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 M $latex \begin{pmatrix}-2\\1\\0\end{pmatrix}=\begin{pmatrix}6\\-3\\0\end{pmatrix}$ 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 x2 sin (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-λ (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 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
(D) both M2 +x M + y I and M2 –x M +y I are non-singular
Ans: (D) both M2 +x M + y I and M2 –x 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 y– x≤2
3 y– x ≥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