GATE 2026 DA Q.22: solution and answer

Official answer: ASource: Official answer key, GATE 2026 DA (IIT Guwahati)

1. Write $x^{T}x=x_1^2+x_2^2+x_3^2$. The condition $x^{T}x\le 16$ describes the solid ball in ${\mathbb{R}}^3$ of radius $\sqrt{16}=4$ centred at the origin.

2. $S_2$ is a subspace of ${\mathbb{R}}^3$ of dimension two, so it is a plane. Every subspace contains the zero vector, so this plane passes through the origin, which is the centre of the ball.

3. Take any point $x$ of the plane. It lies in $S_1$ exactly when $\Vert x\Vert \le 4$. Inside the plane, with an orthonormal basis $u_1,u_2$ of $S_2$ and $x=a{u}_1+b{u}_2$, we have $\Vert x{\Vert }^2=a^2+b^2$. So $S_1\cap S_2=\{(a,b):a^2+b^2\le 16\}$, a full disc of radius $4$. This holds for every two-dimensional subspace, so the answer does not depend on which $S_2$ is chosen.

4. Area of the disc: $$\pi r^2=\pi \cdot 4^2=16\pi .$$ This is option (A).