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### Q.-1

Let A = ${\left(\begin{array}{rrr}1&1&1\3&-1&1\1&5&3\end{array}\right)}$${\left(\begin{array}{rrr}1&1&1\3&-1&1\1&5&3\end{array}\right)}$
 and   V   be the vector space of all $X\in \mathbb{R}^3$$X\in \mathbb{R}^3$ such that $AX=0$$AX=0$. then dim(V) is       (A)     0              (B)     1         (C)          2              (D)       3

### Q.-2

The value of  n  for which the divergence of  the function     $\over\rightarrow{F}=\frac{\vec{r}}{|\vec{r}|^n}, \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} ,|\vec{r}|\neq 0 ,$$\over\rightarrow{F}=\frac{\vec{r}}{|\vec{r}|^n}, \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} ,|\vec{r}|\neq 0 ,$ vanishes is
(A)     1                    (B)  -1            (C)     3             (D)     -3

### Let A and B be subset of $\mathbb{R}$$\mathbb{R}$. which of the following is NOT necessarily true ?

(A)   $({A}\cap{B})^0 \subseteq{A}^0\cap{B}^0$$({A}\cap{B})^0 \subseteq{A}^0\cap{B}^0$ (B)  ${A}^0\cup{B}^0 \subseteq ({A}\cup{B})^0$${A}^0\cup{B}^0 \subseteq ({A}\cup{B})^0$(C)  $\bar{A}\cup \bar{B} \subseteq \overline{{A}\cup{B}}$$\bar{A}\cup \bar{B} \subseteq \overline{{A}\cup{B}}$ (D)  $\bar{A}\cap \bar{B} \subseteq \overline{{A}\cap{B}}$$\bar{A}\cap \bar{B} \subseteq \overline{{A}\cap{B}}$

### Q,-4

#### Let [x ] denoted the greatest integer function of x . the value of $\alpha$$\alpha$ for which the function

$f(n) = \frac{\sin[{-x}^2]}{[{-x}^2]}$$f(n) = \frac{\sin[{-x}^2]}{[{-x}^2]}$, & $x\neq o \neq \alpha$$x\neq o \neq \alpha$, & x = o is continuous   at x=0 is
(A)   0      (B)                sin(-1)       (C)         sin 1        (D)      1

### Q.-5 Let the function is f(x) be defined by

$f(n) = {e}^x$, & x is rational ${e}^1-x$, & x is irrational for x in (0,1).then (A) f is continuous at every point in (0,1) (B) f is discontinuous at every point in (0,1) (C) f is discontinuous only at one point in (0,1) (D) f is continuous only at point in (0,1)

Q.6 The value of integra $\int\limits\int_D\sqrt{x^2+y^2} dxdy, D={(x,y)\in \mathbb{R}^2 :X\leq{x}^2+{y}^2\leq 2x}$ is

(A) 0 (B) $\frac{7}{9}$ (C) $\frac{14}{9}$ (D) $\frac{14}{9}$

Q.8

Let p be the prime number .Let G be the group of all 2*2 matrices over $\mathbb{Z}_P$ with determinant 1 under matric mulliflication.then the order G is

(A) (p-a) p(p+1) (B) ${p}^2(p-1)$ (C) ${p}^3$ (D) ${p}^2(p-1)+p$

Q.7 Let $x_n=\left ( {1}-\frac{1}{3} \right)^2$ $\left ( {1}-\frac{1}{6} \right)^2$ $\left ( {1}-\frac{1}{10} \right)^2.....$ $\left ( {1}-\frac{1}{\frac{n(n-1)}{2}} \right)^2$ , $n\geq2$ then $\lim_{n to \infty}$ IS

(A) $\frac{1}{3}$ (B) $\frac{1}{9}$ ( C) $\frac{1}{81}$ (D) $0$

Q.9

Let v be the vector space of all 2*2 matrices over $\mathbb{R}.$ consider the subspaces

$W_1 =\left(\begin{array}{cc}a&-a\ c&d\end{array}\right):a,c,d \in \mathbb{R}$ and $W_2 =\left(\begin{array}{cc}a&b\ -a&d\end{array}\right):a,b,d \in \mathbb{R}$ \$ if $m = \dim(W_1\cap W_2)$ and $n=\dim(W_1+W_2)$ the pair (m,n) is

(A) (2,3) (B) (2,4) (C) (3,4) (D) (1,3)

Q-10

Let $wp$ be the real vector space of all polynomials of degree at most n. Let $D :wp_n \rightarrow wp_n-1$ and $T:wp_n \rightarrow wp_n+1$ be the linear transformations defined by

D$\left ( a_0+a_1 x +a_2 x^2 +.....+a_n x^n \right )=a_1 +2a_2 x+......+na_n x^{n-1}$

T$\left ( a_0+a_1 x +a_2 x^2 +.....+a_n x^n \right )=a_0 x +a_1 x^2+a_2 x^3+......+na_n x^{n+1},$ respectively

if A is the matrics representation of the transformation DT-TD $: wp \rightarrow wp$ with respect to the standard basis of $wp$ then the trace of A.

(A) n (B) -n (C) (n+1) (D) -(n+1)

FILL IN THE BLANK QUESTION
Q-11
The equation of curve satisfying $\sin y \frac{dy}{dx} = \cos y (1-x\cos y)$ and passing through the origin is
Q-12
Let f be a continuously differentiable function such that
$\int_{0}^{2x^2}$ f(t)dt = $$e^{cosx^2}$$ for all
$x\in(0,\infty )$ the value of $f'(\pi) is$
Q-13
Let $u = \frac{y^2-x^2}{x^2y^2},$ $\frac{z^2-y^2}{y^2z^2}$ for $x\neq o,y\neq 0,z\neq 0.$ Let w=(u,v) ,where f is a real valued function defined on $\mathbb{R}$ having continuous first order partial derivatives. the value of
$x^3\frac{\partial w^3}{\partial x}$+ $x^3\frac{\partial y^3}{\partial x}$ +$x^3\frac{\partial z^3}{\partial x}$ at the point (1,2,3) is
Q-14
The set of points at which the function $f(x,y) = x^4+y^4-x^2-y^2+1,(x,y) \in\mathbb{R}^2$ attains local maximum is
Q-15
Let C be the boundary of the region in the first quadrant by $y=1-x^2$ ,x=0 and y=0, oriented counter-clockwise .the value of $\int_c(xy^2dx-x^2ydy )$ is
Q-16
$f(x) = \begin{cases} {0}, & -1\leq x \leq 0\ {x}^4, & 0<x \leq 1\end{cases}$ . if

$f(x)= \sum_{k=0}^{n}\frac{f^{(k)}f(0)}{k!}x^{k}$+ $\sum_{k=0}^{n}\frac{f^{(n+1)}f(xi)}{n+1!}x^{n+1}$ is the taylor’s formula for f about $x=0$ with maximum possible value of n , then the value of $xi$ for $0 is

Q-17

Let $\vec{F}=2z\hat{i}+4x\hat{j}+5y\hat{k}$, and let C be curve of intersection f the plane $z=x+4$ and the cylinder $x^2+y^2 =4$, oriented counter-clockwise .the value of $o\int_c \vec{F}d\vec{r}$ is

Q-18

Let f and g be an function from $\mathbb{R}$ {o,1} to $\mathbb{R}$ defined by $f(x) = \frac{1}{x}$

$g(x)=\frac{x-1}{x}$ for $x\in \mathbb{R}$ {0,1}.The smallest group of functions from $\mathbb{R}${0,1}

to $\mathbb{R}$ containing f and g under composition of functions isomorphic to

Q-19

The orthogonal trajectory of the family of curves $\frac{x^2}{2}+{y}^2=c ,$ which passes through (1,1 ) is

Q-20

The function to which the power series $\sum_{n=1}^{\infty}{ (-1)}^{n+1} {n} {x}^{2n-2}$ converges is

Q-21

Let $0 $s_1 = \frac{a}{2}$ and for $n\in{N},$ let $s_{n+1} =\frac{1}{2}(s_n^2+a).$

show that the sequence ${{s_n}}$ is convergent , and find its limit.

Q-22

Evaluate

$\int_{\frac{1}{4}}^{1}$ $\int_{\sqrt{x-{x}^2}}^{\sqrt{x}}$ $\frac{x^2-y^2}{x^2}dydx$

by changing the order of integration.

Q-23

Find the general solution of the differential equation

$x^2 \frac {d^3y}{dx^3}+x\frac {d^2y}{dx^2}-6\frac {dy}{dx}+6\frac {y}{x}$ = $${\frac {x\lnx+1}{x^2}}$$ , x>0

Q-24

Let $S_1$ be the hemisphere $x^2+y^2+z^2 =1 ,z >0$ $S_2$be a closed disc $x^2+y^2 \leq 1$ in the xy plane . using gauss’ divergent theorem, evaluate $\int \int_{S} \vec{F}.d\vec{S},$ where

$\vec{F} = z^2 x \hat{i}$ $+ \left ( \frac{y^3}{3}+\tan z \right )\hat{j}$ $+( x^2 z + y^2)\hat{k}$

and $S=S_1 \cup S_2$ also evaluate $\int \int_{s_1} \vec{F}.d\vec{S},$

Q-25

Let

$f(x ,y) = \begin{cases} \frac{2(x^3+y^3)}{x^2+2y}, ( x,y) \neq (0.0)\ 0 , (x,y)=(0,0)\end{cases}$ .
Show that first order partial derivatives of f with respect to x and y exist at (0,0).also show that f is not continuous at (0,0).
Q-26
Let A be an n*n diagonal matrix with characteristics polynomials ${(x-a )^p}{(x-b)^q} ,$ where a and b are distincts real number. Let V be the real vector space of all n*n matrices B such that AB = BA . Determine the dimension of V .
q-27
Let A be an n*n symmetrics matrics with n distinct eigenvalues. prove that there exists an orthogonal matrics P such that AP = PD , where D is a real diagonal matrix.
Q-28
Let K be a compact subset of $\mathbb{R}$ with nonempty interior. prove that K is of the form [a,b ] or of the form [a,b] $\cup I_n$ , where ${{ I_n}}$ is a countable disjoint family of open intervals with end points in K.
Q-29
Let $f: [a,b] \rightarrow \mathbb{R}$ be a continuous function such that f is differentiable in (a,c) and (c,b) , a<c<b.
if $\lim_{x=c} f'(x)$ exists , then prove that f is differentiable at c and $f'(c) = \lim_{x=c} f'(x).$
Q-30
Let G be a finite group , and let $var\phi$ be an automorphism of G such that $var\phi (x)= x$ if and only if x=e ,
where e is the identity element in G prove that every $g\in G$ can be represented as $g= x^{-1} var\phi(x)$
for some $x\in G$ . moreover , if $var\phi(var\phi(x))=x$ for every $x\in G$ , then show that G is abelian.