Height and Distance — Class 12 Mathematics PYQs

Consider the following for the next two (02) items that follow :

A flagstaff $20$ m long standing on a pillar $10$ m high subtends an angle ${\tan }^{-1}(0\cdot 5)$ at a point $P$ on the ground. Let $\theta$ be the angle subtended by the pillar at this point $P$.

If $x$ is the distance of $P$ from bottom of the pillar, then consider the following statements :

1. $x$ can take two values which are in the ratio $1:3$
2. $x$ can be equal to height of the flagstaff

Which of the statements given above is/are correct?

Consider the following for the next two (02) items that follow :

A flagstaff $20$ m long standing on a pillar $10$ m high subtends an angle ${\tan }^{-1}(0\cdot 5)$ at a point $P$ on the ground. Let $\theta$ be the angle subtended by the pillar at this point $P$.

What is a possible value of $\tan \theta$?

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height $h$. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are $\theta$ and $2\theta$ respectively. What is the height of the tower ?

The shadow of a tower becomes $x$ metre longer, when the angle of elevation of sun changes from $60^{\circ }$ to $\theta$. If the height of the tower is $\sqrt{3}x$ metre, then which one of the following is correct ?

A ladder $6$ m long reaches a point $6$ m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is $75^{\circ }$. What is the height of the flagstaff?

The shadow of a tower is found to be $x$ metre longer, when the angle of elevation of the sun changes from $60^{\circ }$ to $45^{\circ }$. If the height of the tower is $5(3+\sqrt{3})$ m, then what is $x$ equal to?