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# MA Mathematics - 2012 GATE Paper (Practice Test)

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MA Mathematics - 2012 GATE Paper (Practice Test) - 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.

MA Mathematics - 2012 GATE Paper (Practice Test) - 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.

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 3

### If (1.001)1259 = 3.52 and (1.001)2062 = 7.85, then (1.001)3321 =

MA Mathematics - 2012 GATE Paper (Practice Test) - 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.

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 5

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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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?

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 11

Q. 11 – Q. 35 carry one mark each.

Q.

The straight lines  are mapped by the transformation
w= z2 into the curves C1 , C2 and C3 respectively. The angle of intersection between the curves at
w =0 is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 12

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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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 ( T1 ) 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
( T2 ) space, then f is a one-one function.
Which of the above statements are correct?

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 14

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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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?

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 16

L et X = {a,b,c} and let  be a topology defined on X . Then which of
the following statements are TRUE?

MA Mathematics - 2012 GATE Paper (Practice Test) - 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  M1 and M2 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:

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 18

A continuous random variable X has the probability density function

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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 19

A simple random sample of size 10 from  gives 98% confidence interval (20.49, 23.51).
Then the null hypothesis H0 : μ = 20.5 against HA   20.5

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 20

F or the linear programming problem

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 21

W hich one of the following statements is TRUE?

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 22

Let α =e2πi/5  and the matrix

Then the trace of the matrix  I +M +M2 is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 23

L et V = C2 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?

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 24

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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 25

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

subject to the conditions  u(e,θ )=1, u(e2 ,θ)=0

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 26

The functional

is path independent if k equals

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 28

The expression  is equal to

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 29

The function φ (x) satisfying the integral equation

is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 30

Given the data:

If the derivative of y(x) is approximated as: hen the value
of y'(2) is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 31

If   then A50 is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 32

If    is assumed to be a solution of the differential equation

x2 y"- xy'-3(1+ x2 )y=0  then the values of r are

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 33

Let the linear transformation T :F2 →F3 be defined by Then the
nullity of T is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 34

The approximate eigenvalue of the matrix

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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 35

The root of the equation xe=1 between 0 and 1, obtained by using two iterations of bisection
method, is

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 37

T he Lebesgue measure of the set   is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 38

Which of the following statements are TRUE?

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 39

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

then E(X) is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 40

The probability density function of the random variable X is

where λ > 0 . For testing the hypothesis H0 :λ = 3 against : HA λ =5 , a test is given as “Reject
H0 if X 4.5 ”. The probability of type I error and power of this test are, respectively,

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 41

The order of the smallest possible non trivial group containing elements x and y such that x7=y=e  and yx= x4y is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 42

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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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)

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 44

Let and G ( M,N) be the group
generated by the matrices M and N under matrix multiplication. Then

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 45

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

is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 46

The integral surface satisfying the partial differential equation  and passing through
the straight line x =1, y = z is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 47

The diffusion equation

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 48

Let  f(x) and xf (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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 49

Let the Legendre equation   have nth degree polynomial solution yn(x) such that yn (I) =3. if    then n is

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 51

The functional  possesses :

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 54

L et  T :P3 →P3 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)

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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 xij 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 x12 = 60, x22 = 10, x23 = 50, x24 = 20,
x31 = 40 and x34 = 60, then the variables entering and leaving the basis, respectively, are

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 57

Consider the system of equations

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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 58

Common Data for Questions 58 and 59:

The optimal table for the primal linear programming problem:

is

Q.

If y1 and y2 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,

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 61

Consider the Fredholm integral equation

Q.

The solution of this integral equation is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 62

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

Q.

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

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 63

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

Cov(X,Y) is

MA Mathematics - 2012 GATE Paper (Practice Test) - Question 64

Consider the functions

Q.

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

MA Mathematics - 2012 GATE Paper (Practice Test) - 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

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