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Q. 1 – Q. 5 carry one mark each.
Q.
A student is required to demonstrate a high level of comprehension of the subject, especially in the
social sciences.
The word closest in meaning to comprehension is
Choose the most appropriate word from the options given below to complete the following
sentence.
One of his biggest ______ was his ability to forgive.
Rajan was not happy that Sajan decided to do the project on his own. On observing his
unhappiness, Sajan explained to Rajan that he preferred to work independently.
Which one of the statements below is logically valid and can be inferred from the above sentences?
If y = 5x^{2} + 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 sixstoreyed building
(the ground floor is numbered 1, the floor above it 2, and so on). Anuj lives on an evennumbered
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 floorperson 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 E_{1} and E_{2} 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 4x^{3 }− 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 X_{1} , X_{2} 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 nonisomorphic 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 nonzero vectors such that
Consider the following linear programming problem:
Minimize x_{1} + x_{2}
Subject to:
2x_{1} + x_{2} ≥ 8
2x_{1} + 5x_{2} ≥ 10
x_{1}, x_{2} ≥ 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 ?
Let d_{1}, d_{2} and d_{3} 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 { tf_{1}, t^{2}f_{2}, t^{3}f_{3}} is linearly independent, then the set { f_{1},f_{2},f_{3}} is linearly
independent, where f_{1},f_{2},f_{3} ∈C [0,1] and tn represents the polynomial function t→ t^{n}, 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 NewtonRaphson method with the initial guess x^{(0)} = 6, the approximate value of the
real root of x log_{10} 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 x^{2} + y^{2} = 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^{−1}XP is a diagonal matrix, is
Using the GaussSeidel iteration method with the initial guess ,the second approximation for the solution to the system of equations
2x_{1}x_{2}=7
x_{1}+2x_{2}x_{3}=1
x_{2}+2x_{3}=1
is
The fourth order RungeKutta 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 H_{0}: θ = 2, against the alternative
hypothesis H_{1}: θ = 3, the following test is used: Reject the null hypothesis if X1 ≥ 1/2 and accept otherwise, where X_{1} is a random sample of size 1 drawn from the above population. Then the power of the test is _____
Suppose that x_{1}, x_{2},…, 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 T_{1}, T_{2} : R^{5} → R^{3} be linear transformations such that rank(T_{1}) = 3 and nullity(T_{2}) = 3. Let
T_{3} : R^{3} → R^{3} be a linear transformation such that T_{3} ° T, = T_{2}. Then rank(T3) is __________
Let F_{3} be the field of 3 elements and let F_{3} × F_{3} be the vector space over F_{3}. The number of
distinct linearly dependent sets of the form {u, v}, where u, v ∈ F_{3} × F_{3} {(0,0)} and u ≠ v
is _____________
Let F_{125} be the field of 125 elements. The number of nonzero elements α ∈ F_{125} such that
α^{5} = α is _______________________
The value of where R is the region in the first quadrant bounded by the curves
y = x^{2}, 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 R^{2} given by the relation (x_{1}, y_{1}) < (x_{2}, y_{2}) EITHER if x_{1} < x_{2} OR
if x_{1} = x_{2} and y_{1} < y_{2}. Then in the order topology on R2 defined by the above order
Consider the following linear programming problem:
Minimize: x^{1} + x^{2} + 2x^{3}
Subject to
The dual to this problem is:
Maximize: 4y_{1} +5y_{2} + 6y_{3}
Subject to
and further subject to:
Let X = C^{1}[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
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