GATE 2023 MA – Question 55

Calculus · Vector Calculus · 2 marks · Numerical answer

C is x²+y²=4,z=2 oriented counterclockwise viewed from above. If $|\oint_C(\sin x+e^x)dx+4x\,dy+e^z\cos^2z\,dz|=\alpha\pi$, find α.

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Correct answer: 16 to 16

Explanation

The dx term is the differential of −cos x+eˣ and integrates to zero on a closed curve. Also dz=0. Green’s theorem gives ∮4xdy=4×area(disk)=4×4π=16π. Therefore **α=16**.

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