Paper 1

+3-I-S-NEP-Major-I-P-1-Sc-Phy .

The trace of $y=3^x$ is symmetrical about ____ axis.
The second term of Taylor series expansion for the function $(1-x)^{-2}$ is ____.
Write the unit vector perpendicular to ($\hat{i} + \hat{j}$) lies in same plane.
For any scaler function $\phi$, the value of curl grad $\phi$ is ____.
The Dirac delta function $\delta(-x)=$ _____.
The function $\tan x $ is not continuous at $x=$_____.
Does the function $f(x)$ which is continuous at $x_0$ will be necessarily differentiable at xo ? (yes/no/can not say)
The value of $\oint_c \vec{r}.\vec{dr}$ is _____.
In spherical polar co-ordinate system $\hat{r}\times\hat{\theta}=$____.
Does the vector triple product is associative ? (yes/no/ can not say)

Solve $dy/dx = 2x-7$ where $y(2)= 0$.
Trace the curve $y = x \sin x$.
If $\vec{a}$ is a constant vector then show that $\hat{\nabla}. (\vec{a}\times \vec{r})=0$.
If $x=r \cos \theta$, and $y=r \sin \theta$ then evaluate $\partial (x, y) / \partial (r, \theta)$.
Prove that, $\hat{i}\times (\vec{a}\times \hat{i})+\hat{j}\times (\vec{a}\times \hat{j})+\hat{k}\times (\vec{a}\times \hat{k})=2\vec{a}$.
Show that $\vec{a} \ne \vec{c}$ although $\vec{a}.\vec{b}-\vec{b}.\vec{c}$.
Evaluate the directional derivative of the function $\phi =x^2-y^2+2z^2$ from P(1,2,3) to Q(2,3,4).
Prove that $\vec{\nabla}\times (\phi \vec{A})=\phi\vec{\nabla}\times \vec{A}+\vec{A}\times \vec{\nabla}\phi$.
For position vector $\vec{r}$ show that $\vec{\nabla}\vec{r}^n = n\vec{r}^{n-1}\hat{r}$.

Soslve the differential equation $(1+x^2) dy - (1+y^2)dx = 0$.
Using complementary function and particular Integral solve $y'' +4y= 2\sin 2x$.
Verify the linear independence of the functions $e^{ax} \sin bx$ and $e^{ax}\cos bx$.
State and prove Green's theorem.
Prove that $\vec{\nabla}\times (\vec{\nabla}\times \vec{F}) =\vec{\nabla}.(\vec{\nabla}.\vec{F})-\nabla^2 \vec{F}$.
Explain the existance and Uniqueness theorem.
Obtain the acceleration expression in circular cylindrical co-ordinate system.
Evaluate $\int\int_R (x^2 + y^2 ) dxdy$ where R is the region in 1st quadrant of XY plane with side 2 and one vertex at origin as a squire.
Prove that $\int_{-\infty}^\infty \delta (x-a) \delta (x-b) dx = \delta (a-b)$.
Show that scalar product of two vectors remains invariant under rotations.

What are Lagrange's multipliers ? Using them show that "The Maximum volume of solid inscribed in a sphere is a cube".
Establish the physical significance of scalar triple product. Obtain the total surface area and volume of a parallelepiped whose edges $(\hat{i}+2\hat{j}+3\hat{k})$, $(3\hat{i}+4\hat{j}-\hat{k})$ and
$(\hat{i}+2\hat{j}+\hat{k})$.
State and prove Gauss divergence theorem. Using it obtain the volume of a sphere having radius r.
Derive the expression for Laplacian $(\nabla^2)$ in spherical polar co-ordinate system.
Solve the differential equation $(x^2+y^2) dx - 2xy dy = 0$ and obtain Dirac delta function as the limitation of Gaussian function.