MA Mathematics - 2015 GATE Paper (Practice Test)

Tanya is older than Eric.

Cliff is polder than Tanya

Eric is older than cliff.

If the first two statements are true, then the third statement is :

Five teams have to compete in a league, with every team playing every other team exactly once, before going to the next round. How many matches will have to be held to completer the league round of matches?

Q No 6 - 10 carry Two marks each

Q.

Select the appropriate option in place of underlined part of the sentnce.

" Increased productivity necessary " reflects greater efforts made by the employee.

Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.

Statements:

I. No manager is leader.

II All leaders are executives.

Conclusions:

I No manager is an executive

II No executive is a manager.

In the given figure angle  is a right angle, PS:QS= 3:1, RT: QT=5:2 and PU: UR = 1:1. if area of triangle QTS is 20 cm².  then the area of triangle PQR in cm² is ________________.

(Important : you should answer only the numeric value)

Right triangle PQR is to be constructed in the xy- plane so that the right angle is at P and line PR is parallel to the x-axis. The x and y coordinates of P,Q, and R are to be integers that satisfy the inequalities -4x5 and 6 y 16. How many diffrent triangles could be constructed with these properties ?

A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occur in three tosses. Based on the  above information, which one of the following statements is TRUE?

Q. 11 – Q. 35 carry one mark each.

Q.

Let T : R⁴ → R⁴ be a linear map defined by

Then the rank of T is equal to _________

(Important : you should answer only the numeric value)

Let M be a 3 x 3 matrix and suppose that 1, 2 and 3 are the eigenvalues of M. If

for some scalar α 0, then α is equal to ___________

(Important : you should answer only the numeric value)

Let M be a 3 x 3 singular matrix and suppose that 2 and 3 are eigenvalues of M. Then the number
of linearly independent eigenvectors of M₃ + 2 M +I₃  is equal to _________

(Important : you should answer only the numeric value)

Let M be a 3 x 3 matrix such that   and suppose that  for
some . Then | α| is equal to _______

(Important : you should answer only the numeric value)

Let be defined by 

         Then the function f is

Consider the power series 

The radius of convergence of the series is equal to __________

(Important : you should answer only the numeric value)

Let C={zC : |z-i|=2} . then  is equal to ____________

(Important : you should answer only the numeric value)

Let   is equal to ___________

(Important : you should answer only the numeric value)

Let the random variable X have the distribution function

Then P(2X<4) is equal to ___________

(Important : you should answer only the numeric value)

Let X be a random variable having the distribution function

Then E(X) is equal to _________

(Important : you should answer only the numeric value)

In an experiment, a fair die is rolled until two sixes are obtained in succession. The probability that
the experiment will end in the fifth trial is equal to

Let x₁ = 2.2, x₂ = 4.3, x₃ = 3.1, x₄ = 4.5, x₅ = 1.1 and x₆ = 5.7 be the observed values of a
random sample of size 6 from a U(θ - 1, θ + 4) distribution, where θ ∈ (0, ∞) is unknown. Then
a maximum likelihood estimate of θ is equal to

Let  be the open unit disc in  with boundary is the solution of the Dirichlet problem

then u(1/2,0) is equal to

Let c ∈ Z₃ be such that  is a field. Then c is equal to __________

(Important : you should answer only the numeric value)
 

Let  Then V is

Let  be defined by  is equal to __________

(Important : you should answer only the numeric value)

Let ₁be the usual topology on R. Let ₂ be the topology on R generated by

Let X be a connected topological space such that there exists a non-constant continuous function
f : X → R, where R is equipped with the usual topology. Let f(X) = { f(x): x ∈ X}. Then

Let d₁ and d₂ denote the usual metric and the discrete metric on R, respectively. Let  be defined by f(x) = x, x ∈ R. Then

If the trapezoidal rule with single interval [0, 1] is exact for approximating the integral 

then the value of c is equal to ________

(Important : you should answer only the numeric value)

Suppose that the Newton-Raphson method is applied to the equation with an
initial approximation x_o sufficiently close to zero. Then, for the root x = 0, the order of convergence of the method is equal to _________

(Important : you should answer only the numeric value)

The minimum possible order of a homogeneous linear ordinary differential equation with real constant coefficients having  x² sin (x) as a solution is equal to  _______

(Important : you should answer only the numeric value)

The Lagrangian of a system in terms of polar coordinates (r,θ) is given by

where m is the mass, g is the acceleration due to gravity and denotes the derivative of s with
respect to time. Then the equations of motion are

If y(x) satisfies the initial value problem 

then y(2) is equal to __________

(Important : you should answer only the numeric value)

It is known that Bessel functions J_n(x). for n 0, satisfy the identity

for all t > 0 and  xR. The value of   is equal to _________

(Important : you should answer only the numeric value)

Q. 36 – Q. 65 carry two marks each.

Q.

Let X and Y be two random variables having the joint probability density function

Then the conditional probability  is equal to

Let Ω = (0,1] be the sample space and let  be a probability function defined by

Then P({1/2})  is equal to __________

(Important : you should answer only the numeric value)

Let X₁, X₂ and X₃ be independent and identically distributed random variables with E(X₁) =0 and  is defined through the conditional expectation

 then   is equal to __________

(Important : you should answer only the numeric value)

Let X ∼ Poisson(λ), where λ> 0 is unknown. If δ(X) is the unbiased estimator of  is equal to ___________

(Important : you should answer only the numeric value)

Let X_1,, … , X_n be a random sample from N(μ,1) distribution, where  For testing the null
hypothesis H_o :μ=0 against the alternative hypothesis consider the critical region

R 

where c is some real constant. If the critical region R has size 0.025 and power 0.7054, then the
value of the sample size n is equal to ___________

(Important : you should answer only the numeric value)

Let X and Y be independently distributed central chi-squared random variables with degrees of
freedom m(3) and n(3) respectively. If E(X/Y) =3 and m+n =14 then E(Y/X ) is equal to 

Let X₁, X₂, … be a sequence of independent and identically distributed random variables with  for n= 1, 2,…,, then   is equal to __________

(Important : you should answer only the numeric value)

Let  be a solution of the initial value problem

Then f(1) is equal to

Let   be the solution of the initial value problem

Then u(2,2) is equal to ________

(Important : you should answer only the numeric value)

Let  be a subspace of the Euclidean space R⁴. Then the
square of the distance from the point (1,1,1,1) to the subspace W is equal to _______

(Important : you should answer only the numeric value)

Let T : R⁴ → R⁴ be a linear map such that the null space of T is 

 and the rank of (T-4I₄) is 3. If the minimal polynomial of T is x(x-4)^α. then α is equal to _______

(Important : you should answer only the numeric value)

Let M be an invertible Hermitian matrix and let x,y  R be such that x² <4y. then

Let   Then the number
of elements in the center of the group G is equal to

The number of ring homomorphisms from   is equal to __________

(Important : you should answer only the numeric value)

Let  and   be two polynomials in
Q [x]. Then, over Q,

Consider the linear programming problem 

Then the maximum value of the objective function is equal to ______

(Important : you should answer only the numeric value)

Let . Under the usual metric on R²,

Let . Then H

Let V be a closed subspace of   be given by f(x) =x and g(x) = x². if     and pg is the orthogonal projection of g on V, then  

Let p(x) be the polynomial of degree at most 3 that passes through the points (-2,12), (-1,1), (0.2) and (2,-8).  Then the coefficient of x³  in p(x) is equal to _____________

(Important : you should answer only the numeric value)

If, for some α,β  R the integration formula

holds for all polynomials p(x) of degree at most 3, then the value of

3(α-β)² is equal to _____.

(Important : you should answer only the numeric value)

Let y(t) be a continuous function on (0, ∞) whose Laplace transform exists. If y(t) satisfies 

 then y(1) is equal to _______

(Important : you should answer only the numeric value)

Consider the initial value problem 

 if y(x) → 0 as x→ 0^+, then α is equal to ______________

(Important : you should answer only the numeric value)

Define  by

Then

Consider the unit sphere   and the unit normal vector 

 at each point (x,y,z) on S. The value of the surface integral

 is equal to _______

(Important : you should answer only the numeric value)

Let   

Define Then the minimum value of f on D is equal to ________

(Important : you should answer only the numeric value)

Let    Then there exists a non-constant analytic function f on D such that for all n = 2, 3, 4, …

Let   be the Laurent series expansion of  in the annulus 3/2 <|z| <5 . Then a₁/a₂  is equal to _________

(Important : you should answer only the numeric value)

The value of  is equal to __________

(Important : you should answer only the numeric value)

Suppose that among all continuously differentiable functions y(x), x R with y(0) =0 and y(1) = 1/2, the function y₀(x)  minimizes the functional

 then y_o (1/2) is equal to