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GATE 2026 DA – Question 22

Linear Algebra · Vector spaces, linear independence, rank and nullity · 1 mark · Multiple choice

Consider a set $S_1 = \{x = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \mid x^Tx \le 16\}$. Let $S_2$ be another set which is a subspace of $\mathbb{R}^3$ with dimension two.

Which of the following gives the area of $S_1 \cap S_2$?

  1. $16\pi$
  2. $4\pi$
  3. $4\pi^2$
  4. $16\pi^2$

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Show answer and explanation

Correct answer: (A) $16\pi$

Explanation

$S_1$ is a solid ball of radius 4 centred at the origin. A two-dimensional subspace is a plane through the origin, which cuts the ball in a disc of the full radius 4. Its area is $\pi \times 4^2 = 16\pi$.