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Q. 1 – Q. 5 carry one mark each.
Q. A number is as much greater than 75 as it is smaller than 117. The number is:
Which of the above underlined parts of the sentence is grammatically incorrect?
Which of the following options is the closest in meaning to the word given below:
HistoricPrimeval
Friendship, no matter how _________it is, has its limitations
Select the pair that best expresses a relationship similar to that expressed in the pair: Medicine: Health
Q. 6 to Q. 10 carry two marks each.
Q. X and Y are two positive real numbers such that 2X + Y ≤ 6 and X + 2Y ≤ 8. For which of the following values of (X, Y) the function /(X, Y) = 3X + 6Y will give maximum value?
If 4X−7=5 then the values of 2 X− −X is:
Following table provides figures (in rupees) on annual expenditure of a firm for two years  2010 and 2011.
In 2011, which of the following two categories have registered increase by same percentage?
A firm is selling its product at Rs. 60 per unit. The total cost of production is Rs. 100 and firm is earning total profit of Rs. 500. Later, the total cost increased by 30%. By what percentage the price should be increased to maintained the same profit level.
Abhishek is elder to Savar. Savar is younger to Anshul. Which of the given conclusions is logically valid and is inferred from the above statements?
Q. 11 – Q. 35 carry one mark each.
Q. The possible set of eigen values of a 4 × 4 skewsymmetric orthogonal real matrix is
The coefficient of (z−π)^{2} in the Taylor series expansion of
around π is
Consider R^{2} with the usual topology. Which of the following statements are TRUE for all A,B ⊆ R^{2}?
Let f:R →R be a continuous function with then g′ (0) is equal to ______
(Important : you should answer only the numeric value)
Let P be a 2×2 complex matrix such that trace(P)=1 anddet(P)=−6. Then, trace(P^{4}−P^{3}) is ______
(Important : you should answer only the numeric value)
Suppose that R is a unique factorization domain and that a,b ∈ R are distinct irreducible elements. Which of the following statements is TRUE?
Let X be a compact Hausdorff topological space and let Y be a topological space. Let f:X→Y be a bijective continuous mapping. Which of the following is TRUE?
Consider the linear programming problem:
If S denotes the set of all solutions of the above problem, then
Which of the following groups has a proper subgroup that is NOT cyclic?
The value of the integral
is _________________
(Important : you should answer only the numeric value)
Suppose the random variable u has uniform distribution on [0,1] and X=−2logu. The density of X is
Let f be an entire function on C such that f(z)≤100logzfor each z withz≥2. If f(i)=2i, then f(1)
The number of group homomorphisms from Z_{3} to Z_{9} is ______
(Important : you should answer only the numeric value)
Let u(x,t) be the solution to the wave equation
Then, the value of u(1,1) is ______
(Important : you should answer only the numeric value)
Suppose X is a random variable with (0,1). For the hypothesis testing problem
consider the test “Reject H_{0} if X ≤ A or if X ≥ B”, where A < B are given positive integers. The typeI error of this test is
Let G be a group of order 231. The number of elements of order 11 in G is ______
(Important : you should answer only the numeric value)
Let The area of the image of the region under the mapping f is
Which of the following is a field?
Let x_{0}=0. Define xn+1=cosxn for every n≥0. Then
Let C be the contour Z=2 oriented in the anticlockwise direction. The value of the integral
For each λ>0, let Xλ be a random variable with exponential density λe^{−λx} on(0,∞). Then, Var(logX_{λ})
Let {a_{n}} be the sequence of consecutive positive solutions of the equation tanx=x and let {b_{n}} be the sequence of consecutive positive solutions of the equationtan√x=x. Then
Let f be an analytic function on Then, which of the following is NOT a possible value of (e^{f})′′ (0)?
The number of nonisomorphic abelian groups of order 24 is ______
(Important : you should answer only the numeric value)
Q. 36 to Q. 65 carry two marks each.
Q.
Let V be the real vector space of all polynomials in one variable with real coefficients and having degree at most 20. Define the subspaces
Then the dimension of is ________________
(Important : you should answer only the numeric value)
Consider the following linear programming problem:
Then the optimal value is ______
(Important : you should answer only the numeric value)
Suppose X is a realvalued random variable. Which of the following values CANNOTbe attained by E[X] and E[X^{2}], respectively?
Which of the following subsets of R^{2} is NOT compact?
Let M be the real vector space of 2×3 matrices with real entries. Let T:M→M be defined by
The determinant of T is ______
(Important : you should answer only the numeric value)
Let H be a Hilbert space and let {e_{n}:n ≥ 1} be an orthonormal basis of H. Suppose T:H→ H is a bounded linear operator. Which of the following CANNOT be true?
Let f: C{3i}→C be defined by f(z)=z−i/(iz +3) . Which of the following statements about f is FALSE?
The matrix A= can be decomposed uniquely into the product A=LU , where The solution of the system LX = [1 2 2]^{t} is
The image of the region {z ∈C: Re(z)>Im(z)>0} under the mapping
Which of the following groups contains a unique normal subgroup of order four?
(Note: correct answer will be updated soon, Temporary marked A)
Let B be a real symmetric positivedefinite n×n matrix. Consider the inner product on R^{n} defined by =Y^{t}BX . Let A be an n × n real matrix and let T:R^{n}→R^{n} be the linear operator defined by T(X)=AX for all X ∈ R^{n}. If S is the adjoint of T,then S(X)=CX for all X∈R^{n},where C is the matrix
Let X be an arbitrary random variable that takes values in{0,1,…,10}. The minimum and maximum possible values of the variance of X are
Let M be the space of all 4x3 matrices with entries in the finite field of three elements. Then the number of matrices of rank three in M is
Let V be a vector space of dimension m≥2. Let T:V→V be a linear transformation such that T^{n+1}=0 and T^{n}≠0 for some n≥1. Then which of the following is necessarily TRUE?
Let X be a convex region in the plane bounded by straight lines. Let X have 7 vertices. Suppose F(x,y)=ax + by + c has maximum valueM and minimum valueN on X and N < M. Let S = {P: P is a vertex of X and N < f (P) <M}. If S has n elements, then which of the following statements is TRUE?
Which of the following statements are TRUE?
Let u be a real valued harmonic function on C. Let g:R^{2}→R be defined by
Which of the following statements is TRUE?
Let S={Z∈C: Z=1} with the induced topology from C and let f:[0,2]→S be defined as f(t)=e2^{πit} . Then, which of the following is TRUE?
Assume that all the zeros of the polynomial have negative real parts. If u(t) is any solution to the ordinary differential equation
then lim_{t→∞ }u(t) is equal to
Common Data for Questions 58 and 59:
Let C_{00} be the vector space of all complex sequences having finitely many nonzero terms. Equip c_{00} with the inner product Let N be the kernel of f
Q. Which of the following is FALSE?
Which of the following is FALSE?
Common Data for Questions 60 and 61:
Let X_{1},X_{2},…,X_{n} be an i.i.d. random sample from exponential distribution with mean μ. In other words, they have density
Q.
Which of the following is NOT an unbiased estimate of μ?
Let X_{1},X_{2},…,X_{n} be an i.i.d. random sample from exponential distribution with mean μ. In other words, they have density
Q.
Consider the problem of estimating μ. The m.s.e (mean square error) of the estimate
is
Statement for Linked Answer Questions 62 and 63:
Let n_{0}=max { k : k< ∞, there are k distinct points p1,…,p_{k }∈ X such that x{p_{1},…,p_{k}} is connected}
Q.
The value of n_{0} is ______
(Important : you should answer only the numeric value)
Let n_{0}=max { k : k< ∞, there are k distinct points p1,…,p_{k }∈ X such that x{p_{1},…,p_{k}} is connected}
Q.
Let Let m be the number of connected components of Y. The maximum possible value of m is ______
(Important : you should answer only the numeric value)
Statement for Linked Answer Questions 64 and 65:
Let Let W(y_{1},y_{2}) be the Wrönskian of two linearly independent solutions y_{1} and y_{2} of the equation y′′+P(x)y′+Q(x)y=0.
Q.
The product equals
Let Let W(y_{1},y_{2}) be the Wrönskian of two linearly independent solutions y_{1} and y_{2} of the equation y′′+P(x)y′+Q(x)y=0.
Q.
If y_{1}=e^{2x} and y_{2}=e^{2x}, then the value of P(0) is
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