Allied & ICAR Exam Bank • ALLIED_0061 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Vector differential operators – del, gradient

Question Statement:

The unit outward normal vector to the surface $x^2 + y^2 + z = 5$ at the point $(1, 2, 0)$ has $z$-component equal to $1/k$. The value of $k$ is:

Answer Key:

4.58

Step-by-Step Solution & Explanation

$\nabla f = 2x\hat{i} + 2y\hat{j} + \hat{k} = 2\hat{i} + 4\hat{j} + \hat{k}$. $|\nabla f| = \sqrt{2^2 + 4^2 + 1^2} = \sqrt{4 + 16 + 1} = \sqrt{21} \approx 4.582$. The $z$-component is $1/\sqrt{21} = 1/4.58$.
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