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# IIT JAM 2013 Question Paper

### Q.-1

Let A = $${\left(\begin{array}{rrr}1&1&1\3&-1&1\1&5&3\end{array}\right)}$$
 and   V   be the vector space of all $$X\in \mathbb{R}^3$$ such that $$AX=0$$. 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     $$\over\rightarrow{F}=\frac{\vec{r}}{|\vec{r}|^n}, \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} ,|\vec{r}|\neq 0 ,$$ vanishes is
(A)     1                    (B)  -1            (C)     3             (D)     -3

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

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

### Q,-4

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

$$f(n) = \frac{\sin[{-x}^2]}{[{-x}^2]}$$, & $$x\neq o \neq \alpha$$, & 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 $$0<x\leq 1$$ 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   $$0<a\leq 1,$$ $$s_1 = \frac{a}{2}$$ 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.

January 5, 2014