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Q. 1 – Q. 5 carry one mark each.
Q. An apple costs Rs. 10. An onion costs Rs. 8.
Select the most suitable sentence with respect to grammar and usage.
The Buddha said, “Holding on to anger is like "grasping" a hot coal with the intent of throwing it at someone else; you are the one who gets burnt.”
Select the word below which is closest in meaning to the word underlined above.
M has a son Q and a daughter R. He has no other children. E is the mother of P and daughterinlaw
of M. How is P related to M?
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, 2x3y =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,XZ} is a basis of .
(Q) : {X+Y+Z, X+2YZ, X3Z} 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 nonzero linear transformation
T : V → V. If dimension(T(V)) < n and T^{2} = 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 a_{1}=1 and then
is equal to _____________________
Maximum is equal to _________________
Let a,b,c such that c^{2}+d^{2} 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 nonnegative 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 x^{4} be approximated by a polynomial of degree 2, which
interpolates x^{4} 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(C_{0}/M) is equal to _______________________
Which of the following statements is TRUE for the function f(x) =xx^{2}/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 X_{1}, X_{2}, X_{3}, … , 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) : XY 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 S_{4}, then S_{4}/H is
abelian.
Which of the above statements hold TRUE?
Let F be a field of order 32. Then the number of nonzero solutions (a,b) ∈ FxF of
the equation x^{2}+xy+y^{2 }=0 is equal to __________________
Q. 36 – Q. 65 carry two marks each.
Q.
Let be oriented in the counterclockwise 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 R^{2}. 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=cx^{2} 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, S_{10 }+ I_{10 } is equal to
For any
Then, is 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 I_{2} is the value of the integral obtained by the composite trapezoidal rule with
two equal subintervals, then I_{2} is exact.
(Q) : If I_{3} is the value of the integral obtained by the composite trapezoidal rule with
three equal subintervals, then I_{3} 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 x^{2} 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: l_{1}l_{2 }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 xz subject to
10x+yz1
2x2y+z2
x,y,z0
Then, the maximum value of w is equal to _________________________
Let X_{1}, X_{2}, X_{3}, … 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(X_{1}+X_{2}+X_{3}) is equal to ____________
Let x_{1} be an exponential random variable with mean 1 and x_{2} a gamma random
variable with mean 2 and variance 2. If x_{1} and x_{2} are independently distributed, then
P(x_{1} < x_{2)} is equal to _________________________
Let x_{1}, x_{2}, x_{3}, … 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 X_{1},X_{2},X_{3}... 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 H_{1}: θ = 0, α = 2.
Which of the following statements is TRUE?
Let X_{1},X_{2},X_{3}... be a sequence of i.i.d. N(μ,1) random variables. Then,
is equal to _____________________________
Let X_{1},X_{2},X_{3}... Xn be a random sample from uniform [1,θ] for some θ >1. if X_{n }= Maximum (X_{1},X_{2},X_{3}... Xn) then the UMVUE of θ is
Let x_{1}=x_{2}=x_{3}=1, x_{4}=x_{5}=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
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