Question : A pole stands vertically on a road, which goes in the north-south direction. P and Q are two points towards the north of the pole, such that $P Q=b$, and the angles of elevation of the top of the pole at $P, Q$ are $\alpha, \beta$ respectively. Then the height of the pole is:

Option 1: $\frac{b}{\tan \beta+\tan \alpha}$

Option 2: $\frac{b}{\tan \beta-\tan \alpha}$

Option 3: $\frac{b}{\cot \beta-\cot \alpha}$

Option 4: $\frac{\mathrm{b} \tan \alpha}{\tan \beta}$

Question : The angles of elevation of the top of a tower from two points on the ground at distances 32 m and 18 m from its base and in the same straight line with it are complementary. The height (in m) of the tower is____________.

Option 1: 20

Option 2: 24

Option 3: 16

Option 4: 28

Question : Two points A and B are on the ground and on opposite sides of a tower. A is closer to the foot of the tower by 42 m than B. If the angles of elevation of the top of the tower, as observed from A and B are 60° and 45°, respectively, then the height of the tower is closest to:

Option 1: 87.6 m

Option 2: 98.6 m

Option 3: 88.2 m

Option 4: 99.4 m

Question : A clock tower stands at the crossing of two roads which point in the north-south and the east-west directions. $P, Q, R$, and $S$ are points on the roads due north, east, south, and west respectively, where the angles of elevation of the top of the tower are respectively, $\alpha, \beta, \gamma$ and $\delta$. Then $\left(\frac{\mathrm{PQ}}{\mathrm{RS}}\right)^2$ is equal to:

Option 1: $\frac{\tan ^2 \alpha+\tan ^2 \beta}{\tan ^2 \gamma+\tan ^2 \delta}$

Option 2: $\frac{\cot ^2 \alpha+\cot ^2 \beta}{\cot ^2 \gamma+\cot ^2 \delta}$

Option 3: $\frac{\cot ^2 \alpha+\cot ^2 \delta}{\cot ^2 \beta+\cot ^2 \gamma}$

Option 4: $\frac{\tan ^2 \alpha+\tan ^2 \delta}{\tan ^2 \beta+\tan ^2 \gamma}$

Question : A pole of length 7 m is fixed vertically on the top of a tower. The angle of elevation of the top of the pole observed from a point on the ground is 60° and the angle of depression of the same point on the ground from the top of the tower is 45°. The height (in m) of the tower is:

Option 1: $7(2 \sqrt{3}-1)$

Option 2: $\frac{7}{2}(\sqrt{3}+2)$

Option 3: $7 \sqrt{3}$

Option 4: $\frac{7}{2}(\sqrt{3}+1)$