Sample Page - Ashirbad Mohanty

1. The value of $m$ so that the vectors $\vec{a}=2\hat{i}-4\hat{j}+5\hat{k}$, $\vec{b}=\hat{i}-m\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+2\hat{j}-5\hat{k}$ are coplanner is

25/26

26/25

46/25

25/46

2. A vector perpendicular to both $\vec{A}=2\hat{i}+\hat{j}-\hat{k}$ and $\vec{B}=\hat{i}+3\hat{j}-2\hat{k}$ is

$\hat{i}+3\hat{j}-2\hat{k}$

$2\hat{i}+\hat{j}-\hat{k}$

$\hat{i}+3\hat{j}+5\hat{k}$

$\hat{i}-3\hat{j}+5\hat{k}$

3. The unit vector perpendicular to the surface $ x^2 + y^2 +z^2=3$ at the point $(1,1,1)$ is

$\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$

$\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

$\frac{\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}$

$\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$

4. If $\vec{r}$ is a position vector, $r^n\vec{r}$ is solenoidal for

$n=-3$

$n=3$

$n=-2$

$n=-1$

5. If $\delta(x)$ is Delta function then

$x\delta(x)=0$

$x\delta(x)=1$

$x\delta(x)=\infty$

$x\delta(x)=x$

6. The volume integral $\int_\nu (r^2+2) \vec{\nabla} \cdot (\frac{\hat{r}}{r^2}) d\tau$ ,where $V$ is the volume of a sphere of radius $R$ centered at the origin, is equal to

$\pi$

$2\pi$

$4\pi$

$8\pi$

7. The Fourier coefficients of the function $f(x)=\left\{\begin{matrix} 0, -\pi<x<0,
\\1, 0<x<\pi

\end{matrix}\right.$ expanded in Fourier series are

$a_0=1$, $a_n=0$, $b_n=\frac{1}{n \pi} [1-(-1)^n]$

$a_0=1$, $a_n=0$, $b_n=\frac{1}{n \pi} [1+(-1)^n]$

$a_0=0$, $a_n=0$, $b_n=\frac{1}{n \pi} [1+(-1)^n]$

$a_0=0$, $a_n=0$, $b_n=\frac{1}{n \pi} [1-(-1)^n]$

8. Let $\vec{r}$ be the position vector of a point on a closed contour C. What is the value of the line integral $\oint\vec{r} \cdot d\vec{r}$ ?

-1

9. The value of the integral $\int_{-\infty}^{\infty} x^2 e^{-ax^2} dx$, where $a>0$ is

$\frac{1}{2}\sqrt{\frac{\pi}{a^3}}$

$\frac{1}{2} \sqrt{\frac{\pi}{a}}$

$\sqrt{\frac{\pi}{2a^3}}$

$\sqrt{\frac{\pi}{2a}}$

10. The trace of a $2\times 2$ matrix is $4$ and its determinant is $8$ . If one of the eigenvalues is $2(1+i)$ , the other eigenvalue is

$2(1-i)$

$2(1+i)$

$1+2i$

$1-2i$

11. The value of the integral $\oint_C \frac{\sin z\,dz}{2z-\pi}$ , with $C$ a circle $|z|=2$ is

$0$

$2i \pi$

$i\pi$

$-i\pi$

12. One of the solutions of the equation $(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+12y=0$ is

$H_4(x)$

$P_3(x)$

$P_4(x)$

$H_3(x)$

13. The Laplace transform of $3t^2+2e^{5t}$ is

$\frac{2}{s^2}+\frac{2}{s-5}$

$\frac{6}{s^3}+\frac{2}{s-5}$

$\frac{2}{s^2}+\frac{3}{s+5}$

$\frac{2}{s}+\frac{2}{s+5}$

14. The Fourier transform of $e^{-|x|}$ is

$\frac{1}{1+k^2}$

$\frac{2}{1+k^2}$

$\frac{2}{1+k}$

$\frac{1}{1+k}$

15. The general solution of the ordinary differential equation $\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=0$ is

$y=A e^{-x} + B e^{x}$

$y=A e^{x} + B x e^{x}$

$y=A e^{x} + B e^{-x}$

$y=A e^{x} + B x $

16. The residue of the complex function $f(z)=e^{1/z}$ at $z=0$ is

-1

17. The modulus and phase of the complex number $(1+i)i$ in polar representation are

$\sqrt{2}$ and $\frac{3\pi}{4}$

$\sqrt{3}$ and $\frac{3\pi}{4}$

$\sqrt{2}$ and $\frac{\pi}{4}$

$\sqrt{3}$ and $\frac{\pi}{4}$

18. The value of $\frac{2}{5}P_3(x)+\frac{3}{5}P_1(x)$ , where $P_l(x)$ and $P_3(x)$ are Legendre polynomials, is

$x^2$

$x^3$

$x$

$0$

19. For the function $f(z)=\frac{z \sin z}{(z-\pi)^3}$, the residue at the pole $z=\pi$ is

-1

-2

20. Consider the transformation to a new set of coordinates $(p,q)$ from rectangular Cartesian coordinates $(x,y)$ , where $p=x+y$ and $q=x-y$. In the $(p,q)$ coordinate system, the area element $dxdy$ is

$\frac{1}{2}dpdq$

$\frac{1}{4}dpdq$

$2dpdq$

$4dpdq$

21. If S is is a closed surface enclosing a volume V, the value of the integral $\oint_S d\vec{s}$ must be

Zero

22. The value of $\Gamma(-3/2)$ is

$\infty$

$\frac{3}{4}\sqrt{\pi}$

$\frac{4}{3}\sqrt{\pi}$

Zero

23. The Bessel function $J_{1/2}(x)$ is

$\sqrt{\frac{2}{\pi x}}\sin x$

$\sqrt{\frac{2}{\pi x}}\cos x$

$\sqrt{\frac{3}{\pi x}}\sin x$

$\sqrt{\frac{3}{\pi x}}\cos x$

24. The directional derivative of $\phi=x^2y+xz$ at (1,2,-1) in the direction $\vec{A}=2\hat{i}-2\hat{j}+\hat{k}$ is

$\frac{5}{3}$

$\frac{2}{3}$

$\frac{1}{3}$

$\frac{5}{2}$

25. If C is a closed contour of unit circle centered at origin, the value of the complex integral $\oint_C \frac{z}{z^2+2z+2}dz$ is

Zero

$2\pi i$

$(1+i)\pi$

$-(1-i)\pi$