MA Mathematics - 2014 GATE Paper (Practice Test)

If y = 5x² + 3, then the tangent at x = 0, y = 3

A foundry has a fixed daily cost of Rs 50,000 whenever it operates and a variable cost of Rs 800Q,
where Q is the daily production in tonnes. What is the cost of production in Rs per tonne for a daily
production of 100 tonnes?

(Important : you should answer only the numeric value)

Q. 6 – Q. 10 carry two marks each.

Q.

Find the odd one in the following group: ALRVX, EPVZB, ITZDF, OYEIK

Anuj, Bhola, Chandan, Dilip, Eswar and Faisal live on different floors in a six-storeyed building
(the ground floor is numbered 1, the floor above it 2, and so on). Anuj lives on an even-numbered
floor. Bhola does not live on an odd numbered floor. Chandan does not live on any of the floors
below Faisal’s floor. Dilip does not live on floor number 2. Eswar does not live on a floor
immediately above or immediately below Bhola. Faisal lives three floors above Dilip. Which of the
following floor-person combinations is correct?

The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The
ratio between the angles of the quadrilateral is 3:4:5:6. The largest angle of the triangle is twice its
smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the
largest angle of the quadrilateral?

(Important : you should answer only the numeric value)

One percent of the people of country X are taller than 6 ft. Two percent of the people of country Y
are taller than 6 ft. There are thrice as many people in country X as in country Y. Taking both
countries together, what is the percentage of people taller than 6 ft?

The monthly rainfall chart based on 50 years of rainfall in Agra is shown in the following figure.
Which of the following are true? (k percentile is the value such that k percent of the data fall below
that value)

(i) On average, it rains more in July than in December
(ii) Every year, the amount of rainfall in August is more than that in January
(iii) July rainfall can be estimated with better confidence than February rainfall
(iv) In August, there is at least 500 mm of rainfall

Q. 1 – Q. 25 carry one mark each.

Q.

The function  is differentiable at

The radius of convergence of the power series  is _____________

Let   E₁ and E₂ be two non empty subsets of a normed linear space X and let 

Then which of the following statements is FALSE:

Let y(x) be the solution to the initial value problem   subject to y(1.2) 2. Using the Euler method with the step size h = 0.05, the approximate value of ??(1.3), correct to two
decimal places, is _____________________

Let α ∈ R. If αx is the polynomial which interpolates the function f (x) = sinπ x on [−1,1]at all the zeroes of the polynomial 4x³ − 3x , then α is ___________

If u(x,t) is the D’Alembert’s solution to the wave equation  with
the condition u(x,0) = 0  is _________________

The solution to the integral equation  

The general solution to the ordinary differential equation  in
terms of Bessel’s functions, ??_v(x), is

The inverse Laplace transform of 

If  X₁ , X₂ is a random sample of size 2 from an N (0,1) population, then follows

 Let be a random variable. Then the value of E[max{Z,0}] is

The number of non-isomorphic groups of order 10 is ___________

Let a,b,c,d be real numbers with a < c < d < b. Consider the ring C[a,b] with pointwise
addition and multiplication. If  then

Let ?? be a ring. If R[x]is a principal ideal domain, then R is necessarily a

Consider the group homomorphism   given by φ ( A) = trace(A) . The kernel of φ
is isomorphic to which of the following groups?

Let X be a set with at least two elements. Let τ and τ′ be two topologies on X such that   Which of the following conditions is necessary for the identity function id : to be
continuous?

Let  be such that det(A− I ) = 0 , where I denotes the 3×3 identity matrix. If the
trace(A) =13 and det(A) = 32, then the sum of squares of the eigenvalues of A is ______

Let V denote the vector space . Then

Let V be a real inner product space of dimension 10 . Let x, y∈V be non-zero vectors such that
= 0 . Then the dimension of   is _________________

Consider the following linear programming problem:
Minimize x₁ + x₂
Subject to:
2x₁ + x₂ ≥ 8
2x₁ + 5x₂ ≥ 10
x₁, x₂ ≥ 0
The optimal value to this problem is _________________________

Let 

be a periodic function of period 2π. The coefficient of sin 3x in the Fourier series expansion of f(x) on the interval [−π, π] is ________________________

For the sequence of functions

consider the following quantities expressed in terms of Lebesgue integrals

Which of the following is TRUE?

Which of the following statements about the spaces   is TRUE ?

The maximum modulus of  is

Let d₁, d₂ and d₃ be metrics on a set X with at least two elements. Which of the following is NOT
a metric on X?

Q. 26 – Q. 55 carry two marks each

Q.

Let   and let ?? be a smooth curve lying in Ω with initial point −1 + 2??
and final point 1 + 2??. The value of  

If ? ? C with a <1, then the value of

where Γ is the simple closed curve |??| = 1 taken with the positive orientation, is _________

Consider C[−1,1] equipped with the supremum norm given by   Define a linear functional T on C[−1,1]by

 Then the value of T is _______

Consider the vector space C[0,1] over R. Consider the following statements:
P: If the set {  tf₁, t²f₂, t³f₃} is linearly independent, then the set { f₁,f₂,f₃} is linearly
independent, where f₁,f₂,f₃  ∈C [0,1] and tn represents the polynomial function t→ tⁿ, n ? N

Q: If F: C[0,1] → R is given by   then F is a linear map

Which of the above statements hold TRUE?

Using the Newton-Raphson method with the initial guess x⁽⁰⁾ = 6, the approximate value of the
real root of x log₁₀ x = 4.77 , after the second iteration, is ____________________

Let the following discrete data be obtained from a curve ?? = ??(??):

Let S be the solid of revolution obtained by rotating the above curve about the ??-axis between x = 0 and ?? = 1 and let V denote its volume. The approximate value of V, obtained using
Simpson’s 1/3 rule, is ______________

The integral surface of the first order partial differential equation

passing through the curve x² + y² = 2x, z = 0 is

The boundary value problem,  ? is converted into the
integral equation 

Then g(2/3) is

If ??1(??) = ?? is a solution to the differential equation then its
general solution is

The solution to the initial value problem  is

The time to failure, in months, of light bulbs manufactured at two plants A and B obey the exponential distribution with means 6 and 2 months respectively. Plant B produces four times as many bulbs as plant A does. Bulbs from these plants are indistinguishable. They are mixed and sold together. Given that a bulb purchased at random is working after 12 months, the probability that it was manufactured at plant A is _____

Let X , Y be continuous random variables with joint density function 

The value of E[X +Y ]is ____________________

Let be the subspace of R, where R is equipped with the usual topology. Which
of the following is FALSE?

Let  A matrix P such that P⁻¹XP is a diagonal matrix, is

Using the Gauss-Seidel iteration method with the initial guess ,the second approximation   for the solution to the system of equations

2x₁-x₂=7

-x₁+2x₂-x₃=1

-x₂+2x₃=1

is

The fourth order Runge-Kutta method given by 

is used to solve the initial value problem  

If u(1) = 1 is obtained by taking the step size h = 1, then the value of 4 K is ______________

A particle P of mass m moves along the cycloid x = (θ − sin θ) and ?? = (1 + cos θ),
0 ≤ θ ≤ 2??. Let g denote the acceleration due to gravity. Neglecting the frictional force, the
Lagrangian associated with the motion of the particle P is:

Suppose that ?? is a population random variable with probability density function

where θ is a parameter. In order to test the null hypothesis H₀: θ = 2, against the alternative
hypothesis H₁: θ = 3, the following test is used: Reject the null hypothesis if X1 ≥ 1/2 and accept otherwise, where  X₁ is a random sample of size 1 drawn from the above population. Then the power of the test is _____

Suppose that x₁, x₂,…, x_n is a random sample of size n drawn from a population with probability
density function

where θ is a parameter such that θ > 0. The maximum likelihood estimator of θ is

Let F be a vector field defined on   Let   be defined by ??(t) = (8 cos 2πt, 17 sin 2πt) and ??(t) = (26 cos 2πt, −10 sin 2πt).

 then m is _________________________

Let g: R3 → R3 be defined by g(x, y, z) = (3y + 4z, 2x − 3z, x + 3y) and let 

S = {(x, y, z) ∈ R3 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 }. If

 then α is _____________________

Let T₁, T₂ : R⁵ → R³ be linear transformations such that rank(T₁) = 3 and nullity(T₂) = 3. Let
T₃ : R³ → R³ be a linear transformation such that T₃ ° T, = T₂. Then rank(T3) is __________

Let F₃ be the field of 3 elements and let F₃ × F₃ be the vector space over F₃. The number of
distinct linearly dependent sets of the form {u, v}, where u, v ∈ F₃ × F₃ {(0,0)} and u ≠ v
is _____________

Let F₁₂₅ be the field of 125 elements. The number of non-zero elements α ∈ F₁₂₅ such that
α⁵ = α is _______________________

The value of  where R is the region in the first quadrant bounded by the curves
y = x², y + x = 2 and x = 0 is ______________

Consider the heat equation 

with the boundary conditions u(0, t) = 0, u(π, t) = 0 for t > 0, and the initial condition
 is ___________________

Consider the partial order in R² given by the relation (x₁, y₁) < (x₂, y₂) EITHER if x₁ < x₂ OR
if x₁ = x₂ and y₁ < y₂. Then in the order topology on R2 defined by the above order

Consider the following linear programming problem:
Minimize: x¹ + x² + 2x³
Subject to

The dual to this problem is:
Maximize: 4y₁ +5y₂ + 6y₃
Subject to

and further subject to:

Let X = C¹[0,1]. For each f ∈ X , define

Which of the following statements is TRUE?

If the power series  converges at 5i and diverges at −3i, then the power series