UPSC NDA I 2025 Mathematics Paper

Given,
(x+y)p+q=xpyq‌ and ‌p+q=10
Differentiate both sides with respect to X implicitly:
‌

d

dx

((x+y)p+q)=‌

d

dx

(xpyq)
Left side:
(p+q)(x+y)p+q−1(1+‌

dy

dx

)
Right side (product rule):
pxp−1yq+qxpyq−1‌

dy

dx

Rearrange to collect ‌

dy

dx

terms and use
(x+y)p+q=xpyq⟹(x+y)p+q−1=‌

xpyq

x+y

After cancellation of the common factor py−qx, you obtain:
∴‌

dy

dx

=‌

y

x