Sample Page 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 is25/2626/2546/2525/46 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 $\oint\vec{r} \cdot d\vec{r}$ ?10-13 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}$$y=A e^{x} + B x $ 16. The residue of the complex function $f(z)=e^{1/z}$ at $z=0$ is10-12 17. The modulus and phase of the complex number $(1+i)i$ in polar representation are$\sqrt{2}$ and $\frac{3\pi}{4}$$\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$$0$ 19. For the function $f(z)=\frac{z \sin z}{(z-\pi)^3}$, the residue at the pole $z=\pi$ is12-1-2 20. Consider the transformation to a new set of coordinates $(p,q)$ from rectangular Cartesian coordinates $(x,y)$ , where $p=x+y$ and $q=x-y$. In the $(p,q)$ coordinate system, the area element $dxdy$ is$\frac{1}{2}dpdq$$\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 beZeroSV3V 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 $\oint_C \frac{z}{z^2+2z+2}dz$ isZero$2\pi i$$(1+i)\pi$$-(1-i)\pi$ Loading...