GATE 2026 MA – Question 56

Calculus · Vector Calculus · 2 marks · Numerical answer

$\alpha=\iint_S\mathbf F\cdot\hat n\,dS$, with $\mathbf F=(2x+3z)\mathbf i+(xz-y)\mathbf j+(y^2+2z)\mathbf k$. S is the radius-9 sphere centered at (3,−1,2), with outward normal. Find α/(36π) (integer).

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 81 to 81

Explanation

Divergence is 2−1+2 = 3. By Gauss’s theorem, flux is $\alpha=3(4\pi9^3/3)=2916\pi$. Hence α/(36π) = 2916/36 = **81**; the center has no effect on this constant-divergence calculation.

More Vector Calculus questions

GATE 2025 MA Q32 – F=(y−z)i+(z−x)j+(x−y)k. S is the outward upper hemisphere x²+y²+(z−1)²=9, 1≤z≤4. Find -1…
GATE 2024 MA Q35 – Outward flux of F=(x³,y³,z³) through the unit sphere is απ. Find α to two decimals.
GATE 2024 MA Q61 – A C¹ path runs from (0,2) to (3,0). For F(x,y)=(1−2y,1−2x), find ∫F·dr to two decimals.
GATE 2023 MA Q31 – The portion of z=2x+2y−100 inside x²+y²=1 has surface area απ. Find α.
GATE 2023 MA Q55 – C is x²+y²=4,z=2 oriented counterclockwise viewed from above. If C(…
GATE 2022 MA Q45 – Find work of F=(x+y)i−(x²+y²)j along the upper unit semicircle from (1,0) to (−1,0).

All Vector Calculus questions →