**Q.-1**

**Let A = **

and V be the vector space of all such that . 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 vanishes is(A) 1 (B) -1 (C) 3 (D) -3

**Q.-3**

**Let A and B be subset of . which of the following is NOT necessarily true ?**

(A) (B)(C) (D)

### Q,-4

**Let [x ] denoted the greatest integer function of x . the value of for which the function**

, & , & x = ois 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**

, & x is rational , & 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 ** is**

** (A) 0 (B) (C) (D) **

**Q.8**

*Let p be the prime number .Let G be the group of all 2*2 matrices over with determinant 1 under matric mulliflication.then the order G is *

* (A) (p-a) p(p+1) (B) (C) (D) *

**Q.7** **Let , then IS **

(A) (B) ( C) (D)

Q.9

Let v be the vector space of all 2*2 matrices over 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 and the pair (m,n) is**

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

Q-10

Let be the real vector space of all polynomials of degree at most n. Let and be the linear transformations defined by

D

T respectively

if A is the matrics representation of the transformation DT-TD ** with respect to the standard basis of 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 and passing through the origin is Q-12 Let f be a continuously differentiable function such thatf(t)dt = \( e^{cosx^2}\) for all the value of Q-13 Let for Let w=(u,v) ,where f is a real valued function defined on having continuous first order partial derivatives. the value of + + at the point (1,2,3) is Q-14 The set of points at which the function attains local maximum is Q-15 Let C be the boundary of the region in the first quadrant by ,x=0 and y=0, oriented counter-clockwise .the value of is Q-16 \[ f(x) = \begin{cases} {0}, & -1\leq x \leq 0\ {x}^4, & 0<x \leq 1\end{cases}\] . if

+ is the taylor’s formula for f about with maximum possible value of n , then the value of for is

Q-17

Let , and let C be curve of intersection f the plane and the cylinder , oriented counter-clockwise .the value of is

Q-18

Let f and g be an function from {o,1} to defined by

for {0,1}.The smallest group of functions from {0,1}

to containing f and g under composition of functions isomorphic to

Q-19

The orthogonal trajectory of the family of curves which passes through (1,1 ) is

Q-20

The function to which the power series converges is

Q-21

Let and for let

show that the sequence is convergent , and find its limit.

Q-22

Evaluate

by changing the order of integration.

Q-23

Find the general solution of the differential equation

= \( {\frac {x\lnx+1}{x^2}} \) , x>0

Q-24

Let be the hemisphere be a closed disc in the xy plane . using gauss’ divergent theorem, evaluate where

and also evaluate

Q-25

Let

. 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 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 with nonempty interior. prove that K is of the form [a,b ] or of the form [a,b] , where is a countable disjoint family of open intervals with end points in K. Q-29 Let be a continuous function such that f is differentiable in (a,c) and (c,b) , a<c<b. if exists , then prove that f is differentiable at c and Q-30 Let G be a finite group , and let be an automorphism of G such that if and only if x=e , where e is the identity element in G prove that every can be represented as for some . moreover , if for every , then show that G is abelian.