GATE AG 2025 • Q38 MCQ • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Partial derivatives

Question Statement:

If $u = x^4 + y^4 + 3x^2y^2$, find the value of $\left(x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y}\right)^{-1}$:

Options:

  • (A) 4u
  • (B) 0
  • (C) 1/(4u)
  • (D) 2u

Official Answer Key:

C

Step-by-Step Solution & Explanation

u is homogeneous of degree 4 in x, y. By Euler's theorem, x·∂u/∂x + y·∂u/∂y = 4u (verified directly: ∂u/∂x=4x³+6xy², ∂u/∂y=4y³+6x²y; x·∂u/∂x+y·∂u/∂y = 4x⁴+6x²y²+4y⁴+6x²y² = 4x⁴+4y⁴+12x²y² = 4u). So the required value is 1/(4u).

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