UPSC NDA I 2025 Mathematics Paper

The expression to simplify is p2+q2+r2, where:
‌p2+q2+r2
‌q=sin‌(25∘)
‌r=sin‌(−95∘)=−cos(5∘)
We are given that p+q+r=0. Using the identity for squaring a sum:
(p+q+r)2=p2+q2+r2+2(pq+qr+rp)
Since p+q+r=0, we substitute this into the identity:
0=p2+q2+r2+2(pq+qr+rp)
From previous calculations, we know that:
pq+qr+rp=−‌

3

4

Substitute this value into the equation:
0=p2+q2+r2+2(−‌

3

4

)
Simplifying:
‌0=p2+q2+r2−‌

3

2

‌p2+q2+r2=‌

3

2

.