Problem 34
A necessary and sufficient condition that the line integral \(\int \mathbf{F} . d \mathbf{R}\) for every closed \(C\) vanishes, is (a) \(\operatorname{curl} \mathbf{F}=0\) (b) \(\operatorname{div} \mathbf{F}=0\) (c) \(\operatorname{curl} \mathbf{F} \neq 0\) (d) div \(\mathbf{F}=0 .\)
Problem 39
The value of the line integral \(\int_{C}\left(y^{2} d x+x^{2} d y\right)\) where \(C\) is the boundary of the square \(-1 \leq x \leq 1,-1 \leq y \leq 1\) is $\begin{array}{lllll}\text { (a) } 0 & \text { (b) } 2(x+y) & \text { (c) } 4 & \text { (d) } 4 / 3 .
Problem 43
The magnitude of the vector drawn perpendicular to the surface \(x^{2}+2 y^{2}+z^{2}=7\) at the point \((1,-1,2)\) is (a) 28 (b) \(3 / 2\) (c) 3 (d) 6 .
Problem 46
Value of \(\int_{e}\left(y^{2} d x+x^{2} d y\right)\) where \(e\) in the boundary of the square \(-1 \leq x \leq 1,-1 \leq y \leq 1\), is (a) 4 (b) 0 (c) \(2(x+y)\) (d) \(4 / 3\).
Problem 49
If \(u=1 / r\) where \(r^{2}=x^{2}+y^{2}\), then \(\nabla^{2} u=0\).
Problem 50
\(\mathbf{F}=(x+3 y) \mathbf{I}+(z-3 y) \mathbf{J}+(x+2\\}) \mathbf{K}\) is a solenoidal vector function.
Problem 51
\(\mathbf{F}=y z \mathbf{I}+\mathbf{z x}_{\boldsymbol{J}}+\boldsymbol{x y K}\) is irrotational.
