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GATE 2026 EC – Question 12

Engineering Mathematics · Vector Analysis · 1 mark · Multiple choice

A surface is given by $z^2 = 2x^2 - y^2$ and $\vec n$ and $-\vec n$ are unit normal vectors to the surface at the point $\vec P = \hat i + \sqrt2\,\hat k$. Which of the following vectors can be $\vec n$, where $\hat i,\hat j$ and $\hat k$ are the unit vectors along x, y and z axes, respectively?

  1. $\hat i - \sqrt2\,\hat k$
  2. $\frac{2}{3}\hat i - \frac{1}{3}\hat k$
  3. $\sqrt2\,\hat i - \sqrt3\,\hat k$
  4. $\frac{\sqrt2\,\hat i - \hat k}{\sqrt3}$

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Correct answer: (D) $\frac{\sqrt2\,\hat i - \hat k}{\sqrt3}$

Explanation

With $F=z^2-2x^2+y^2$, $\nabla F=(-4x,2y,2z)=(-4,0,2\sqrt2)$ at P. Normalising gives $\pm(\sqrt2\,\hat i-\hat k)/\sqrt3$.