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QUESTION: 1

**Q. 1 – Q. 5 carry one mark each**

**Q.**

Choose the most appropriate word from the options given below to complete the following

sentence:

Given the seriousness of the situation that he had to face, his ___ was impressive.

Solution:

QUESTION: 2

Choose the most appropriate alternative from the options given below to complete the following

sentence:

If the tired soldier wanted to lie down, he ___ the mattress out on the balcony.

Solution:

QUESTION: 3

If (1.001)^{1259} = 3.52 and (1.001)^{2062} = 7.85, then (1.001)^{3321 }=

Solution:

QUESTION: 4

One of the parts (A, B, C, D) in the sentence given below contains an ERROR. Which one of the

following is INCORRECT?

I requested that he should be given the driving test today instead of tomorrow.

Solution:

QUESTION: 5

Which one of the following options is the closest in meaning to the word given below?

Latitude

Solution:

QUESTION: 6

**Q. 6 - Q. 10 carry two marks each.**

**Q.**

**There are eight bags of rice looking alike, seven of which have equal weight and one is slightly
heavier. The weighing balance is of unlimited capacity. Using this balance, the minimum number
of weighings required to identify the heavier bag is**

Solution:

QUESTION: 7

Raju has 14 currency notes in his pocket consisting of only Rs. 20 notes and Rs. 10 notes. The total

money value of the notes is Rs. 230. The number of Rs. 10 notes that Raju has is

Solution:

QUESTION: 8

One of the legacies of the Roman legions was discipline. In the legions, military law prevailed

and discipline was brutal. Discipline on the battlefield kept units obedient, intact and fighting,

even when the odds and conditions were against them.

Which one of the following statements best sums up the meaning of the above passage?

Solution:

QUESTION: 9

A and B are friends. They decide to meet between 1 PM and 2 PM on a given day. There is a

condition that whoever arrives first will not wait for the other for more than 15 minutes. The

probability that they will meet on that day is

Solution:

QUESTION: 10

The data given in the following table summarizes the monthly budget of an average household.

The approximate percentage of the monthly budget NOT spent on savings is

Solution:

QUESTION: 11

**Q. 11 – Q. 35 carry one mark each.**

**Q.**

The straight lines are mapped by the transformation

w= z^{2} into the curves C_{1} , C_{2} and C_{3} respectively. The angle of intersection between the curves at

w =0 is

Solution:

QUESTION: 12

In a topological space, which of the following statements is NOT always true :

Solution:

QUESTION: 13

C onsider the following statements:

P: The family of subsets satisfies the finite intersection property.

Q: On an infinite set X , a metric

The metric space (X,d) is compact.

R: In a Frechet ( T_{1} ) topological space, every finite set is closed.

S: If f : R→X is continuous, where R is given the usual topology and (X, ) is a Hausdorff

( T_{2} ) space, then f is a one-one function.

Which of the above statements are correct?

Solution:

QUESTION: 14

L et H be a Hilbert space and denote the orthogonal complement of a set . Which of

the following is INCORRECT?

Solution:

QUESTION: 15

L et H be a complex Hilbert space, T :H →H be a bounded linear operator and let T * denote

the adjoint of T . Which of the following statements are always TRUE?

Solution:

QUESTION: 16

L et X = {a,b,c} and let be a topology defined on X . Then which of

the following statements are TRUE?

Solution:

QUESTION: 17

C onsider the statements

P: If X is a normed linear space and is a subspace, then the closure is also a subspace

of X.

Q: If X is a Banach space and is an absolutely convergent series in X , then is convergent.

R: Let M_{1} and M_{2} be subspaces of an inner product space such that Then

S: Let f :X →Y be a linear transformation from the Banach Space X into the Banach space Y .

If f is continuous, then the graph of f is always compact.

The correct statements amongst the above are:

Solution:

QUESTION: 18

A continuous random variable X has the probability density function

The probability density function of Y= 3X + 2 is

Solution:

QUESTION: 19

A simple random sample of size 10 from gives 98% confidence interval (20.49, 23.51).

Then the null hypothesis H_{0} : μ = 20.5 against H_{A }:μ_{ } 20.5

Solution:

QUESTION: 20

F or the linear programming problem

Solution:

QUESTION: 21

W hich one of the following statements is TRUE?

Solution:

QUESTION: 22

Let α =e^{2πi/5} and the matrix

Then the trace of the matrix I +M +M^{2} is

Solution:

QUESTION: 23

L et V = C^{2} be the vector space over the field of complex numbers and B={(1, i), (i,1)}be a given

ordered basis of V. Then for which of the following, is a dual basis of B over C?

Solution:

QUESTION: 24

Let R = ZxZxZ and I = ZxZx{0}. Then which of the following statement is correct?

Solution:

QUESTION: 25

T he function u(r,θ) satisfying the Laplace equation

subject to the conditions u(e,θ )=1, u(e^{2} ,θ)=0

Solution:

QUESTION: 26

The functional

is path independent if k equals

Solution:

QUESTION: 27

If a transformation y = uv transforms the given differential equation

f (x)y"- 4 f '(x)y'+ g(x)y = 0 into the equation of the form v''+ h(x)v = 0, then

Solution:

QUESTION: 28

The expression is equal to

Solution:

QUESTION: 29

The function φ (x) satisfying the integral equation

is

Solution:

QUESTION: 30

Given the data:

If the derivative of y(x) is approximated as: hen the value

of y'(2) is

Solution:

QUESTION: 31

If then A^{50} is

Solution:

QUESTION: 32

If is assumed to be a solution of the differential equation

x^{2} y"- xy'-3(1+ x^{2} )y=0 then the values of r are

Solution:

QUESTION: 33

Let the linear transformation T :F^{2} →F^{3} be defined by Then the

nullity of T is

Solution:

QUESTION: 34

The approximate eigenvalue of the matrix

obtained after two iterations of Power method, with the initial vector [1 1 1]^{T} , is

Solution:

QUESTION: 35

The root of the equation xe^{x }=1 between 0 and 1, obtained by using two iterations of bisection

method, is

Solution:

QUESTION: 36

**Q. 36 to Q. 65 carry two marks each.**

**Q.**

Let where the close curve C is the triangle having vertices at

the integral being taken in anti-clockwise direction. Then one value of a is

Solution:

QUESTION: 37

T he Lebesgue measure of the set is

Solution:

QUESTION: 38

Which of the following statements are TRUE?

Solution:

QUESTION: 39

If a random variable X assumes only positive integral values, with the probability

then E(X) is

Solution:

QUESTION: 40

The probability density function of the random variable X is

where λ > 0 . For testing the hypothesis H_{0} :λ = 3 against : H_{A} λ =5 , a test is given as “Reject

H_{0} if X 4.5 ”. The probability of type I error and power of this test are, respectively,

Solution:

QUESTION: 41

The order of the smallest possible non trivial group containing elements x and y such that x^{7}=y^{2 }=e and yx= x^{4}y is

Solution:

QUESTION: 42

The number of 5-Sylow subgroup(s) in a group of order 45 is

Solution:

QUESTION: 43

The solution of the initial value problem

where δ (t) denotes the Dirac-delta function, is

**(Correct Answer will be updated Soon, Temporary marked A)**

Solution:

QUESTION: 44

Let and G ( M,N) be the group

generated by the matrices M and N under matrix multiplication. Then

Solution:

QUESTION: 45

The flux of the vector field flowing out through the surface of the ellipsoid

is

Solution:

QUESTION: 46

The integral surface satisfying the partial differential equation and passing through

the straight line x =1, y = z is

Solution:

QUESTION: 47

The diffusion equation

admits the solution

Solution:

QUESTION: 48

Let *f*(x) and x*f* (x) be the particular solutions of a differential equation

y"+R(x)y'+S(x)y =0

Then the solution of the differential equation y"+R(x)y'+S(x)y =*f*(x) is

Solution:

QUESTION: 49

Let the Legendre equation have n^{th} degree polynomial solution y_{n}(x) such that y_{n }(I) =3. if then n is

Solution:

QUESTION: 50

The maximum value of the function *f*(x,y,z) = xyz subject to the constraint xy+yz+zx-a=0, a>0 is

Solution:

QUESTION: 51

The functional possesses :

Solution:

QUESTION: 52

A particle of mass m is constrained to move on a circle with radius a which itself is rotating about

its vertical diameter with a constant angular velocity ω . Assume that the initial angular velocity is

zero and g is the acceleration due to gravity. If θ be the inclination of the radius vector of the

particle with the axis of rotation and θ denotes the derivative of θ with respect to t , then the

Lagrangian of this system is

Solution:

QUESTION: 53

For the matrix

which of the following statements are correct?

P : M is skew-Hermitian and iM is Hermitian

Q : M is Hermitian and iM is skew Hermitian

R : eigenvalues of M are real

S : eigenvalues of iM are real

Solution:

QUESTION: 54

L et T :P_{3} →P_{3} be the map given by If the matrix of T relative to the

standard bases is M and M' denotes the transpose of the matrix M , then M +M' is

**(Note: Correct Answer will be updated Soon, Temporary marked A)**

Solution:

QUESTION: 55

U sing Euler’s method taking step size =0.1 the approximate value of y obtained corresponding to

x = 0.2 for the initial value problem and y(0)=1, is

Solution:

QUESTION: 56

The following table gives the unit transportation costs, the supply at each origin and the demand of

e ach destination for a transportation problem

Let x_{ij} denote the number of units to be transported from origin i to destination j. If the u-v method

is applied to improve the basic feasible solution given by x_{12} = 60, x_{22} = 10, x_{23} = 50, x_{24} = 20,

x_{31} = 40 and x_{34} = 60, then the variables entering and leaving the basis, respectively, are

Solution:

QUESTION: 57

Consider the system of equations

Using Jacobi’s method with the initial guess the approximate solution after two iterations, is

Solution:

QUESTION: 58

**Common Data for Questions 58 and 59:**

**The optimal table for the primal linear programming problem:**

is

**Q.**

If y_{1} and y_{2} are the dual variables corresponding to the first and second primal constraints, then

their values in the optimal solution of the dual problem are, respectively,

Solution:

QUESTION: 59

**The optimal table for the primal linear programming problem:**

is

**Q.**

If the right hand side of the second constraint is changed from 8 to 20, then in the optimal solution

of the primal problem, the basic variables will be

Solution:

QUESTION: 60

**Common Data for Questions 60 and 61:**

**Consider the Fredholm integral equation **

**Q.**

**The resolvent kernel R(x, t;λ) for this integral equation is**

Solution:

QUESTION: 61

**Consider the Fredholm integral equation **

**Q.**

The solution of this integral equation is

Solution:

QUESTION: 62

**Statement for Linked Answer Questions 62 and 63:**

**The joint probability density function of two random variables X and Y is given as**

Q.

E(X) and E(Y) are, respectively

Solution:

QUESTION: 63

The joint probability density function of two random variables X and Y is given as

Cov(X,Y) is

Solution:

QUESTION: 64

**Statement for Linked Answer Questions 64 and 65:**

**Consider the functions **

**Q.**

**The residue of f (z) at its pole is equal to 1. Then the value of α is**

Solution:

QUESTION: 65

**Consider the functions **

**Q.**

**For the value of α obtained in Q.54, the function g(z) is not conformal at a point**

Solution:

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