1. The function $e^{\cos x}$ is Taylor expanded about $x=0$. The coefficient of $x^2$ is

2. Let M be a $2\times 2$ matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are

3. The solution $y(x)$ of the differential equation $y\frac{dy}{dx}+3x=0$, $y(1)=0$ is described by

4. The volume integral $\int_V e^{-(r/R)^2}\vec{\nabla}\cdot(\frac{\hat{r}}{r^2})d^3 r$, where $V$ is the volume of a sphere of radius $R$  centered at the origin, is equal to

5. If $P$ and $Q$ are ermitian matrices, which of the following is/are true?

A. $PQ+QP$ is always Hermitian.

B. $i(PQ-QP)$ is always Hermitian.

C. $PQ$ is always Hermitian.

D. $PQ-QP$ is always Hermitian.

Choose the correct answer from the options given below:

6. The line integral of the vector function $\vec{A}(x,y)=2y\hat{i}+x\hat{j}$ along the straight line from (0, 0) to (2, 4) is

7. The function $f(x)=\frac{8x}{x^2+9}$ is continuous everywhere except at

8. If $\phi(x,y,z)$ is a scalar function which satisfies the Laplace equation, then the gradient of $\phi$ is

9. The eigenvalues of $\begin{pmatrix}
3 & i & 0\\
-i & 3 & 0\\
0 & 0 & 6
\end{pmatrix}$ are

10. The gradient of a scalar field $S(x,y,z)$ has the following characteristic(s).

A. Line integral of the gradient is path independent.

B. Closed line integral of the gradient is zero.

C. Gradient is a measure of the maximum rate of change.

D. Gradiet is scalar quantity.

Choose the most appropriate answer from the options given below:

11. Let $f(x,y)=x^3-2y^3$. The curve along which $\vec{\nabla}^2f=0$ is

12. A curve is given by $\vec{r}(t)=t\hat{i}+t^2\hat{j}+t^3\hat{k}$. The unit vector of the tangent to the curve at $t=1$ is

13. The function $f(x)=\left\{\begin{matrix}x, -\pi<x<0
& \\-x, 0<x<\pi
\end{matrix}\right.$ is expanded as a Fourier series of the form $a_0 + \sum_{n=1}^{\infty}a_n \cos(nx)+ \sum_{n=1}^{\infty}b_n \sin(nx)$. Which of the following is true?

14. Let $f(x)=3x^6-2x^2-8$. Which of the following statements are true?

A. The sum of all its roots is zero.

B. The product of its roots is $-\frac{8}{3}$.

C. The sum of all its roots is $\frac{2}{3}$.

D. Complex roots are conjugates of each other.

15. For the Fourier series of the following function of period $2\pi$,

$f(x)=\left\{\begin{matrix} 0, -\pi<x<0,
& \\1, 0<x<\pi

\end{matrix}\right.$, the ratio of the Fourier coefficients of the first and the third harmonic is:

16. If $\lambda$ is an eigen value of a non-singular matrix $A$, then the eigen value of $A^{-1}$ is

17. A unit vector perpendicular to the plane containing $\vec{A}=\hat{i}+\hat{j}-2\hat{k}$ and $\vec{B}=2\hat{i}-\hat{j}+\hat{k}$ is

18. A hemispherical shell is placed on the x-y plane centered at the origin. For a vector field $\vec{E}=\frac{-y\hat{i}+x\hat{j}}{x^2+y^2}$, the value of the integral $\int_S(\vec{\nabla}\times\vec{E})\cdot d\vec{s}$ over the hemispherical surface is

19. Which of the following is not true for Hermite polynomials?

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

21. Consider the differential equation $y^{\prime\prime}+2y^{\prime}+y=0$. If $y(0)=0$ and $y^{\prime}(0)=1$, then the value of $y(2)$ is

22. Consider a $2\times 2$ matrix M=$\begin{pmatrix}
0 & a\\
a & b
\end{pmatrix}$, where $a,b>0$. Then

A. $M$ is real symmetric matrix.

B. One of the eigenvalues of $M$ is greater than $b$.

C. One of the eigenvalues of $M$ is negative.

D. Product of eigenvalues of $M$ is $b$.

Choose the most appropriate answer from the options given below:

23. The equation $z^2+\bar{z}^2=4$ in the complex plane (where $\bar{z}$ is the complex conjugate of $z$) represents

24. Consider a unit circle $C$ in the xy plane, centered at the origin. The value of the integral $\oint [(\sin x-y) dx - (\sin y- x) dy]$ over the circle $C$, traversed anticlockwise is

25. Consider a vector field $\vec{F}=y\hat{i}+xz^3\hat{j}-zy\hat{k}$. Let C be the circle $x^2+y^2=4$ on the plane $z=2$, oriented counter-clockwise. The value of the contour integral $\oint \vec{F}\cdot\vec{dr}$ is

26. Let $(x,y)$ denote the coordinates in a rectangular Cartesian coordinate system $C$. Let $(x^{\prime}, y^{\prime})$ denote the coordinates in another coordinate system $C^{\prime}$, defined by

$x^{\prime}=2x+3y$

$y^{\prime}=-3x+4y$.

The area element in $C^{\prime}$ is

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

28. What is the equation of the plane which is tangent to the surface $xyz=4$ at the point (1,2,2)?

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

30. Three vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$ are given by $\vec{A}=\alpha\hat{i}-2\hat{j}+2\hat{k}$, $\vec{B}=6\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{C}=-3\hat{i}-2\hat{j}-4\hat{k}$. The value of $\alpha$ for which the vectors will be coplanar is

31. If $\vec{F}$ is a constant vector and $\vec{r}$ is the position vector then $\vec{\nabla}(\vec{F}\cdot\vec{r})$ would be

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

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

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

35. 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

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

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

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

39. If  $J_{1/2}(x)$ are $J_{-1/2}(x)$ are Bessel's functions, the value of $[J_{1/2}(x)]^2+[J_{-1/2}(x)]^2$ is

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

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

42. If $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is the position vector, then the value of $\vec{\nabla}(\log r)$ is

43. The value of $\alpha$ so that $e^{\alpha y^2}$ is an integrating factor of the differential equation $(e^{-{y^2}/2}-xy)dy-dx=0$ is

44. The value of $\sqrt{i}+\sqrt{-i}$, where $i=\sqrt{-1}$ is

45. The value of the integral $\int_C \frac{z^2+1}{(z+1)(z+2)}dz$, where $C$ is $|z|=\frac{3}{2}$ is

46. Let $P_n(x)$ be the Legendre polynomial of degree $n>1$, then the value of the integral $\int_{-1}^{1}(1+x)P_n(x)dx$ is equal to

47. Laplace transform of $e^{-2t}\sin 4t$ is

48. The value of the integral $\int_{-\pi/2}^{\pi/2} \sin^2\theta\, \delta(3\theta+\pi)\,d\theta$ is

49. Out of the given equations, the only equation which is an exact differential is

50. The Laplace transform of $t^3 \delta(t-4)$ is