UPSC NDA I 2025 Mathematics Paper

For the right-angled triangle △MPQ the tangent of the angle of elevation at point M is
⇒tan‌45∘=PQ∕PM=‌

50+h

100

where h is the additional height at point Q due to the smoke
Since tan‌45∘=1
‌⇒1=‌

50+h

100

=100=50+h
‌⇒h=50m
So, the height at point Q due to the smoke is 50 meters.
Finding the angle ∠RMQ
∠RMQ, is formed by the line PR (the straight line from P to R ) and the line PM .
Using the definition of the tangent function, we have
‌⇒tan‌∠R‌M‌Q=‌

50

100

=‌

1

2

‌⇒∠RMQ=tan−1‌

1

2