UPSC NDA Math Model Paper 10

The height of the lighthouse AC = 120 m. The observer is at C.
The ship at B makes an angle of depression 15°
Taking AB = x
In right-angled triangle BAC
tan‌15°=

120

x

tan(45−30)=

120

x

tan‌45−tan‌30

1+tan‌45‌tan‌30

=

120

x

⇒

1− 1 √3

1+ 1 √3

=

120

x

⇒x(√3−1)=(√3+1)120
⇒

√3−1

√3+1

=

120

x

⇒x=

120(√3+1)

√3−1

120(√3+1)

√3−1

×

√3+1

√3+1

=

120(3+1+2√3)

3−1

=

120(4+2√3)

2

=120(2+√3)
=120×3.7
=444m
Hence, option (3) is correct.