UPSC NDA I 2025 Mathematics Paper

Given,
The line equation is: ‌

x+1

p

=‌

y−1

q

=‌

z−2

r

, where p=2q=3r.
The line makes an angle θ with the positive direction of the y -axis.
Using the relation between the direction cosines
1=‌

p

√p2+q2+r2

,m=‌

q

√p2+q2+r2

,n=‌

r

√p2+q2+r2

we substitute p=3r and q=‌

3r

2

Direction cosine with y -axis is given by m=‌

q

√p2+q2+r2

.
m=‌

‌ 3r 2

√(3r)2+(‌ 3r 2 )2+r2

=‌

‌ 3r 2

√9r2+‌ 9r2 4 +r2

=‌

‌ 3r 2

√‌ 49r2 4

=‌

3

7

.
Now, using the double angle identity for cosine‌cos‌2‌θ=2cos2θ−1 :
cos‌2‌θ=2(‌

3

7

)2−1=2×‌

9

49

−1=‌

18

49

−1=‌

−31

49

.
∴ The value of cos‌2‌θ is −‌

31

49

.