Allied & ICAR Exam Bank • ALLIED_0782 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Vector differential operators – del, gradient

Question Statement:

For the vector field $\vec{F} = (2x + 3y)\hat{i} + (k y - z)\hat{j} + (4z)\hat{k}$ to be solenoidal, the divergence $\nabla \cdot \vec{F} = 2 + k + 4 = 0$. The value of $k$ is:

Answer Key:

-6

Step-by-Step Solution & Explanation

$\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(2x + 3y) + \frac{\partial}{\partial y}(ky - z) + \frac{\partial}{\partial z}(4z) = 2 + k + 4 = 0 \implies k = -6$.
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