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 $\varphi (x,y,z)=x^{2}yz+4x{z}^2$ at the point (1,-2,-1) in the direction of the vector $2\overrightarrow{i}-\overrightarrow{j}-2\overrightarrow{k}$.

a) ${\displaystyle \frac{37}{9}}$
b) ${\displaystyle \frac{37}{3}}$
c) ${\displaystyle \frac{37}{6}}$
d) ${\displaystyle \frac{37}{7}}$

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

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

a) $-6$
b) $6$
c) $3$
d) $-3$

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

a) 1 b) 0 c) ${\displaystyle \frac{3}{r^4}}$
d) ${\displaystyle \frac{-3}{r^4}}$

5) Evaluate the line integral ${\int }_C(y^{2}dx-x^{2}dy)$ around the triangle whose vertices are (1,0),(0,1) (-1,0) in the positive sense.

a) $=-{\displaystyle \frac{2}{3}}$
b) $=-{\displaystyle \frac{4}{3}}$
c) $=-{\displaystyle \frac{-2}{3}}$
d) $=-{\displaystyle \frac{-4}{3}}$

6) Evaluate ${\int }_C[(\sin x-y)dx-\cos xdy\mathrm{l},$ where $C$ is the triangle with vertices $(0,0),({\displaystyle \frac{\pi }{2}},0)$ and $({\displaystyle \frac{\pi }{2}},1)$.

a) ${\displaystyle \frac{2}{\pi }}+{\displaystyle \frac{\pi }{4}}$
b) ${\displaystyle \frac{2}{\pi }}+{\displaystyle \frac{\pi }{2}}$
c) ${\displaystyle \frac{2}{\pi }}+{\displaystyle \frac{\pi }{3}}$
d) ${\displaystyle \frac{4}{\pi }}+{\displaystyle \frac{\pi }{4}}$

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

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

8) Evaluate ${\iint }_S\overrightarrow{F}\cdot \overrightarrow{n}dS$ if $\overrightarrow{F}=yz\overrightarrow{i}+zx\overrightarrow{j}+xy\overrightarrow{k}$ and $S$ is part of the surface $x^2+y^2+z^2=1$ which lies in the first octant.

a) ${\displaystyle \frac{3}{8}}$
b) ${\displaystyle \frac{3}{4}}$
c) ${\displaystyle \frac{5}{8}}$
d) ${\displaystyle \frac{5}{5}}$

9) Using divergence theorem, evaluate $\iint \overrightarrow{F}\cdot \overrightarrow{n}dS,$ where $\overrightarrow{F}=4xz\overrightarrow{i}-y^2\overrightarrow{j}+yz\overrightarrow{k}$ 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 ${\iint }_S{x}^{3}dydz+x^{2}ydzdx+x^{2}zdxdy$ over the surface $z=0,z=h,x^2+y^2=a^2$.

a) ${\displaystyle \frac{5}{4}}\pi a^{4}h$
b) ${\displaystyle \frac{3}{4}}\pi a^{4}h$ c) ${\displaystyle \frac{7}{4}}\pi a^{4}h$
d) ${\displaystyle \frac{1}{4}}\pi a^{4}h$

11) Evaluate ${\int }_C(xydx+x{y}^{2}dy)$ by Stoke’s theorem, where $C$ is the square in the $xy-$ plane with vertices $(1,0)$, $(-1,0)$, $(0,1)$, $(0,-1)$.

a) ${\displaystyle \frac{1}{3}}$
b) ${\displaystyle \frac{2}{3}}$ c) $1$
d) ${\displaystyle \frac{4}{3}}$

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

a) ${\displaystyle \frac{1}{3}}$
b) ${\displaystyle \frac{2}{3}}$ c) $1$
d) ${\displaystyle \frac{4}{3}}$

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

a) ${\displaystyle \frac{1}{r^2}}$ b) ${\displaystyle \frac{\overrightarrow{r}}{r^3}}$ c) ${\displaystyle \frac{\overrightarrow{r}}{r^2}}$
d) ${\displaystyle \frac{1}{r}}$

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

a) $2$
b) $1$
c) $3$
d) $4$

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

a) $n=3$
b) $n=-3$
c) $n=1$
d) $n=-1$