Mains
MCQ
14 Mar 2026
From the top $\mathrm{A}$ of a vertical wall $\mathrm{AB}$ of height $30 \mathrm{~m}$, the angles of depression of the top $\mathrm{P}$ and bottom $\mathrm{Q}$ of a vertical tower $\mathrm{PQ}$ are $15^{\circ}$ and $60^{\circ}$ respectively, $\mathrm{B}$ and $\mathrm{Q}$ are on the same horizontal level. If $\mathrm{C}$ is a point on $\mathrm{AB}$ such that $\mathrm{CB}=\mathrm{PQ}$, then the area (in $\mathrm{m}^{2}$ ) of the quadrilateral $\mathrm{BCPQ}$ is equal to :
A.
$200(3-\sqrt{3})$
B.
$300(\sqrt{3}-1)$
C.
$300(\sqrt{3}+1)$
D.
$600(\sqrt{3}-1)$
2022
Q3
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of the top of a tower from a point A due north of it is $\alpha$ and from a point B at a distance of 9 units due west of A is $\cos ^{-1}\left(\frac{3}{\sqrt{13}}\right)$. If the distance of the point B from the tower is 15 units, then $\cot \alpha$ is equal to :
A.
$\frac{6}{5}$
B.
$\frac{9}{5}$
C.
$\frac{4}{3}$
D.
$\frac{7}{3}$
2022
Q4
JEE Mains
MCQ
14 Mar 2026
A horizontal park is in the shape of a triangle $\mathrm{OAB}$ with $\mathrm{AB}=16$. A vertical lamp post $\mathrm{OP}$ is erected at the point $\mathrm{O}$ such that $\angle \mathrm{PAO}=\angle \mathrm{PBO}=15^{\circ}$ and $\angle \mathrm{PCO}=45^{\circ}$, where $\mathrm{C}$ is the midpoint of $\mathrm{AB}$. Then $(\mathrm{OP})^{2}$ is equal to :
A.
$\frac{32}{\sqrt{3}}(\sqrt{3}-1)$
B.
$\frac{32}{\sqrt{3}}(2-\sqrt{3})$
C.
$\frac{16}{\sqrt{3}}(\sqrt{3}-1)$
D.
$\frac{16}{\sqrt{3}}(2-\sqrt{3})$
2022
Q5
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is $45^{\circ}$. Let R be a point on AQ and from a point B, vertically above $\mathrm{R}$, the angle of elevation of $\mathrm{P}$ is $60^{\circ}$. If $\angle \mathrm{BAQ}=30^{\circ}, \mathrm{AB}=\mathrm{d}$ and the area of the trapezium $\mathrm{PQRB}$ is $\alpha$, then the ordered pair $(\mathrm{d}, \alpha)$ is :
A.
$(10(\sqrt{3}-1), 25)$
B.
$\left(10(\sqrt{3}-1), \frac{25}{2}\right)$
C.
$(10(\sqrt{3}+1), 25)$
D.
$\left(10(\sqrt{3}+1), \frac{25}{2}\right)$
2022
Q6
JEE Mains
MCQ
14 Mar 2026
Let a vertical tower $A B$ of height $2 h$ stands on a horizontal ground. Let from a point $P%$ on the ground a man can see upto height $h$ of the tower with an angle of elevation $2 \alpha$. When from $P$, he moves a distance $d$ in the direction of $\overrightarrow{A P}$, he can see the top $B$ of the tower with an angle of elevation $\alpha$. If $d=\sqrt{7} h$, then $\tan \alpha$ is equal to
A.
$\sqrt{5}-2$
B.
$\sqrt{3}-1$
C.
$ \sqrt{7}-2$
D.
$\sqrt{7}-\sqrt{3}$
2022
Q7
JEE Mains
MCQ
14 Mar 2026
A tower PQ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the tower in two parts such that $Q R=15 \mathrm{~m}$. If from a point $A$ on the ground the angle of elevation of $R$ is $60^{\circ}$ and the part $P R$ of the tower subtends an angle of $15^{\circ}$ at $A$, then the height of the tower is :
A.
$5(2 \sqrt{3}+3) \,\mathrm{m}$
B.
$5(\sqrt{3}+3) \,\mathrm{m}$
C.
$10(\sqrt{3}+1) \,\mathrm{m}$
D.
$10(2 \sqrt{3}+1) \,\mathrm{m}$
2022
Q8
JEE Mains
MCQ
14 Mar 2026
From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60$^\circ$. The pole subtends an angle 30$^\circ$ at the top of the tower. Then the height of the tower is :
A.
$15\sqrt 3 $
B.
$20\sqrt 3 $
C.
20 + $10\sqrt 3 $
D.
30
2022
Q9
JEE Mains
MCQ
14 Mar 2026
Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let ${\pi \over 8}$ and $\theta$ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan2$\theta$ is equal to
A.
${{3 - 2\sqrt 2 } \over 2}$
B.
${{3 + \sqrt 2 } \over 2}$
C.
${{3 - 2\sqrt 2 } \over 4}$
D.
${{3 - \sqrt 2 } \over 4}$
2021
Q10
JEE Mains
MCQ
14 Mar 2026
A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is :
A.
12$\sqrt {15} $
B.
12$\sqrt {10} $
C.
8$\sqrt {10} $
D.
6$\sqrt {10} $
2021
Q11
JEE Mains
MCQ
14 Mar 2026
Two poles, AB of length a metres and CD of length a + b (b $\ne$ a) metres are erected at the same horizontal level with bases at B and D. If BD = x and tan$\angle$ACB = ${1 \over 2}$, then :
A.
x2 + 2(a + 2b)x $-$ b(a + b) = 0
B.
x2 + 2(a + 2b)x + a(a + b) = 0
C.
x2 $-$ 2ax + b(a + b) = 0
D.
x2 $-$ 2ax + a(a + b) = 0
2021
Q12
JEE Mains
MCQ
14 Mar 2026
A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = $\sqrt 5 $ inches and $\angle$PCB = tan-1(2). The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is :
A.
${\tan ^{ - 1}}\left( {{3 \over 4}} \right)$
B.
tan$-$1(1)
C.
${\tan ^{ - 1}}\left( {{4 \over 3}} \right)$
D.
${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$
2021
Q13
JEE Mains
MCQ
14 Mar 2026
A spherical gas balloon of radius 16 meter subtends an angle 60$^\circ$ at the eye of the observer A while the angle of elevation of its center from the eye of A is 75$^\circ$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :
A.
$8(2 + 2\sqrt 3 + \sqrt 2 )$
B.
$8(\sqrt 6 + \sqrt 2 + 2)$
C.
$8(\sqrt 2 + 2 + \sqrt 3 )$
D.
$8(\sqrt 6 - \sqrt 2 + 2)$
2021
Q14
JEE Mains
MCQ
14 Mar 2026
Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin$\theta$ is equal to :
A.
${{\sqrt 5 + 1} \over 4}$
B.
${{\sqrt 5 - 1} \over 2}$
C.
${{\sqrt 2 - 1} \over 2}$
D.
${{\sqrt 5 - 1} \over 4}$
2021
Q15
JEE Mains
MCQ
14 Mar 2026
A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be ${\pi \over 3}$. If the radius of the circumcircle of $\Delta$ABC is 2, then the height of the pole is equal to :
A.
${{1 \over {\sqrt 3 }}}$
B.
2${\sqrt 3 }$
C.
${\sqrt 3 }$
D.
${{{2\sqrt 3 } \over 3}}$
2021
Q16
JEE Mains
MCQ
14 Mar 2026
A man is observing, from the top of a tower, a boat speeding towards the lower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30$^\circ$ (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45$^\circ$. Then the time taken (in seconds) by the boat from B to reach the base of the tower is :
A.
$10(\sqrt 3 + 1)$
B.
$10(\sqrt 3 - 1)$
C.
10
D.
$10\sqrt 3$
2021
Q17
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of a jet plane from a point A on the ground is 60$^\circ$. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30$^\circ$. If the jet plane is flying at a constant height, then its height is :
A.
$3600\sqrt 3 $ m
B.
$1200\sqrt 3 $ m
C.
$1800\sqrt 3 $ m
D.
$2400\sqrt 3 $ m
2021
Q18
JEE Mains
MCQ
14 Mar 2026
Two vertical poles are 150 m apart and the height of one is three times that of the other. If
from the middle point of the line joining their feet, an observer finds the angles of elevation of
their tops to be complementary, then the height of the shorter pole (in meters) is :
A.
30
B.
25
C.
20$\sqrt 3 $
D.
25$\sqrt 3 $
2020
Q19
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of the summit of a
mountain from a point on the ground is 45°.
After climbing up one km towards the summit
at an inclination of 30° from the ground, the
angle of elevation of the summit is found to be
60°. Then the height (in km) of the summit from
the ground is :
A.
${1 \over {\sqrt 3 - 1}}$
B.
${{\sqrt 3 + 1} \over {\sqrt 3 - 1}}$
C.
${1 \over {\sqrt 3 + 1}}$
D.
${{\sqrt 3 - 1} \over {\sqrt 3 + 1}}$
2020
Q20
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of a cloud C from a point P, 200 m above a still lake is 30°. If the angle of depression of the image of C in the lake from the point P is 60°,then PC (in m) is equal to :
A.
$200\sqrt 3 $
B.
400
C.
100
D.
$400\sqrt 3 $
2019
Q21
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45o from
a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the
top of the tower from B be 30o, then the distance (in m) of the foot of the tower from the point A is :
A.
$15\left( {1 + \sqrt 3 } \right)$
B.
$15\left( {3 - \sqrt 3 } \right)$
C.
$15\left( {3 + \sqrt 3 } \right)$
D.
$15\left( {5 - \sqrt 3 } \right)$
${x \over d} = \tan {30^o} \Rightarrow {x \over d} = {1 \over {\sqrt 3 }} \Rightarrow d = \sqrt 3 x$
2019
Q22
JEE Mains
MCQ
14 Mar 2026
ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If
the angles of elevation of the top of the tower at A and B are cot–1
(3$\sqrt 2 $ ) and cosec–1
(2$\sqrt 2 $ ) respectively,
then the height of the tower (in metres) is :
A.
${{100} \over {3\sqrt 3 }}$
B.
25
C.
20
D.
10$\sqrt 5 $
2019
Q23
JEE Mains
MCQ
14 Mar 2026
Two poles standing on a horizontal ground are of
heights 5m and 10 m respectively. The line joining
their tops makes an angle of 15º with ground. Then
the distance (in m) between the poles, is :-
A.
$5\left( {2 + \sqrt 3 } \right)$
B.
${5 \over 2}\left( {2 + \sqrt 3 } \right)$
C.
$10\left( {\sqrt3 - 1 } \right)$
D.
$5\left( {\sqrt3 + 1 } \right)$
2019
Q24
JEE Mains
MCQ
14 Mar 2026
Two vertical poles of heights, 20 m and 80 m stand
a part on a horizontal plane. The height (in meters)
of the point of intersection of the lines joining the
top of each pole to the foot of the other, from this
horizontal plane is :
A.
12
B.
16
C.
15
D.
18
2019
Q25
JEE Mains
MCQ
14 Mar 2026
If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30o and the angle of depression of reflection of the cloud in the lake from P be 60o, then the height of the cloud (in meters) from
the surface of the lake is :
A.
45
B.
42
C.
50
D.
60
2019
Q26
JEE Mains
MCQ
14 Mar 2026
Consider a triangular plot ABC with sides AB = 7m, BC = 5m and CA = 6m. A vertical lamp-post at the mid point D of AC subtends an angle 30o at B. The height (in m) of the lamp-post is -
A.
$2\sqrt {21} $
B.
${3 \over 2}\sqrt {21} $
C.
$7\sqrt {3} $
D.
${2 \over 3}\sqrt {21} $
2018
Q27
JEE Mains
MCQ
14 Mar 2026
A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from 30o to 45o ; then after this, the time taken (in min.) by the car to reach the foot of the tower, is :
A.
$9\left( {1 + \sqrt 3 } \right)$
B.
$18\left( {1 + \sqrt 3 } \right)$
C.
$18\left( {\sqrt 3 - 1} \right)$
D.
${9 \over 2}\left( {\sqrt 3 - 1} \right)$
2018
Q28
JEE Mains
MCQ
14 Mar 2026
PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles
of elevation of the top of the tower at P, Q and R are respectively 45$^\circ $, 30$^\circ $ and 30$^\circ $, then the height of the tower (in m) is :
A.
$50\sqrt 2 $
B.
100
C.
50
D.
$100\sqrt 3 $
2018
Q29
JEE Mains
MCQ
14 Mar 2026
A tower T1 of height 60 m is located exactly opposite to a tower T2 of height 80 m on a straight road. Fromthe top of T1, if the angle of depression of the foot of T2 is twice the angle of elevation of the top of T2, then the width (in m) of the road between the feetof the towers T1 and T2 is :
A.
$10\sqrt 2 $
B.
$10\sqrt 3 $
C.
$20\sqrt 3 $
D.
$20\sqrt 2 $
2018
Q30
JEE Mains
MCQ
14 Mar 2026
An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt 3 $ kmabove it, is obsered at an elevation of ${60^o}$ from a point on the ground. If, after five seconds, its elevation from the same point, is ${30^o}$, then the speed (in km / hr) of the aeroplane, is :
A.
1500
B.
1440
C.
750
D.
720
2017
Q31
JEE Mains
MCQ
14 Mar 2026
Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point
on the ground such that AP = 2AB. If $\angle $BPC = $\beta $, then tan$\beta $ is equal to:
A.
${1 \over 4}$
B.
${2 \over 9}$
C.
${4 \over 9}$
D.
${6 \over 7}$
2016
Q32
JEE Mains
MCQ
14 Mar 2026
The angle of elevation of the top of a vertical tower from a point A, due east of it is 45o. The angle of elevation of the top of the same tower from a point B, due south of A is 30o. If the distance between A and B is $54\sqrt 2 \,m,$ then the height of the tower (in metres), is :
A.
$36\sqrt 3 $
B.
54
C.
$54\sqrt 3 $
D.
108
2016
Q33
JEE Mains
MCQ
14 Mar 2026
A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30o. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60o. Then the time taken (in minutes) by him, from B to reach the pillar, is :
A.
6
B.
10
C.
20
D.
5
2015
Q34
JEE Mains
MCQ
14 Mar 2026
If the angles of elevation of the top of a tower from three collinear points $A, B$ and $C,$ on a line leading to the foot of the tower, are ${30^ \circ }$, ${45^ \circ }$ and ${60^ \circ }$ respectively, then the ratio, $AB:BC,$ is :
A.
$1:\sqrt 3 $
B.
$2:3$
C.
$\sqrt 3 :1$
D.
$\sqrt 3 :\sqrt 2 $
2014
Q35
JEE Mains
MCQ
14 Mar 2026
A bird is sitting on the top of a vertical pole $20$ m high and its elevation from a point $O$ on the ground is ${45^ \circ }$. It files off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to ${30^ \circ }$. Then the speed (in m/s) of the bird is :
A.
$20\sqrt 2 $
B.
$20\left( {\sqrt 3 - 1} \right)$
C.
$40\left( {\sqrt 2 - 1} \right)$
D.
$40\left( {\sqrt 3 - \sqrt 2 } \right)$
2013
Q36
JEE Mains
MCQ
14 Mar 2026
$ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC \bot CD.$ If $\angle ADB = \theta ,\,BC = p$ and $CD = q,$ then AB is equal to:
A.
${{\left( {{p^2} + {q^2}} \right)\sin \theta } \over {p\cos \theta + q\sin \theta }}$
B.
${{{p^2} + {q^2}\cos \theta } \over {p\cos \theta + q\sin \theta }}$
C.
${{{p^2} + {q^2}} \over {{p^2}\cos \theta + {q^2}\sin \theta }}$
D.
${{\left( {{p^2} + {q^2}} \right)\sin \theta } \over {{{\left( {p\cos \theta + q\sin \theta } \right)}^2}}}$
2008
Q37
JEE Mains
MCQ
14 Mar 2026
$AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. $A$ man finds that the angle of elevation of the point $A$ from a certain point $C$ on the ground is ${60^ \circ }$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD=7$ m. From $D$ the angle of elevation of the point $A$ is ${45^ \circ }$. Then the height of the pole is :
A.
${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 - 1} }}m$
B.
${{7\sqrt 3 } \over 2}\left( {\sqrt {3 } + 1 } \right)m$
C.
${{7\sqrt 3 } \over 2}\left( {\sqrt {3 } - 1 } \right)m$
D.
${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 + 1} }}m$
2007
Q38
JEE Mains
MCQ
14 Mar 2026
A tower stands at the centre of a circular park. $A$ and $B$ are two points on the boundary of the park such that $AB(=a)$ 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 ${30^ \circ }$. The height of the tower is :
A.
$a/\sqrt 3 $
B.
$a\sqrt 3 $
C.
$2a/\sqrt 3 $
D.
$2a\sqrt 3 $
2004
Q39
JEE Mains
MCQ
14 Mar 2026
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 ${60^ \circ }$ and when he retires $40$ meters away from the tree the angle of elevation becomes ${30^ \circ }$. The breadth of the river is :
A.
$60\,\,m$
B.
$30\,\,m$
C.
$40\,\,m$
D.
$20\,\,m$
