NDA Math Mock Test 03 – Question 114

The top of a hill observed from the top and bottom of a building of height \(h\) is at angles of elevation \(\alpha\) and \(\beta\) respectively. The height of the hill is:

• \(\frac{h\cot\beta}{\cot\beta – \cot\alpha}\)
• \(\frac{h\cot\alpha}{\cot\alpha – \cot\beta}\)
• \(\frac{h\cot\alpha}{\cot\beta – \cot\alpha}\)
• None of the above

Let the height of the hill be \(H\) and the horizontal distance between building and hill be \(d\).
From the bottom of the building:
\[d = H\cot\beta\]
From the top of the building (height \(h\)):
\[d = (H – h)\cot\alpha\]
Equating both expressions for \(d\):
\[H\cot\beta = H\cot\alpha – h\cot\alpha \implies H(\cot\alpha – \cot\beta) = h\cot\alpha\]
\[H = \frac{h\cot\alpha}{\cot\alpha – \cot\beta}\]
Hence, the correct option is (b).