GATE 2015 Question Paper with Answer Keys for Mathematics (MA) - Grad Plus

# GATE 2015 Question Paper with Answer Keys for Mathematics (MA)

Q. 1 – Q. 25 carry one mark each.

Q.1 Let $latex T\;:\;\mathbb{R}^4\rightarrow\mathbb{R}^4\;$ be a linear map defined by

$latex T\left(x,y,z,w\right)\;=\;\left(x\;+\;z,\;2x\;+y\;+3z,\;2y\;+\;2z,\;w\right).$

Then the rank of T is equal to _________

Ans: 3

Q.2 Let M be a 3 × 3 matrix and suppose that 1, 2 and 3 are the eigenvalues of M. If

$latex M^{-1}=\frac{M^2}\alpha-M\;+\frac{11}\alpha I_3$

for some scalar α ≠ 0, then α is equal to ________

Ans: 6

Q.3 Let M be a 3 × 3 singular matrix and suppose that 2 and 3 are eigenvalues of M. Then the number of linearly independent eigenvectors of M3 +2M +I3 is equal to __________

Ans: 3

Q.4 Let M be a 3 × 3 matrix such that $latex \begin{pmatrix}-2\\1\\0\end{pmatrix}=\begin{pmatrix}6\\-3\\0\end{pmatrix}$ and suppose that M3 $latex \begin{pmatrix}1\\-1/2\\0\end{pmatrix}=\begin{pmatrix}\alpha\\\beta\\\gamma\end{pmatrix}$ for some $latex \alpha,\;\beta,\;\gamma\;\in\mathbb{R}.\;Then\;\left|\alpha\right|$ is equal to _______

Ans: 27

Q.5 Let f\;:\;\lbrack\;0,\;\infty\;)\;\rightarrow\mathbb{R} be defined by

$latex f\left(x\right)\;=\int_0^x\sin^2\left(t^2\right)\;dt.$

Then the function f is

(A) uniformly continuous on [0, 1) but NOT on (0, ∞)

(B) uniformly continuous on (0, ∞) but NOT on [0, 1)

(C) uniformly continuous on both [0, 1) and (0, ∞)

(D) neither uniformly continuous on [0, 1) nor uniformly continuous on (0, ∞)

Ans: (C) uniformly continuous on both [0, 1) and (0, ∞)

Q.6 Consider the power series $latex \textstyle\sum_{n=0}^\infty a_nz^n$ , where $latex a_n=\left\{\begin{array}{l}\frac1{3^n}\;if\;n\;is\;even\\\frac1{5^n}\;if\;n\;is\;odd\end{array}\right.$

The radius of convergence of the series is equal to __________

Ans: 3

Q.7 Let $latex C\;=\;\left\{\;z\;\in\mathbb{C}\;:\left|\;z-i\;\right|=2\right\}.\;$ Then $latex \frac1{2\pi}\oint c\;\frac{z^2\;-4}{z^2\;+\;4}dz$ is equal to ____________

Ans: -2

Q.8 Let $latex X\;\sim\;B\left(5\;,\frac12\right)\;and\;Y\sim U\left(\;0,1\right).\;Then\;\frac{P\left(X+Y\;\leq2\right)}{P\left(X\;+\;Y\geq5\right)}$  is equal to ___________

Ans: 6

Q.9 Let the random variable X have the distribution function

……

Then P\left(2\leq x<4\right) is equal to ______

Ans: 4

Q.10 Let X be a random variable having the distribution function

Then E (X)  is equal to _________

Ans: 2.25

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) $latex \frac{125}{6^5}$

(B) $latex \frac{150}{6^5}$

(C) $latex \frac{175}{6^5}$

(D) $latex \frac{200}{6^5}$

Ans: (C) $latex \frac{175}{6^5}$

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 $latex U\;\left(\;\theta-1,\;\theta\;+4\right)$ distribution, where $latex \theta\;\in\;\left(0,\;\infty\right)$. Then a maximum likelihood estimate of  is equal to θ is equal to

(A) 1.8

(B) 2.3

(C) 3.1

(D) 3.6

Ans: (A) 1.8

Q.13 Let $latex \Omega\;=\;\left\{\left(x,\;y\right)\;\in\mathbb{R}^2\;\vert x^2\;+\;y^2\;<1\;\right\}$ be the open unit disc in $latex \mathbb{R}^2$ with boundary ∂ Ω. If $latex u\left(x,y\right)$ is
the solution of the Dirichlet problem

$latex u_{xx}\;+\;u_{yy\;}=0$ in Ω

$latex u\;\left(x,y\right)\;=1-2y^2$ on ∂ Ω,

then $latex u\;\left(\frac12,0\right)$ is equal to

(A) -1

(B) $latex \frac{-1}4$

(C) $latex \frac{1}4$

(D) 1

Ans: (C) $latex \frac{1}4$

Q.14 Let $latex c\;\in\;{\mathbb{Z}}_3\;$ be such that $latex \frac{{\mathbb{Z}}_3\left[X\right]}{\left\langle x^3\;+\;c\;X\;+1\right\rangle}$ is a field. Then c is equal to __________

Ans: 2

Q.15 Let $latex V\;=\;C^1\left[0,1\right],\;X\;=\left(C\left[0,1\right],\;\parallel\;\;\;\parallel_2\right).\;$. Then V is

(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

Ans: (C) dense in both X and Y

Q.16 Let $latex T\;:\;\left(C\;\left[\;0,1\right],\parallel\;\;\;\;\parallel_\infty\right)\;\rightarrow\;\mathbb{R}\;$ be defined by $latex T\left(f\right)\;=\int_0^12xf\left(x\right)\;dx\;for\;all\;f\in C\left[0,1\right].\;Then\;\left\|T\right\|$

is equal to __________

Ans: 1

Q.17 Let τ1 be the usual topology on $latex \mathbb{R}$ Let τ2 be the topology on $latex \mathbb{R}$ generated by $latex \mathcal B\;=\left\{\;\lbrack\;a,\;b)\;\subset\mathbb{R}\;:\;-\;\infty<a<b<\infty\right\}.$ Then the set $latex \left\{x\;\in\mathbb{R}\;:4\;\sin^2x\leq1\right\}\;\cup\left\{\frac{\mathrm\pi}2\right\}$ is

(A) closed in ($latex \mathbb{R}$, τ1 ) but NOT in ($latex \mathbb{R}$, τ2)

(B) closed in ($latex \mathbb{R}$, τ2) but NOT in closed in ($latex \mathbb{R}$, τ1 )

(C) closed in both ($latex \mathbb{R}$, τ1 ) and ($latex \mathbb{R}$, τ2)

(D) neither closed in ($latex \mathbb{R}$, τ1 ) nor closed in ($latex \mathbb{R}$, τ2)

Ans: (C) closed in both ($latex \mathbb{R}$, τ1 ) and ($latex \mathbb{R}$, τ2)

Q.18 Let X be a connected topological space such that there exists a non-constant continuous function $latex f\;:\;X\;\rightarrow\mathbb{R},\;$ ,where $latex \mathbb{R}$ is equipped with the usual topology. Let $latex f\;\left(X\right)\;=\left\{\;f\left(x\right)\;:\;x\;\in\;X\right\}$ Then

(A) X is countable but f (X) is uncountable

(B) f (X) is countable but X is uncountable

(C) both f (X) and X are countable

(D) both f (X) and X are uncountable

Ans: (D) both f (X) and X are uncountable

Q.19 Let d1 and d2denote the usual metric and the discrete metric on $latex \mathbb{R}$, respectively. Let $latex f\;:\;\left(\mathbb{R},\;d_1\right)\;\rightarrow\left(\mathbb{R},\;d_2\right)$ be defined by $latex f\left(x\right)\;=x,\;x\in\mathbb{R}\;$. Then

(A) f is continuous but f-1 is NOT continuous

(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

Ans: (B) f-1 is continuous but f is NOT continuous

Q.20 If the trapezoidal rule with single interval $latex \left[0,1\right]$ is exact for approximating the integral $latex \int_0^1\left(x^3-c\;x^2\right)\;dx$ then the value of c is equal to ________

Ans: 1.5

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 _________

Ans: 1

Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real constant coefficients having xsin (x) as a solution is equal to __________

Ans: 6

Q.23 The Lagrangian of a system in terms of polar coordinates (r,θ ) is given by

$latex L\;=\frac12\;m\;r^2\;+\frac12m\;\left(r^2\;+r^2\theta^2\right)\;-m\;g\;r\;\left(1-\cos\left(\theta\right)\right),$

(A) $latex 2\;\ddot r\;=r\;\dot{\theta^2\;}-\;g\;\left(1-\cos\left(\theta\right)\right),\;\frac d{dt}\;\left(r^2\;\dot\theta\right)\;=-g\;r\;\sin\;\left(\theta\right)$

(B) $latex 2\;\ddot r\;=r\;\dot{\theta^2\;}+\;g\;\left(1-\cos\left(\theta\right)\right),\;\frac d{dt}\;\left(r^2\;\dot\theta\right)\;=-g\;r\;\sin\;\left(\theta\right)$

(C) $latex 2\;\ddot r\;=r\;\dot{\theta^2\;}-\;g\;\left(1-\cos\left(\theta\right)\right),\;\frac d{dt}\;\left(r^2\;\dot\theta\right)\;=g\;r\;\sin\;\left(\theta\right)$

(D) $latex 2\;\ddot r\;=r\;\dot{\theta^2\;}+\;g\;\left(1-\cos\left(\theta\right)\right),\;\frac d{dt}\;\left(r^2\;\dot\theta\right)\;=g\;r\;\sin\;\left(\theta\right)$

Ans: (A) $latex 2\;\ddot r\;=r\;\dot{\theta^2\;}-\;g\;\left(1-\cos\left(\theta\right)\right),\;\frac d{dt}\;\left(r^2\;\dot\theta\right)\;=-g\;r\;\sin\;\left(\theta\right)$

Q.24 If y (x)  satisfies the initial value problem

(x2 +y) dx = x dy,     y(1)

then y (2) is equal to __________

Ans: 6

Q.25 It is known that Bessel functions Jn (x), for  n ≥0, satisfy the identity

$latex e^{\frac x2\left(t-\frac1t\right)}\;=J_0\;\left(x\right)\;+\sum_{n\;=1}^\infty\;J_n\;\left(x\right)\;\left(t^{\;n}\;+\frac{\left(-1\right)^n}{t^n}\right)$

for all t > 0 and $latex x\;\in\;\mathbb{R}$ The value of $latex J_0\left(\frac{\mathrm\pi}3\right)\;+\;2\;{\textstyle\sum_{n=1}^\infty}\;J_{2n}\left(\frac{\mathrm\pi}3\right)$ is equal to _________

Ans: 1

Q. 26 – Q. 55 carry two marks each.

Q.26 Let X and Y be two random variables having the joint probability density function

$latex f\;\left(x,y\right)\;=\left\{\begin{array}{l}2\\0\end{array}\right.\;\;\;if\;0<x<y<1\;otherwise.$

Then the conditional probability $latex P\left(X\leq\frac23\;\vert\;Y\;=\frac34\;\right)\;$ is equal to

(A) $latex \frac59$

(B) $latex \frac23$

(C) $latex \frac79$

(D) $latex \frac89$

Ans: (D) $latex \frac89$

Q.27 Let Ω = (0,1] be the sample space and let P (.) be a probability function defined by

……

Then $latex P\left(\left\{\frac12\right\}\right)$ is equal to __________

Ans: 0.25

Q.28 Let X1, X2 and X3 be independent and identically distributed random variables with E (X1)=0 0 and $latex E\;\left(X_1^2\right)\;=\frac{15}4.\;If\;\psi\;:\left(0,\infty\right)\rightarrow\left(0,\infty\right)\;$ is defined through the conditional expectation $latex \psi\;\left(t\right)\;=E\left(X_1^2\;\vert\;X_1^2\;+X_2^2\;+X_3^2\;=t\;\right),\;t>0$ then $latex E\left(\psi\left(\left(X_1\;+X_2\right)^2\right)\right)$ is equal to __________

Ans: 2.5

Q.29 Let X ∼ Poisson (λ), where λ >0 is unknown. If δ (X) is the unbiased estimator of

g(λ) = e (3λ2 +2λ +1), then $latex {\textstyle\sum_{k=0}^\infty}\;\partial\left(k\right)$ is equal to ___________

Ans: 9

Q.30 Let X1, … , Xn be a random sample from N (μ,1)distribution, where $latex \mu\;\in\;\left\{0,\frac12\right\}$ For testing the null hypothesis H0 : μ = 0 against the alternative hypothesis H1 : μ =1/2, consider the critical region

R=\left{\left(x_1,x_2,…,x_n\right):\sum_{i=1}^nx_i>c\right},

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 ___________

Ans: 25

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 $latex E\;\left(\frac xy\right)\;=3$ and m + n =14, then $latex E\;\left(\frac xy\right)\;$ is equal to

(A) $latex \frac27$

(B) $latex \frac37$

(C) $latex \frac47$

(D) $latex \frac57$

Ans: (D) $latex \frac57$

Q.32 Let X1, X2,… be a sequence of independent and identically distributed random variables with $latex P\left(X_1=1\right)\;=\frac14\;$ and $latex P\left(X_1=2\right)\;=\frac34\;$. If $latex \overline{X_n}\;=\frac1n{\textstyle\sum_{i=1}^n}\;X_i,$ n=1,2,…, then $latex \lim_{n\rightarrow\infty}\;P\left({\overline X}_n\;\leq1.8\right)\;\;\;\;$ is equal to __________

Ans: 1

Q.33 Let $latex u\;\left(x,y\right)\;=2\;f\;\left(y\right)\;\cos\;\left(x-2y\right),\;\left(x,y\right)\in\;\mathbb{R}^2$ be a solution of the initial value problem $latex 2u_x\;+\;u_y\;=u$ $latex u\;\left(\;x,\;0\right)\;=\cos\left(x\right)$

Then f (1) is equal to

(A) $latex \frac12$

(B) \frac e2

(C) e

(D) $latex \frac3e2$

Ans: $latex \frac e2$

Q.34 Let $latex u\;\left(x,t\;\right),\;x\in\mathbb{R},\;t\geq0$ 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 ________

Ans: 4

Q.35 Let $latex W\;=Span\;\left\{\frac1{\sqrt2}\left(0,0,1,1\right),\;\frac1{\sqrt2}\left(1,\;-1,0,0\right)\right\}$ be a subspace of the Euclidean space $latex \mathbb{R}^4$. Then the square of the distance from the point (1,1,1,1) to the subspace W is equal to ________

Ans: 2

Q.36 Let $latex T\;:\;\mathbb{R}^4\;\rightarrow\mathbb{R}^4$ be a linear map such that the null space of T is $latex \left\{\left(x,y,z,w\right)\;\in\mathbb{R}^4\;:x\;+y+\:z\;+\;w\;=0\right\}$ and the rank of (T-4 I4 ) is 3. If the minimal polynomial of T is x (x-4)a then α is equal to _______

Ans: 1

Q.37 Let M be an invertible Hermitian matrix and let be such that $latex x,y\;\in\mathbb{R}$ be such that x2 < 4y. Then

(A) both M2 + x M +y I and M2x M + y I are singular

(B) M2+x M+ y I is singular but M2x M + y I is non-singular

(C) M2 +x M+ y I is non-singular but M2x M +y I is singular

(D) both M2 +x M + y I and M2x M +y I are non-singular

Ans: (D) both M2 +x M + y I and M2x M +y I are non-singular

Q.38 Let $latex G=\left\{e,x,x^2,x^3,y,xy,x^2y,x^3y\right\}$ with 0(x)=4,0 (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

(D) 8

Ans: (B) 2

Q.39 The number of ring homomorphisms from $latex {\mathbb{Z}}_2\;\times{\mathbb{Z}}_2\;to\;{\mathbb{Z}}_4$ is equal to __________

Ans: 1

Q.40 Let $latex p\left(x\right)\;=9x^5+10x^3\;+5x\;+15$ $latex q\left(x\right)\;=x^3+x^2\;+x\;+2$ be two polynomials in $latex \mathbb{Q}\left[x\right].$ Then, over $latex \mathbb{Q}$,

(A) p(x) q(x)and  are both irreducible

(B) p(x) is reducible but q(x) is irreducible

(C) p(x) is irreducible but q(x) is reducible

(D) p(x) and q(x)are both reducible

Ans: (C) p(x) is irreducible but q(x) is reducible

Q.41 Consider the linear programming problem

Maximize 3 x+9 y,

subject to 2 yx≤2

3 yx ≥0

2 x+3 y≤10

x, y ≥0

Then the maximum value of the objective function is equal to ______

Ans: 24

Q.42 Let $latex S=\left\{\left(x,\sin\;\frac1x\right)\;:0<x\leq1\right\}$ and $latex T=S\cup\left\{\left(0,0\right)\right\}$ Under the usual metric on $latex \mathbb{R}^2$

(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

Ans: (D) neither S nor T is closed

Q.43 Let $latex H\;=\left\{\left(x_n\right)\;\in\;\mathcal l\;:{\textstyle\sum_{n=1}^\infty}\frac{x_n}n=1\right\}$ Then H

(A) is bounded

(B) is closed

(C) is a subspace

(D) has an interior point

Ans: (B) is closed

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)=x2If V⊥ =Span { f } = 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) $latex \frac34x$

(B) $latex \frac14x$

(C) $latex \frac34x^2$

(D) $latex \frac14x^2$

Ans: (A) $latex \frac34x$

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 _________

Ans: -2

Q.46 If, for some $latex \alpha,\;\beta\in\mathbb{R}$ the integration formula
$latex \int_0^2\;p\left(x\right)\;dx=p\;(a)+\;p\;(\beta)$
holds for all polynomials p (x) of degree at most 3, then the value of 3 ( α-β)2 is equal to _____

Ans: 4

Q.47 Let y ( t ) be a continuous function on [0,∞) whose Laplace transform exists. If y (t) satisfies
$latex \int_0^t\;\left(1-\cos\left(t-\;\tau\right)\right)\;y\;(\tau)\;d\tau\;=t^4$
then y (1) is equal to _______

Ans: 28

Q.48 Consider the initial value problem
x 2 y” – 6y=0 y(1)= α, y'(1)=6
If y (x) →0 as x→ 0+,  then α is equal to __________

Ans: 2

Q.49 Define f1, f2: [0,1]→ R by
$latex f_1\left(x\right)\;=\sum_{n=1}^\infty\;\frac{x\;\sin\left(n^2\;x\right)}{n^2}\;\;$ and
$latex f_{2\;}\left(\;x\;\right)\;=\sum_{n=1}^\infty\;x^2\left(1-x^2\right)^{n-1}$
Then
(A) f1is continuous but f2 is NOT continuous

(B) f2 is continuous but f1 is NOT continuous

(C) both f1 and f2 are continuous

(D) neither f1 nor f2 is continuous

Ans: (A) f1is continuous but f2 is NOT continuous

Q.50 Consider the unit sphere $latex S\;=\left\{\left(x,y,z\right)\;\in\mathbb{R}^3\;:\;x^2\;+y^2\;+z^2=1\right\}$ and the unit normal vector $latex \widehat{n\;}=\left(\;x,y,z\;\right)$ at each point (x, y, z)  on S. The value of the surface integral
\int\int_s\;\left\{\left(\frac{2\;x}\pi\;+\sin\left(y^2\right)\right)\;x\;+\left(e^z\;-\;\frac y\pi\;\right)\;y\;+\;\left(\frac{2\;z}{\mathrm\pi}\;+\sin^2y\right)z\right\}\;d\delta
is equal to _______

Ans: 4

Q.51 Let D\;=\left\{\;\left(x,y\right)\;\in\mathbb{R}^2\;:\;1\leq x\leq1000,1\leq y\leq1000\right\}\;. Define
$latex f\left(x,y\right)=\frac{x\;y}2\;+\frac{500}x+\frac{500}y$
Then the minimum value of f  on D is equal to_______

Ans: 150

Q.52 Let D= \left\{z\in\mathbb{C}\;:\left|z\right|<1\right\} Then there exists a non-constant analytic function f on D such that for all n =2,3,4….

(A) f\;\left(\frac{\sqrt{-1}}n\right)=0

(B) f\;\left(\frac1n\right)=0

(C) f\;\left(1-\frac1n\right)=0

(D) f\;\left(\frac12-\frac1n\right)=0

Ans: (C) f\;\left(1-\frac1n\right)=0

Q.53 Let {\textstyle\sum_{n=-\infty}^\infty}\alpha_nz^n be the Laurent series expansion of f\;(z)\;=\frac1{2\;z^2-13\;z+15} in the annulus \frac32<\left|z\right|<5 \frac{\alpha_1}{\alpha_2} is equal to _________

Ans: 5

Q.54 The value of \frac i{4-\mathrm\pi}\int_{\left|z\right|=4}\;\frac{dz}{z\;\cos\left(z\right)} is equal to __________

Ans: 2

Q.55 Suppose that among all continuously differentiable functions y\;\left(x\right),\;x\;\in\;\mathbb{R} with y (0) =0 and y (1)=1/2, the function  y0(x) minimizes the functional
\int_0^1\left(e^{-\left(y'-x\right)}\;+\left(1\;+y\right)\;y'\right)\;dx
Then y0 (½) is equal to
(A) 0

(B) 1/8

(C) ¼

(D) ½

Ans: (B) 1/8

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