GATE 2025 IN Q.36: solution and answer

GATE 2025 · INQ.36 · 2 marks · MCQ

The value of the surface integral $\iint_{S} (2 x + z) d y d z + (2 x + z) d x d z + (2 z + y) d x d y$ over the sphere $S : x^{2} + y^{2} + z^{2} = 9$ is

(A)
$72 π$
(B)
$144 π$
(C)
$36 π$
(D)
$432 π$

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Answer and solution

ANSWER

The surface integral equals $144 π$ , which is option (B).

Official answer: BSource: Official answer key, GATE 2025 IN (IIT Roorkee)

The integral is in flux form, $\iint_{S} F_{1} d y d z + F_{2} d z d x + F_{3} d x d y = \iint_{S} F \cdot \overset{n}{^} d S$ , with $F = (2 x + z) \hat{i} + (2 x + z) \hat{j} + (2 z + y) \hat{k}$ . The sphere is a closed surface, so we take the outward normal.
The paper prints the middle term as $d x d z$ , not the standard $d z d x$ . As an oriented area element, $d x d z = - d z d x$ , so the most this can do is flip the sign of $F_{2}$ . Since $F_{2} = 2 x + z$ does not depend on $y$ , $\partial F_{2} / \partial y = 0$ with either sign, and the answer is the same.
Apply the Gauss divergence theorem: $\oint_{S} F \cdot \overset{n}{^} d S = ∭_{V} (\nabla \cdot F) d V$ , where $V$ is the solid ball $x^{2} + y^{2} + z^{2} \leq 9$ .
Divergence: $\nabla \cdot F = \frac{\partial}{\partial x} (2 x + z) + \frac{\partial}{\partial y} (2 x + z) + \frac{\partial}{\partial z} (2 z + y) = 2 + 0 + 2 = 4$
The divergence is a constant, so the volume integral is $4$ times the volume of the ball. The radius is $r = 3$ : $V = \frac{4}{3} π r^{3} = \frac{4}{3} π (27) = 36 π$
So $\oint_{S} F \cdot \overset{n}{^} d S = 4 \times 36 π = 144 π$ which is option (B).

KEY IDEA

Read $F_{1} d y d z + F_{2} d z d x + F_{3} d x d y$ over a closed surface as the outward flux of $F = (F_{1}, F_{2}, F_{3})$ . Then use the divergence theorem. Here $\nabla \cdot F = 4$ is constant, so the flux is just $4 \times$ the volume of the ball of radius 3.

WATCH OUT FOR

Using the surface area of the sphere ( $4 π r^{2} = 36 π$ ) where the enclosed volume ( $\frac{4}{3} π r^{3} = 36 π$ ) is needed. For $r = 3$ these happen to be the same number, which hides the error, and it gives the wrong answer for any other radius. Other slips: taking the radius as 9 instead of $9 = 3$ (which gives a far larger value), writing $\partial (2 x + z) / \partial y = 1$ (divergence 5, so $180 π$ , which is not an option), and dropping one of the two 2's to get $72 π$ (option A).

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GATE 2025 IN Q.36: solution and answer

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