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Firstranker's choice
GATE 2015
List of Symbols, Notations and Data
B(n, p): Binomial distribution with n trials and success probability p; n ? {1,2, ... } and p? (0,1)
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U(a, b): Uniform distribution on the interval (a, b), -8 < a <b<8
?(µ, s²): Normal distribution with mean u and variance s², µ? (-8,8), s > 0
P(A): Probability of the event A
Poisson(?): Poisson distribution with mean ?, ? > 0
E(X): Expected value (mean) of the random variable X
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If Z ~ N(0,1), then P(Z = 1.96) = 0.975 and P(Z = 0.54) = 0.7054
Z: Set of integers
Q: Set of rational numbers
IR: Set of real numbers
C: Set of complex numbers
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Zn: The cyclic group of order n
F[x]: Polynomial ring over the field IF
C[0, 1]: Set of all real valued continuous functions on the interval [0, 1]
C¹ [0, 1]: Set of all real valued continuously differentiable functions on the interval [0, 1]
l2: Normed space of all square-summable real sequences
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L2 [0, 1]: Space of all square-Lebesgue integrable real valued functions on the interval [0, 1]
(C [0,1], || ||2): The space C[0, 1] with ||f || = (?0¹ |f(x)|² dx)1/2
(C[0, 1], || ||8): The space C[0, 1] with ||f||8 = sup{|f(x)|: x ? [0, 1]}
V?: The orthogonal complement of V in an inner product space
Rn: n-dimensional Euclidean space
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Usual metric on Rn is given by d((x1,x2, ..., x?), (y1, y2, ..., y?)) = (?=1(x? – y?)²)1/2
I?: The n×n identity matrix ( I : the identity matrix when order is NOT specified)
o(g): The order of the element g of a group
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Firstranker's choice
GATE 2015
Q. 1 – Q. 25 carry one mark each.
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Q.1
Let T : R4 ? R4 be a linear map defined by
T(x, y, z, w) = (x + z, 2x + y + 3z, 2y + 2z, w).
Then the rank of T is equal to
Q.2
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Let M be a 3 × 3 matrix and suppose that 1, 2 and 3 are the eigenvalues of M. If
M² + M-1 = a M + 11/6 I3
for some scalar a ? 0, then a is equal to
Q.3
Let M be a 3 x 3 singular matrix and suppose that 2 and 3 are eigenvalues of M. Then the number of linearly independent eigenvectors of M³ + 2 M + I3 is equal to
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Q.4
Let M be a 3 x 3 matrix such that M (7, -2, 1)T = (-3, 1, 0)T and suppose that M³ (-1/2, 0, 0)T = (18, a, ß, ?)T for some a, ß,? ? R. Then | a | is equal to
Q.5
Let f: [0,8) ? R be defined by
f(x) = ?0? sin²(t²) dt.
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Then the function f is
(A) uniformly continuous on [0, 1) but NOT on (0, 8)
(B) uniformly continuous on (0, 8) but NOT on [0, 1)
(C) uniformly continuous on both [0, 1) and (0,8)
(D) neither uniformly continuous on [0, 1) nor uniformly continuous on (0,8)
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Q.6
Consider the power series ?=0 a?zn, where a? = { 1 if n is even, 3n if n is odd.
The radius of convergence of the series is equal to
Q.7
Let C = {z ? C : |z - i|= 2}. Then
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?C 1/(z²-4) dz is equal to
Q.8
Let X ~ B(5,½) and Y ~ U(0,1). Then
P(X+Y =2) / P(X+Y=5)
is equal to
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Q.9 Let the random variable X have the distribution function
F(x) = { 0 if x < 0, x/4 if 0 = x < 1, 3/8 + (x-1)/8 if 1 < x < 2, 5/8 + (x-2)/8 if 2 = x <3, 1 if x = 3.
Then P(2 < X < 4) is equal to
Q.10 Let X be a random variable having the distribution function
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F(x) = { 0 if x < 0, 1/4 if 0 = x < 1, 1/3 if 1 = x < 2, 11/24 if 2 = x < 3, 1 if x=3.
Then E(X) is equal to
Q.11 In an experiment, a fair die is rolled until two sixes are obtained in succession. The probability that the experiment will end in the fifth trial is equal to
(A) 125/65
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(B) 150/65
(C) 200/65
Q.12 Let x1 = 2.2, x2 = 4.3, x3 = 3.1, x4 = 4.5, x5 = 1.1 and x6 = 5.7 be the observed values of a random sample of size 6 from a U(? - 1, ? + 4) distribution, where ? ? (0,8) is unknown. Then a maximum likelihood estimate of @ is equal to
(A) 1.8
(B) 2.3
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(C) 3.1
(D) 3.6
Q.13 Let O = {(x,y) ? R²|x² + y² < 1} be the open unit disc in R² with boundary ?O. If u(x, y) is the solution of the Dirichlet problem
Uxx + Uyy = 0 in O
u(x, y) = 1 – 2 y² on TO,
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then u (1/2,0) is equal to
(A) -1/4
(B) 1/4
(C) 3/4
(D) 1
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Q.14 Let c? Z3 be such that
Z3 [X] / (X³+cX+1)
is a field. Then c is equal to
Q.15 Let V = C¹[0, 1], X = ( C[0, 1], || || 8) and Y = ( C[0, 1], || ||2). Then V is
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(A) dense in X but NOT in Y
(B) dense in Y but NOT in X
(C) dense in both X and Y
(D) neither dense in X nor dense in Y
Q.16
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Let T : (C[0, 1], || ||8) ? R be defined by T(f) = ?0¹ 2xf(x) dx for all f ? C[0, 1]. Then ||T|| is equal to
Q.17 Let t1be the usual topology on R. Let t2 be the topology on R generated by
B = {[a, b) ? R : -8 < a < b < 8}. Then the set {x ? R : 4 sin²x = 1} is
(A) closed in (R, t1) but NOT in (R, t2)
(B) closed in (R, t2) but NOT in (R, t1)
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(C) closed in both (R, t1) and (R, t2)
(D) neither closed in (R, t1) nor closed in (R, t2)
Q.18 Let X be a connected topological space such that there exists a non-constant continuous function f : X ? IR, where R is equipped with the usual topology. Let f(X) = { f(x): x ? X}. Then
(A) X is countable but f(X) is uncountable
(B) f(X) is countable but X is uncountable
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(C) both f(X) and X are countable
(D) both f(X) and X are uncountable
Q.19 Let d1 and d2 denote the usual metric and the discrete metric on R, respectively.
Let f: (R, d1) ? (R, d2) be defined by f(x) = x, x ? R. Then
(A) f is continuous but f?¹ is NOT continuous
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(B) f?¹ is continuous but f is NOT continuous
(C) both f and f?¹ are continuous
(D) neither f nor f?¹ is continuous
Q.20 If the trapezoidal rule with single interval [0, 1] is exact for approximating the integral ?0¹(x³ - c x²)dx, then the value of c is equal to
Q.21 Suppose that the Newton-Raphson method is applied to the equation 2x² + 1 - ex² = 0 with an initial approximation x0 sufficiently close to zero. Then, for the root x = 0, the order of convergence of the method is equal to
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Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real constant coefficients having x² sin(x) as a solution is equal to
Q.23 The Lagrangian of a system in terms of polar coordinates (r, ?) is given by
L = ½ m(r?² + r²?²) - mgr (1-cos(?)),
where m is the mass, g is the acceleration due to gravity and s? denotes the derivative of s with respect to time. Then the equations of motion are
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(A) 2r¨ = r?² - g (1 – cos(?)), d/dt (r²?) = -gr sin(?)
(B) 2r¨ = r?² + g (1 - cos(?)), d/dt (r²?) = -gr sin(?)
(C) 2r¨ = r?² - g (1 – cos(?)), d/dt (r²?) = gr sin(?)
(D) 2 r¨ = r?² + g (1 – cos(?)), d/dt (r²?) = gr sin(?)
Q.24 If y(x) satisfies the initial value problem
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(x² + y)dx = x dy, y(1) = 2,
then y(2) is equal to
Q.25 It is known that Bessel functions J?(x), for n = 0, satisfy the identity
ex/2(t-1/t) = J0(x) + ?=1 J?(x)(tn + (-1)n/tn)
for all t > 0 and x ? R. The value of J0 (7) is equal to
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Q. 26 – Q. 55 carry two marks each.
Q.26 Let X and Y be two random variables having the joint probability density function
f(x,y) = { 2 if 0 < x < y < 1, 0 otherwise.
Then the conditional probability P (X = ½ | Y = ?) is equal to
(A) 4/9
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(B) 5/9
(C) 7/9
(D) 8/9
Q.27 Let O = (0,1] be the sample space and let P(·) be a probability function defined by
P((0,x]) = { x/2 if 0 < x = ½, 2x - x² if ½ = x = 1.
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Then P((?,?]) is equal to
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Q.28 Let X1, X2 and X3 be independent and identically distributed random variables with E(X1) = 0 and E(X14) = 15/4. If ? : (0,8) ? (0,8) is defined through the conditional expectation
?(t) = E(X1² | X1² + X2² + X3² = t), t > 0,
then E(?((X1 + X2)²)) is equal to
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Q.29 Let X ~ Poisson(?), where ? > 0 is unknown. If d(X) is the unbiased estimator of g(?) = e?(3?² + 2? + 1), then d(k) is equal to
Q.30 Let X1, ..., X? be a random sample from N(µ, 1) distribution, where µ? {0,3}. For testing the null hypothesis ?0: µ = 0 against the alternative hypothesis H1: µ = 3, consider the critical region
R = {(X1, X2, ..., X?): ?=1n X? > C}
where c is some real constant. If the critical region R has size 0.025 and power 0.7054, then the value of the sample size n is equal to
Q.31 Let X and Y be independently distributed central chi-squared random variables with degrees of freedom m (= 3) and n (= 3), respectively. If E(X/Y) = 3/5 and m + n = 14, then E(Y/X) is equal to
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(A) 2/7
(B) 3/7
(C) 4/7
(D) 5/7
Q.32 Let X1, X2, ... be a sequence of independent and identically distributed random variables with P(X1 = 1) = 1/4 and P(X1 = 2) = 3/4. If X¯? = 1/n ?=1n X?, for n = 1, 2, ..., then
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lim?8 P(X¯? = 1.8) is equal to
Q.33 Let u(x, y) = 2f (y) cos(x – 2y), (x, y) ? R², be a solution of the initial value problem
2ux + uy = u
u(x, 0) = cos(x).
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Then f(1) is equal to
(A) e/2
(B) e/3
(C) 2e/3
(D) 3e/2
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Q.34 Let u(x,t), x ? R, t = 0, be the solution of the initial value problem
utt = uxx
u(x, 0) = x
ut(x, 0) = 1.
Then u(2,2) is equal to
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Q.35 Let W = Span {(0,0,1,1), (1,-1,0,0)} be a subspace of the Euclidean space R4. Then the square of the distance from the point (1,1,1,1) to the subspace W is equal to
Q.36 Let T: R4 ? R4 be a linear map such that the null space of T is
{(x, y, z, w) ? R4: x + y + z + w = 0}
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and the rank of (T – 4 I4) is 3. If the minimal polynomial of T is xa(x - 4), then a is equal to
Q.37 Let M be an invertible Hermitian matrix and let x, y ? R be such that x² < 4y. Then
(A) both M² + x M + y I and M² - x M + y I are singular
(B) M² + x M + y I is singular but M² - x M + y I is non-singular
(C) M² + x M + y I is non-singular but M² - x M + y I is singular
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(D) both M² + x M + y I and M² - x M + y I are non-singular
Q.38 Let G = { e,x,x², x³, y, xy, x²y, x³y} with o(x) = 4, o(y) = 2 and xy = yx³. Then the number of elements in the center of the group G is equal to
(A) 1
(B) 2
(C) 4
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(D) 8
Q.39 The number of ring homomorphisms from Z2 × Z2 to Z4 is equal to
Q.40 Let p(x) = 9 x5 + 10 x³ + 5 x + 15 and q(x) = x² - x - 2 be two polynomials in Q [x]. Then, over Q,
(A) p(x) and q(x) are both irreducible
(B) p(x) is reducible but q(x) is irreducible
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(C) p(x) is irreducible but q(x) is reducible
(D) p(x) and q(x) are both reducible
Q.41 Consider the linear programming problem
Maximize 3 x + 9 y,
subject to
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2y-x=2
3y-x=0
2 x + 3 y = 10
x, y = 0.
Then the maximum value of the objective function is equal to
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Q.42 Let S = {(x, sin(1/x)) : 0 < x = 1} and T = S ? {(0,0)}. Under the usual metric on R²,
(A) S is closed but T is NOT closed
(B) T is closed but S is NOT closed
(C) both S and T are closed
(D) neither S nor T is closed
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Q.43 Let H = {(x?) ? l2: ?=18 x?/n = 1}. Then H
(A) is bounded
(B) is closed
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(C) is a subspace
(D) has an interior point
Q.44 Let V be a closed subspace of L²[0,1] and let f, g? L²[0, 1] be given by f(x) = x and g(x) = x². If V? = Span {f} and Pg is the orthogonal projection of g on V, then
(g – Pg)(x), x? [0, 1], is
(A) x² - 3/4 x
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(B) x² + 3/4 x
(C) x² - 1/4 x
(D) x² + 1/4 x
Q.45 Let p(x) be the polynomial of degree at most 3 that passes through the points (-2, 12), (-1, 1), (0,2) and (2, -8). Then the coefficient of x³ in p(x) is equal to
Q.46 If, for some a, ß? R, the integration formula
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?0² p(x)dx = p(a) + p(ß)
holds for all polynomials p(x) of degree at most 3, then the value of 3(a – b)² is equal to
Q.47 Let y(t) be a continuous function on [0,8) whose Laplace transform exists. If y(t) satisfies
?0? (1 - cos(t - t)) y (t) dt = t4,
then y(1) is equal to
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Q.48 Consider the initial value problem
x²y" - 6 y = 0,
y(1) = a, y'(1) = 6.
If y(x) ? 0 as x ? 0+, then a is equal to
Q.49 Define f1, f2: [0,1] ? R by
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f1(x) = ?=18 x sin(n²x)/n²
and f2(x) = ?=18 x²(1 - x²)n?¹ .
Then
(A) f1 is continuous but f2 is NOT continuous
(B) f2 is continuous but f1 is NOT continuous
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(C) both f1 and f2 are continuous
(D) neither f1 nor f2 is continuous
Q.50 Consider the unit sphere S = {(x, y, z) ? R³: x² + y² + z² = 1} and the unit normal vector n = (x, y, z) at each point (x, y, z) on S. The value of the surface integral
?S {(ey + sin(x²)) x + (ez - 2y) y + (2 x + sin² y) z} ds
is equal to
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Q.51 Let D = {(x, y) ? R²: 1 = x = 1000, 1 = y = 1000}. Define
f(x, y) = x/y + y/x + x/500 + 500/x + y/500 + 500/y.
Then the minimum value of f on D is equal to
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Q.52 Let D = { z ? C : |z| < 1}. Then there exists a non-constant analytic function f on D such that
(A) f(1/n) = 0
(B) f(1/2n) = 0
(C) f(1/n²) = 0
(D) f(-1/n) = 0
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for all n = 2, 3, 4, ...
Q.53 Let ?=-8 a?zn be the Laurent series expansion of f(z) = 1/(2 z²-13 z+15) in the annulus 2 < |z| < 5. Then a2 is equal to
Q.54 The value of
?|z|=4 i dz/(4-p z cos(z))
is equal to
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Q.55 Suppose that among all continuously differentiable functions y(x), x? R, with y(0) = 0 and y(1) = ½, the function y0(x) minimizes the functional
?0¹ (e?(y'-x) + (1 + y)y')dx.
Then y0 is equal to
(A) 0
(B) x/8
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(C) x/4
(D) x/2
END OF THE QUESTION PAPER
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