Firstranker's choice
GATE 2016
Q. 1 – Q. 5 carry one mark each.
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Q.1 An apple costs Rs. 10. An onion costs Rs. 8.
Select the most suitable sentence with respect to grammar and usage.
(A) The price of an apple is greater than an onion.
(B) The price of an apple is more than onion.
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(C) The price of an apple is greater than that of an onion.
(D) Apples are more costlier than onions.
Q.2 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.
(A) burning (B) igniting (C) clutching (D) flinging
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Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in- law of M. How is P related to M?
(A) P is the son-in-law of M.
(B) P is the grandchild of M.
(C) P is the daughter-in law of M.
(D) P is the grandfather of M.
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Q.4 The number that least fits this set: (324, 441 and 64)
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
(A) 2.0 (B) 10.0 (C) 12.0 (D) 22.0
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Firstranker's choice
GATE 2016
Q. 6 – Q. 10 carry two marks each.
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Q.6 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)?
V (m/s)
(A) 0 (B) 3 (C) 4 (D) 5
Q.7 The overwhelming number of people infected with rabies in India 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 with rabies.
Which of the following can be logically inferred from the above sentences?
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(A) The number of people in India infected with rabies is high.
(B) The number of people in other parts of the world who are infected with rabies is low
(C) Rabies can be eradicated in India by vaccinating 70% of stray dogs.
(D) Stray dogs are the main source of rabies worldwide.
Q.8 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?
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(A) Manu (B) Sravan (C) Trideep (D) Pavan
Q.9 Find the area bounded by the lines 3x+2y=14, 2x-3y=5 in the first quadrant.
(A) 14.95 (B) 15.25 (C) 15.70 (D) 20.35
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Firstranker's choice
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GATE 2016
Q.10 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?
(A) -0.030 (B) -0.014 (C) 0.014 (D) 0.030
END OF THE QUESTION PAPER
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3/3
Firstranker's choice
GATE 2016
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List of Symbols, Notations and Data
Mathematics - MA
i.i.d. : independent and identically distributed
?(µ, s²): Normal distribution with mean u and variance s², µe (-8,8), s > 0
E(X): Expected value (mean) of the random variable X
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x2
(t) = edx
1
v2p
[x]: the greatest integer less than or equal to x
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Z: Set of integers
Zn: Set of integers modulo n
R: Set of real numbers
C: Set of complex numbers
Rn: n- dimensional Euclidean space
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Usual metric d on Rn is given by d((X1,X2, ..., Xn), (1, 2, ..., Yn)) = (S=1(xi – yi)2)1/2
l2: Normed linear space of all square-summable real sequences
C[0,1]: Set of all real valued continuous functions on the interval [0,1]
B(0,1) : = {(x, y) ? R2: x² + y² = 1}
M*: Conjugate transpose of the matrix M
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MT: Transpose of the matrix M
Id: Identity matrix of appropriate order
R(M): Range space of M
N(M): Null space of M
W¹: Orthogonal complement of the subspace W
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MA
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Firstranker's choice
GATE 2016
Mathematics - MA
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Q. 1 - Q. 25 carry one mark each.
Q.1 Let {X, Y, Z} be a basis of R3. Consider the following statements P and Q:
(P) : {X + Y, Y + Z, X - Z} is a basis of R3.
(Q) : {X + Y + Z,X + 2Y – Z,X – 3Z} is a basis of R3.
Which of the above statements hold TRUE?
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(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
Q.2 Consider the following statements P and Q:
[1 1 1]
(P): If M = 1 2 4, then M is singular .
139
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(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?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
Q.3 Consider the following statements P and Q:
(P): If M is an n × n complex matrix, then R(M) = (N(M*))£.
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(Q): There exists a unitary matrix with an eigenvalue a such that |?| <1.
Which of the above statements hold TRUE?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
MA
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Firstranker's choice
GATE 2016
Mathematics - MA
Q.4 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 = ?T, for some? R\{0}, then which of the following statements is TRUE?
(A) determinant(T) = |?
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(B) There exists a non-trivial subspace V1 of V such that T (X) = 0 for all XE V1
(C) T is invertible
(D) is the only eigenvalue of T
Q.5 Let S = [0,1)U [2,3] and f:S ? R be a strictly increasing function such that f (S) is connected. Which of the following statements is TRUE?
(A) f has exactly one discontinuity
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(B) f has exactly two discontinuities
(C) f has infinitely many discontinuities
(D) f is continuous
Q.6 Let a1 = 1 and an = an-1 + 4, n = 2. Then,
1 1 1
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lim + + +
n?8 a1a2 A2A3 An-1an
is equal to
Q.7 Maximum {x + y : (x, y) ? B(0,1) } is equal to
Q.8 Let a, b, c, d? R such that c² + d² ? 0. Then, the Cauchy problem
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aux + buy = ex+y, x, y ? R,
u(x,y) = 0 on cx + dy = 0
has a unique solution if
(A) ac + bd ? 0 (B) ad - bc ? 0 (C) ac - bd ? 0 (D) ad + bc ? 0
MA
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3/16
Firstranker's choice
GATE 2016
Mathematics - MA
Q.9 Let u(x, t) be the d'Alembert's solution of the initial value problem for the wave equation
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Utt c² Uxx = 0
u(x, 0) = f(x), ut(x, 0) = g(x),
where c is a positive real number and f, g are smooth odd functions. Then, u(0,1) is equal to
Q.10 Let the probability density function of a random variable X be
f(x) = X 0=x<=
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c(2x - 1)² =< x = 1
otherwise.
Then, the value of c is equal to
Q.11 Let V be the set of all solutions of the equation y" + a y'+ b y = 0 satisfying y(0) = y(1), where a, b are positive real numbers. Then, dimension(V ) is equal to
Q.12 Let y" + p(x) y' + q(x)y = 0,x ? ,8), where p(x) and q(x) are continuous functions. If y1(x) = sin (x) 2 cos (x) and y2(x) = 2 sin (x) + cos (x) are two linearly independent solutions of the above equation, then | 4p(0) + 2 g (1) is equal to
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Q.13 Let P(x) be the Legendre polynomial of degree n and I = f1xk P(x)dx, 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.
MA
Which of the above statements hold TRUE?
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(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
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Firstranker's choice
GATE 2016
Mathematics - MA
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Q.14 Consider the following statements P and Q:
(P): x²y" + xy' + (x² -)y y = 0 has two linearly independent Frobenius series solutions near x = 0.
(Q): x²y" + 3 sin(x) y' + y = 0 has two linearly independent Frobenius series solutions near x 0.
Which of the above statements hold TRUE?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
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Q.15 Let the polynomial x4 be approximated by a polynomial of degree = 2, which interpolates x at x = -1,0 and 1. Then, the maximum absolute interpolation error over the interval [-1,1] is equal to
Q.16 Let (zn) be a sequence of distinct points in D(0,1) = {z? C: |z| < 1} with limn?8 Zn = 0. Consider the following statements P and Q:
(P) : There exists a unique analytic function f on D(0,1) such that f (zn) = sin(zn) for all n.
(Q) : There exists an analytic function fon D(0,1) such that f (zn) and f (Zn) = 1 if n is oda
Which of the above statements hold TRUE?
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(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
= 0 if n is even
Q.17 Let (R, t) be a topological space with the cofinite topology. Every infinite subset of R is
(A) Compact but NOT connected
(B) Both compact and connected
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(C) NOT compact but connected
(D) Neither compact nor connected
MA
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Firstranker's choice
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GATE 2016
Mathematics - MA
Q.18 Let co = {(xn): xn ? R, xn ? 0 } and M = {(xn) ? Co : X1 + X2 + … + X10 = 0 }. Then, dimension(C/M) is equal to
Q.19 Consider (R2, ||·||8), where ||(x, y) ||8 = maximum{|x|, |y|}. Let f: R2 ? R be defined by f (x, y) = ** and ƒ the norm preserving linear extension of f to (R3, ||·||8). Then, ƒ(1,1,1) is equal to
Q.20 f: [0,1] ? [0,1] is called a shrinking map if |f(x) - f(y)| < |x - y| for all x,y ? [0,1] and a contraction if there exists an a < 1 such that |f(x) - f(y)| = a|x - y| for all x, y ? [0,1]. Which of the following statements is TRUE for the function f (x) = x-x2/2?
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(A) f is both a shrinking map and a contraction
(B) f is a shrinking map but NOT a contraction
(C) f is NOT a shrinking map but a contraction
(D) f is Neither a shrinking map nor a contraction
Q.21 Let M be the set of all n x n real matrices with the usual norm topology. Consider the following statements P and Q:
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(P) : The set of all symmetric positive definite matrices in M is connected.
(Q): The set of all invertible matrices in M is compact.
Which of the above statements hold TRUE?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
MA
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6/16
Firstranker's choice
GATE 2016
Mathematics - MA
Q.22 Let X1, X2, X3, ..., Xn be a random sample from the following probability density function for 0 < µ < 8, 0 < a < 1,
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1
f(x; µ,a) = G(a) (x - µ)a-1?-(x-µ); ? > µ
0 otherwise.
Here a and u are unknown parameters. Which of the following statements is TRUE?
(A) Maximum likelihood estimator of only µ exists
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(B) Maximum likelihood estimator of only a exists
(C) Maximum likelihood estimators of both u and a exist
(D) Maximum likelihood estimator of Neither unor a exists
Q.23 Suppose X and Y are two random variables such that ax + bY is a normal random variable for all a, b ? R. Consider the following statements P, Q, R and S:
(P): X is a standard normal random variable.
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(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?
(A) both P and Q (B) both Q and R (C) both Q and S (D) both P and S
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Q.24 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.
(Q) : If Q = {±1, ±i, ±j, ± k} is the quaternion group, then 2/{-1,1} {-1,1} is abelian.
Which of the above statements hold TRUE?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
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Q.25 Let F be a field of order 32. Then the number of non-zero solutions (a, b) ? F × F of the equation x² + xy + y² = 0 is equal to
MA
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Firstranker's choice
GATE 2016
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Mathematics - MA
Q. 26 – Q. 55 carry two marks each.
Q.26 Let y = {z ? C : |z| = 2} be oriented in the counter-clockwise direction. Let
1 z7
1= 2p? $ z7 cos (2) dz.
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Y
Then, the value of I is equal to
Q.27 Let u(x, y) = x³ + ax²y + bxy² + 2 y³ be a harmonic function and v(x, y) its harmonic conjugate. If v(0,0) = 1, then |a + b + v(1,1)| is equal to
Q.28 Let y be the triangular path connecting the points (0,0), (2,2) and (0,2) in the counter- clockwise direction in R². Then
I = sin (x3)dx + 6x y dy
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Y
is equal to
Q.29 Let y be the solution of
y'+ x y = 0,
(-1) = 0.
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XER
Then y(1) is equal to
(A)-2e (B)2e2 (C)2-2e (D)22e
Q.30 Let X be a random variable with the following cumulative distribution function:
MA
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0 x < 0
x2
F(x) = 3-41
0=x<=
=< x < 1
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x = 1.
Then P (< X < 1) is equal to
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Firstranker's choice
GATE 2016
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Mathematics - MA
Q.31 Let y be the curve which passes through (0,1) and intersects each curve of the family y = c x² orthogonally. Then y also passes through the point
(A) (v2, 0) (B) (0,v2) (C) (1,1) (D) (-1,1)
Q.32 Let S(x) = ao + 2n=1(an cos (n x) + bn sin (n x)) be the Fourier series of the 2 p periodic function defined by f(x) = x² + 4 sin (x) cos(x), -p = x = p. Then is equal to
San2-Sbn2
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Q.33 Let y(t) be a continuous function on [0,8). If
t t
y(t) = t (1 4 y(x)dx) + 4 x y(x) dx,
0 0
then /2
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fo/2y(t) dt is equal to
Q.34 Let Sn = Sk=11k and In = Sn nx-xdx/x2 Then, S10 + 110 is equal to
(A) ln 10 + 1 (B) ln 10 1 (C) ln 10 (D) ln 10++
Q.35 For any (x,y) ? R² \ B(0,1), let f(x, y) = distance ((x, y), B(0,1)) = infimum {v(x - x1)² + (y - y1)2 : (x1,y1) ? B(0,1)}. Then, || ? f (3,4) || is equal to
Q.36 Let f(x) = (fe-t² dt) and g(x) = fe1 e-x²(1+t²) dt. Then f?(vp) + g'(vp) is 1 + t2 equal to
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MA
9/16
Firstranker's choice
GATE 2016
Mathematics - MA
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Q.37 Let M = bear be a real matrix with eigenvalues 1, 0 and 3. If the eigenvectors corresponding to 1 and 0 are (1,1,1) and (1, -1,0)" respectively, then the value of 3f is equal to
Q.38 [110
Let M = 0 1 1 and eM L001 = Id + M + 1 M² + 1 M³ + …. If eM 2! 3! = [bij], then 1
SS bij
i=1 j=1
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is equal to
{4 0 0 = x = 2
Q.39 Let the integral I = fof(x)dx, where f(x) = Consider the following statements P and Q:
14-x 2= x = 4.
(P): If I2 is the value of the integral obtained by the composite trapezoidal rule with two equal sub-intervals, then I2 is exact.
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(Q): If I2 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?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
Q.40 The difference between the least two eigenvalues of the boundary value problem
y" + ?y = 0, 0 < x < p
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y(0) = 0,
?(p) = 0,
is equal to
Q.41 The number of roots of the equation x2 – cos(x) = 0 in the interval pp 2 2 is equal to
MA
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10/16
Firstranker's choice
GATE 2016
Mathematics - MA
Q.42 For the fixed point iteration xk+1 = g(xk), k = 0, 1, 2, ... ..., consider the following statements P and Q:
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(P) : If g(x) = 1 + then the fixed point iteration converges to 2 for all xo ? [1, 100]. X (Q) : If g(x) = v2 + x then the fixed point iteration converges to 2 for all xo ? [0,100]. Which of the above statements hold TRUE?
(A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
Q.43 Let T: l2 ? l2 be defined by
T ((X1, X2, …, ?, …)) = (X2 X1, X3 - X2, …, Xn+1 - Xn, … ). Then
(A) ||T|| = 1 (B) ||T|| > 2 but bounded (C) 1 < ||T|| = 2 (D) ||T|| is unbounded
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Q.44 Minimize w = x + 2y subject to
2x + y = 3
x+y =2
x = 0, y = 0.
Then, the minimum value of w fw is equal to
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Q.45 Maximize w = 11 x z subject to
10 x+y-z =1
2x - 2y + z =2
Then, the maximum value of w is equal to
x, y, z = 0.
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MA
11/16
Firstranker's choice
GATE 2016
Mathematics - MA
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Q.46 Let X1, X2, X3, be a sequence of i.i.d. random variables with mean 1. If N is a 1 geometric random variable with the probability mass function P(N = k) = ; k = 1,2,3, ... and it is independent of the Xi's, then E(X1 + X2 + … + X4) is equal to
Q.47 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
Q.48 Let X1, X2, X3, ... be a sequence of i.i.d. uniform (0,1) random variables. Then, the value of
lim P(- ln(1 - X1) - … – ln(1 - Xn) = n)
is equal to
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Q.49 Let X be a standard normal random variable. Then, P (X < 0 |[X]] = 1) is equal to
F(1)-1 F(2)-1 (A) (B)
F(2)+1 F(1)+1 F(2)+1 F(1)+1
(C) (D)
Q.50 Let X1, X2, X3, ..., Xn be a random sample from the probability density function
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{?a e-ax + - ?)2 a e-2 ax;
f(x) = x = 0 {0 otherwise,
where a > 0, 05 Fandom sample from the probability density function
0 = 0 = 1 are parameters. Consider the following testing problem:
?: ? = 1 a = 1 versus H1: 0 = 0, a = 2.
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Which of the following statements is TRUE?
(A) Uniformly Most Powerful test does NOT exist
(B) Uniformly Most Powerful test is of the form ?i=1 Xi > c, for some 0 < c < 8
(C) Uniformly Most Powerful test is of the form ?i=1 Xi < c, for some 0 < c < 8
(D) Uniformly Most Powerful test is of the form c1 < S=1Xi < C2, for some 0 < C1 < C2 < 8
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MA
12/16
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GATE 2016
Mathematics - MA
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Q.51 Let X1, X2, X3, ... be a sequence of i.i.d. N(µ, 1) random variables. Then,
vp
lim S ?(?1 – µ])
n?8 2n i=1
is equal to
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Q.52 Let X1, X2, X3, ..., Xn be a random sample from uniform [1, 0], for some 0 > 1. If X(n) = Maximum (X1, X2, X3, ..., X?), then the UMVUE of dis
(A) n+1 X(n) +1 (B) n+1X(n) - 1
n n n n
(C) n+1X(n) + 1 (D) X(n) + n+1
n n n n+1
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Q.53 Let x1 = X2 = X3 = 1, X4 = X5 = X6 = 2 be a random sample from a Poisson random variable with mean 0, where ? {1,2}. Then, the maximum likelihood estimator of ? is equal to
Q.54 The remainder when 98! is divided by 101 is equal to
Q.55 Let G be a group whose presentation is
G = { x, y | = y² = e,
x²y = y x}.
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Then G is isomorphic to
(A) Z5 (B) Z10 (C) Z2 (D) Z30
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GATE 2016
END OF THE QUESTION PAPER
Mathematics - MA
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MA
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Firstranker's choice
GATE 2016
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Mathematics - MA
MA
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Firstranker's choice
GATE 2016
Mathematics - MA
MA
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16/16
This download link is referred from the post: GATE Previous Last 10 Years 2010-2020 Question Papers With Solutions And Answer Keys
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