MA Mathematics - 2016 GATE Paper (Practice Test)

The number that least fits this set: (324, 441, 97 and 64) is ________.

Note: Question may have more than one correct answer)

It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to
completely pass a telegraph post. The length of the first train is 120 m and that of the second train is
150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________.

Q. 6 – Q. 10 carry two marks each.

Q.

The velocity V of a vehicle along a straight line is measured in m/s and plotted as shown with respect to time in seconds. At the end of the 7 seconds, how much will the odometer reading increase by (in m)?

The overwhelming number of people infected with rabies in In dia has been flagged by the World  Health Organization as a source of concern. It is estimated that inoculating 70% of pets and stray  dogs against rabies can lead to a significant reduction in the number of people infected w ith rabies.

Which of the fo llowing can be logically inferred from the above sentences?

A flat is shared by four first year undergraduate students. They agreed to allow the oldest of them to enjoy some extra space in the flat. Manu is two months older than Sravan, who is three months younger than Trideep. Pavan is one month older than Sravan. Who should occupy the extra space in the flat?

Find the area bounded by the lines 3x+2 y=14, 2x-3y =5 in the first quadrant

A straight line is fit to a data set (ln x, y) . This line intercepts the abscissa at ln x = 0.1 and has a slope of −0.02. What is the value of y at x = 5 from the fit?

Q. 11 – Q. 35 carry one mark each.

Q.

Let {X,Y,Z} be a basis of . Consider the following statements P and Q:
(P) : {X+Y, Y+Z,X-Z} is a basis of .
(Q) : {X+Y+Z, X+2Y-Z, X-3Z} is a basis of .
Which of the above statements hold TRUE?

Consider the following statements P and Q:
(P) : If  M= then M is singular

(Q) : Let S be a diagonalizable matrix. If T is a matrix such that S + 5 T = Id, then T is
diagonalizable

Which of the above statements hold TRUE?

Consider the following statements P and Q:

(P) : If M is an n x n complex matrix, then  

(Q) : There exists a unitary matrix with an eigenvalue λ such that |λ| <1.

Which of the above statements hold TRUE?

Consider a real vector space V of dimension n and a non-zero linear transformation
T : V → V. If dimension(T(V)) < n and T² = λ T, for some then which of the following statements is TRUE?

Let  and  and   be a strictly increasing function such that f(S) is connected. Which of the following statements is TRUE?

Let a₁=1 and   then

 is equal to _____________________

Maximum   is equal to _________________

Let a,b,c such that c²+d² 0. Then, the Cauchy problem

has a unique solution if

Let u(x,t)  be the d'Alembert's solution of the initial value problem for the wave
equation

where c is a positive real number and f, g are smooth odd functions. Then, u(0,1) is
equal to ___________

Let the probability density function of a random variable X be

Then, the value of c is equal to ________________________

Let V be the set of all solutions of the equation y" +a y'+by=0 satisfying  y(0) =y(1), where a, b are positive real numbers. Then, dimension(V ) is equal to
_____________________

Let  where p(x) and q(x)are continuous functions. If   cos(x) are two linearly independent solutions of the above equation, then |4p(0)+2q(1)| is equal to ____________________

Let P_n(x) be the Legendre polynomial of degree n and   where k
is a non-negative integer. Consider the following statements P and Q:

(P) : I = 0 if k < n.
(Q) : I = 0 if n - k is an odd integer.
Which of the above statements hold TRUE?

Consider the following statements P and Q:

(P) :   has two linearly independent Frobenius series
solutions near x=0.

(Q) : has two linearly independent Frobenius series
solutions near x=0.

Which of the above statements hold TRUE?

Let the polynomial x⁴ be approximated by a polynomial of degree 2, which
interpolates x⁴ at x=-1, 0 and 1. Then, the maximum absolute interpolation error
over the interval [-1, 1] is equal to ______________________

Let  (z_n)be a sequence of distinct points in  with    

Consider the following statements P and Q:

(P) : There exists a unique analytic function f on D(0,1) such that f(z_n) = sin(z_n) for
all n.
(Q) : There exists an analytic function f on D(0,1) such that f(zn) = 0 if n is even
and f(z_n) = 1 if n is odd.

Which of the above statements hold TRUE?

Let be a topological space with the cofinite topology. Every infinite subset of
is

Let 

Then, dimension(C₀/M) is equal to _______________________

Which of the following statements is TRUE for the function f(x)  =x-x²/2 ?

Note: Question may have more than one correct answer)

Let be the set of all n x  n real matrices with the usual norm topology. Consider the
following statements P and Q:
(P) : The set of all symmetric positive definite matrices in is connected.
(Q) : The set of all invertible matrices in is compact.

Which of the above statements hold TRUE?

Let X₁, X₂, X₃, … , X_n be a random sample from the following probability density
function for 0 < μ < ∞, 0 < α < 1,

Here α and μ are unknown parameters. Which of the following statements is TRUE?

Suppose X and Y are two random variables such that aX+bY  is a normal random a,b 
variable for all. Consider the following statements P, Q, R and S:

(P) : X is a standard normal random variable.
(Q) : The conditional distribution of X given Y is normal.
(R) : The conditional distribution of X given X+Y is normal.
(S) : X-Y has mean 0.
Which of the above statements ALWAYS hold TRUE?

Consider the following statements P and Q:
(P) : If H is a normal subgroup of order 4 of the symmetric group S₄, then S₄/H is
abelian.

Which of the above statements hold TRUE?

Let F be a field of order 32. Then the number of non-zero solutions (a,b) ∈ FxF of
the equation x²+xy+y² =0 is equal to __________________

Q. 36 – Q. 65 carry two marks each.

Q.

Let   be oriented in the counter-clockwise direction. Let

Then, the value of I is equal to __________________________

Let  be a harmonic function and v(x,y) its harmonic conjugate. If  is equal to _______________

Let λ be the triangular path connecting the points (0,0), (2,2) and (0,2) in the counterclockwise
direction in R². Then

is equal to _____________________

Let y be the solution of 

Then y(1) is equal to

Let X be a random variable with the following cumulative distribution function:

Then P(1/4 <X<1) is equal to ___________________

Let γ be the curve which passes through (0,1) and intersects each curve of the family y=cx² orthogonally. Then γ also passes through the point

Let  be the Fourier series of the
2 π periodic function defined by   then 

 is equal to ________________

Let y(t) be a continuous function on [0, ∞). If 

 then   is equal to _______________.

Let  then, S₁₀ + I₁₀  is equal to 

For any 

Then, is equal to ____________________

Let 

 equal to ______________

Let M= be a real matrix with eigenvalues 1, 0 and 3. If the eigenvectors
corresponding to 1 and 0 are (1,1,1)^T and (1, -1,0)^T respectively, then the value of
3f is equal to _________________

Let M=  and   then

 is equal to ________________________

Let the integral 

Consider the following statements P and Q:
(P) : If I₂ is the value of the integral obtained by the composite trapezoidal rule with
two equal sub-intervals, then I₂ is exact.
(Q) : If I₃ is the value of the integral obtained by the composite trapezoidal rule with
three equal sub-intervals, then I₃ is exact.

Which of the above statements hold TRUE?

The difference between the least two eigenvalues of the boundary value problem

is equal to ______________________________

The number of roots of the equation x²- cos(x)= 0 in the interval  

 is equal to ____________

For the fixed point iteration   consider the following statements P and Q:

(P) : if g(x) =1+2/x then the fixed point iteration converges to 2 for all  

(Q) : if g(x) =  then the fixed point iteration converges to 2 for all 

Which of the above statements hold TRUE?

Let T: l₁-l₂ be defined by

 Then

Minimize w =x+2y subject to

2x+y 3

x+y2

x0, y0

Then, the minimum value of w is equal to _________________________

Maximize w=11 x-z subject to

10x+y-z1

2x-2y+z2

x,y,z0

Then, the maximum value of w is equal to _________________________

Let X₁, X₂, X₃, … be a sequence of i.i.d. random variables with mean 1. If N is a geometric random variable with the probability mass function P(N=K) =1/2^K.  K=1,2,3, .... and it is independent of the X_I's then E(X₁+X₂+X₃) is equal to ____________

Let x₁ be an exponential random variable with mean 1 and x₂ a gamma random
variable with mean 2 and variance 2. If x₁ and x₂ are independently distributed, then
P(x₁ < x₂₎ is equal to _________________________

Let x₁, x₂, x₃, … be a sequence of i.i.d. uniform (0,1) random variables. Then, the value
of 

is equal to ____________________

Let X be a standard normal random variable. Then  is equal to

Let X₁,X₂,X₃... Xn be a random sample from the probability density function

where α>0, 0 θ 1 are parameters. Consider the following testing problem:

H_o: θ = 1, α = 1 versus H₁: θ = 0, α = 2.

Which of the following statements is TRUE?

Let X₁,X₂,X₃... be a sequence of i.i.d. N(μ,1) random variables. Then,

is equal to _____________________________

Let X₁,X₂,X₃... Xn be a random sample from uniform [1,θ] for some  θ >1. if X_n  = Maximum (X₁,X₂,X₃... Xn) then the UMVUE of θ is

Let x₁=x₂=x₃=1, x₄=x₅=x_{6 =2} be a random sample from a Poisson random
variable with mean θ, where  Then, the maximum likelihood estimator of θ
is equal to ____________________

The remainder when 98! is divided by 101 is equal to ____________________________

Let G be a group whose presentation is  

Then G is isomorphic to