Allied & ICAR Exam Bank • ALLIED_0025 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Partial derivatives

Question Statement:

If $u(x, y) = x^3 - 3x y^2$, the value of $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$ is:

Options:

  • (A) $0$
  • (B) $6x$
  • (C) $-6x$
  • (D) $6y$

Answer Key:

A

Step-by-Step Solution & Explanation

$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \implies \frac{\partial^2 u}{\partial x^2} = 6x$. $\frac{\partial u}{\partial y} = -6xy \implies \frac{\partial^2 u}{\partial y^2} = -6x$. Sum $= 6x - 6x = 0$. $u$ is harmonic.
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