Heights and Distances — JEE Main PYQs

From the top A of a vertical wall AB of height 30 m , the angles of depression of the top P and bottom Q of a vertical tower PQ are $15^{\circ }$ and $60^{\circ }$ respectively. B and Q are on the same horizontal level. If C is a point on AB such that …

The angle of elevation of the top 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) …

A horizontal park is in the shape of a triangle OAB with $\mathrm{AB}=16$. A vertical lamp post OP is erected at the point O such that $\angle \mathrm{PAO}=\angle \mathrm{PBO}=15^{\circ }$ and $\angle \mathrm{PCO}=45^{\circ }$, where C is the midpoint of AB. Then …

Let a vertical tower AB of height $2h$ 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 $\overrightarrow{AP}$, he can see the t …

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 $QR=15$ m. If from a point A on the ground the angle of elevation of R is $60^{\circ }$ and the part PR of the tower subtends an angle of $15^{\circ }$ at A, the …

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 R, the angle of elevation of P is $60^{\circ }$. If $\angle BAQ=30^{\circ }$, $AB=d$ and the …

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 $\frac{\pi }{8}$ and $\theta$ be the angles of elevation from C to P and A , respectively. If the height of pole PQ is twice the heig …

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:

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 …

Two vertical poles are 150 m apart and the height of one is three times that of 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:

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:

Two poles $AB$ of length a meter 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 $\tan \angle ACB=\frac{1}{2}$, then:

A man is observing, from the top of a tower, a boat speeding towards the tower 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 th …

A vertical pole fixed to the horizontal ground is divided in the ratio 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) …

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' …

A pole stands vertically inside a triangular park ABC. Let the angel of elevation of the top of the pole from each corner of the park be $\frac{\pi }{3}$. If the radius of the circumcircle of $△ABC$ is 2, then the height of the pole is equal to :

The angle of elevation of the summit of a mountain from a point on the ground is $45^{\circ }$. After climbing up one km towards the summit at an inclination of $30^{\circ }$ from the ground, the angle of elevation of the summit is found to be $60^{\circ }$. Then the height (in km ) …

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $45^{\circ }$. After walking a distance of 80 meters towards the top, up a slope inclined at an angle of $30^{\circ }$ to the horizontal plane, the a …

Two vertical poles $\mathrm{AB}=15\mathrm{m}$ and $\mathrm{CD}=10\mathrm{m}$ are standing apart on a horizontal ground with points $A$ and $C$ on the ground. If $P$ is the point of intersection of $BC$ and $AD$, then the height of $P$ (in $m$ ) above the lin …

The angle of elevation of a cloud C from a point $\mathrm{P},200\mathrm{m}$ above a still take is $30^{\circ }$. If the angle of depression of the image of C in the lake from the point P is $60^{\circ }$, then PC (in m ) is equal to

Two vertical poles of heights, 20 m and 80 m stand apart 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:

The angle of elevation of the top of vertical tower standing on a horizontal plane is observed to be $45^{\circ }$ 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 $30^{\circ }$, then …

If the angle of elevation of a cloud from a point P which is 25 m above a lake be $30^{\circ }$ and the angle of depression of reflection of the cloud in the lake from P be $60^{\circ }$, then the height of the cloud (in meters) from the surface of the lake is :

Two poles standing on a horizontal ground are of heights 5 m and 10 m respectively. The line joining their tops makes an angle of $15^{\circ }$ with ground. Then the distance (in m) between the poles, is

ABC is a triangular park with $\mathrm{AB}=\mathrm{AC}=100$ 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 ${\mathrm{cosec}}^{-1}(2\sqrt{2})$ respec …

PQR is a triangular park with $\mathrm{PQ}=\mathrm{PR}=200\mathrm{m}$. 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 (i …

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 $\mathrm{AP}=2\mathrm{AB}$. If $\angle \mathrm{BPC}=\beta$, then $\tan \beta$ is equal to :

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 $30^{\circ }$. After walking for 10 minutes from A in the same direction, at a point B, he obser …

If the angles of elevation of the top of a tower from three collinear points $\mathrm{A},\mathrm{B}$ and C , on a line leading to the foot of the tower, are $30^{\circ },45^{\circ }$ and $60^{\circ }$ respectively, then the ratio, $\mathrm{AB}:\mathrm{BC}$, is :

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 flies 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/ …

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 …

A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that $\mathrm{AB}(=\mathrm{a})$ 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 …

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 meter away from the tree the angle of elevation becomes $30^{\circ }$. The breadth of the river is