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January 5, 2014

IIT JAM 2013 Question Paper

Q.-1

Let A = </strong></span><span style="color: #000000;"><strong>{</strong></span><span style="color: #000000;"><strong>\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

Q.-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.
  
 

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