GATE 2023 EC – Question 37
The value of the line integral $\int_P^Q(z^2\,dx+3y^2\,dy+2xz\,dz)$ along the straight line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is
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Correct answer: (B) 24
Explanation
The field $(z^2,3y^2,2xz)$ is conservative (its curl is zero: $\partial F_x/\partial z=2z=\partial F_z/\partial x$), with potential $\phi=xz^2+y^3$. So the integral is $\phi(Q)-\phi(P)=(2\cdot1+27)-(1\cdot4+1)=29-5=24$.