Test 4: Vector Analysis

Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom

Total Marks: 75

1) Find the directional derivative of \(\phi(x, y, z)=x^{2} y z+4 x z^{2}\) at the point (1,-2,-1) in the direction of the vector \(2 \vec{i}-\vec{j}-2 \vec{k}\).

a) \(\dfrac{37}{9}\)
b) \(\dfrac{37}{3}\)
c) \(\dfrac{37}{6}\)
d) \(\dfrac{37}{7}\)

2) Find the angle between the surfaces \(x^{2}+y^{2}+z^{2}=9\) and \(x^{2}+y^{2}-z=3\) at the point \((2,-1,2)\).

a) \(\cos ^{-1}\left(\dfrac{8}{3 \sqrt{21}}\right)\)
b) \(\cos ^{-1}\left(\dfrac{8}{3 \sqrt{22}}\right)\)
c) \(\cos ^{-1}\left(\dfrac{8}{3 \sqrt{23}}\right)\)
d) \(\cos ^{-1}\left(\dfrac{8}{3 \sqrt{24}}\right)\)

3) Find the value of \(a\) if the vector \(\overrightarrow{\mathrm{F}}=\left(2 x^{2} y+y z\right) \vec{i}+\left(x y^{2}-x z^{2}\right) \vec{j}+\left(a x y z-2 x^{2} y^{2}\right) \vec{k}\) is solenoidal.

a) \(-6\)
b) \(6\)
c) \(3\)
d) \(-3\)

4) The value of \(\nabla \cdot\left(r \nabla\left(\dfrac{1}{r^{3}}\right)\right)\) is given by:

a) 1 b) 0 c) \(\dfrac{3}{r^{4}}\)
d) \(\dfrac{-3}{r^{4}}\)

5) Evaluate the line integral \(\int_{C}\left(y^{2} d x-x^{2} d y\right)\) around the triangle whose vertices are (1,0),(0,1) (-1,0) in the positive sense.

a) \(=-\dfrac{2}{3}\)
b) \(=-\dfrac{4}{3}\)
c) \(=-\dfrac{-2}{3}\)
d) \(=-\dfrac{-4}{3}\)

6) Evaluate \(\int_{C}\left[(\sin x-y) d x-\cos x d y \mathrm{l},\right.\) where \(C\) is the triangle with vertices \((0,0),\left(\dfrac{\pi}{2}, 0\right)\) and \(\left(\dfrac{\pi}{2}, 1\right)\).

a) \(\dfrac{2}{\pi}+\dfrac{\pi}{4}\)
b) \(\dfrac{2}{\pi}+\dfrac{\pi}{2}\)
c) \(\dfrac{2}{\pi}+\dfrac{\pi}{3}\)
d) \(\dfrac{4}{\pi}+\dfrac{\pi}{4}\)

7) Evaluate \(\iint_{S} \vec{F} \cdot \vec{n} d S\) if \(\vec{F}=4 y \vec{i}+18 z \vec{j}-x \vec{k}\) and \(S\) is the surface of the plane \(3 x+2 y+6 z=6\) contained in the first octant.

a) 9
b) 10
c) 11
d) 12

8) Evaluate \(\iint_{S} \vec{F} \cdot \vec{n} d S\) if \(\vec{F}=y z \vec{i}+z x \vec{j}+x y \vec{k}\) and \(S\) is part of the surface \(x^{2}+y^{2}+z^{2}=1\) which lies in the first octant.

a) \(\dfrac{3}{8}\)
b) \(\dfrac{3}{4}\)
c) \(\dfrac{5}{8}\)
d) \(\dfrac{5}{5}\)

9) Using divergence theorem, evaluate \(\iint \vec{F} \cdot \vec{n} d S,\) where \(\vec{F}=4 x z \vec{i}-y^{2} \vec{j}+y z \vec{k}\) and \(S\) is the surface of the cube bounded by the planes \(x=0, x=2, y=0, y=2, z=0, z=2\).

a) \(24\)
b) \(26\)
c) \(28\)
d) \(30\)

10) Evaluate \(\iint_{S} x^{3} d y d z+x^{2} y d z d x+x^{2} z d x d y\) over the surface \(z=0, z=h, x^{2}+y^{2}=a^{2}\).

a) \(\dfrac{5}{4} \pi a^{4} h\)
b) \(\dfrac{3}{4} \pi a^{4} h\) c) \(\dfrac{7}{4} \pi a^{4} h\)
d) \(\dfrac{1}{4} \pi a^{4} h\)

11) Evaluate \(\int_{C}\left(x y d x+x y^{2} d y\right)\) by Stoke’s theorem, where \(C\) is the square in the \(x y-\) plane with vertices \((1,0)\), \((-1,0)\), \((0,1)\), \((0,-1)\).

a) \(\dfrac{1}{3}\)
b) \(\dfrac{2}{3}\) c) \(1\)
d) \(\dfrac{4}{3}\)

12) Using Stoke’s theorem, evaluate \(\int_{C} \vec{F} \cdot d \vec{r},\) where \(\vec{F}=y^{2} \vec{i}+x^{2} \vec{j}-(x+z) \vec{k}\) and \(C\) is the boundary of the triangle with vertices at \((0,0,0)\), \((1,0,0)\), \((1,1,0)\).

a) \(\dfrac{1}{3}\)
b) \(\dfrac{2}{3}\) c) \(1\)
d) \(\dfrac{4}{3}\)

13) If \(\vec{r}+x \vec{i}+y \vec{j}+z \vec{k}\) and \(r=\vert \vec{r} \vert\), then \(\nabla(\log r)\) is equal to:

a) \(\dfrac{1}{r^{2}}\) b) \(\dfrac{\vec{r}}{r^{3}}\) c) \(\dfrac{\vec{r}}{r^{2}}\)
d) \(\dfrac{1}{r}\)

14) The value of \(b\) if the surfaces \(a x^{2}-b y z=(a+2) x\) and \(4 x^{2} y+z^{3}=4\) cut orthogonally at the point \((1,-1,2)\) is given by:

a) \(2\)
b) \(1\)
c) \(3\)
d) \(4\)

15) If \(\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}\) and \(r=\vert \vec{r} \vert\), then \(r^{n} \vec{r}\) is solenoidal for:

a) \(n=3\)
b) \(n=-3\)
c) \(n=1\)
d) \(n=-1\)