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GATE (Graduate Aptitude Test in Engineering) Mathematics is an online examination conducted to get admission in M-Tech (Masters in Technology), ME (Master of Engineering ) and PhD at IITs (Indian Institutes of Technology), NITs (National Institutes of Technology), IIITs (Indian Institutes of Information Technology) and other universities in India. GATE exam is jointly conducted by IIT Bombay, IIT Delhi, IIT Guwahati, IIT Kanpur, IIT Kharagpur, IIT Madras, IIT Roorkee and Indian Institute for Science, Bangalore (IISC Bangalore). The exam is divided into two different parts General Aptitude (GA) and Candidate’s Selected Subject (Core Subjects). GATE Mathematics online mock tests help candidates in preparing for the GATE exam. By attempting GATE online mock tests candidates get familiar with the pattern of actual question paper. BrPaper provides all GATE question papers completely free. Here you can practice free online mock tests of Main exam of GATE Mathematics. We advise candidates to solve the online GATE Mathematics mock tests to get good marks in the exam. Start practicing MathematicsGATE 2019 Mock tests here online for free

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Category: GATE Mathematics 2019

Question 1:

For a balanced transportation problem with three sources and three destinations where costs, availabilities and demands are all finite and positive, which one of the following statements is FALSE?

The transportation problem does not have unbounded solution

The number of non-basic variables of the transportation problem is 4

The dual variables of the transportation problem are unrestricted in sign

The transportation problem has at most 5 basic feasible solutions

Category: GATE Mathematics 2019

Question 2:

Let

f : [a, b] \to R

(the set of all real numbers) be any function that is twice differentiable in

(a, b)

with only one root

α

(a, b)

. Let

f' (x)

and

f'' (x)

denote the first and second-order derivatives of

f (x)

with respect to x. It

α

is a simple root and is computed by the Newton-Raphson method, then the method converges if

A	$\|f (x) f'' (x)\| < {\|f' (x)\|}^{2}, f o r a l l x \in (a, b)$	B	$\|f (x) f' (x)\| < {\|f'' (x)\|}^{2}, f o r a l l x \in (a, b)$
C	$\|f' (x) f'' (x)\| < {\|f (x)\|}^{2}, f o r a l l x \in (a, b)$	D	$\|f (x) f'' (x)\| < \|f' (x)\|, f o r a l l x \in (a, b)$

Category: GATE Mathematics 2019

Question 3:

Let

f : C \to C

(the set of all complex numbers) be defined by

f (x + i y) = x^{3} + 3 x y^{2} + i (y^{3} + 3 x^{2} y), i = \sqrt{- 1 .}

Let

f' (z)

denote the derivative of f with respect to z. Then which one of the following statements is TRUE?

A	$f' (1 + i) e x i s t s a n d \|f' (1 + i)\| = 3 \sqrt{5}$	B	$f$ is analytic at the origin
C	$f$ is not differentiable at i	D	$f$ is differentiable at 1

Category: GATE Mathematics 2019

Question 4:

The partial differential equation

(x^{2} + y^{2} - 1) \frac{\partial^{2} u}{\partial x^{2}} + \partial \frac{\partial^{2} u}{\partial x \partial y} + (x^{2} + y^{2} - 1) \frac{\partial^{2} u}{\partial y^{2}} = 0

A	parabolic in the region $x^{2} + y^{2} > 2$	B	hyperbolic in the region $x^{2} + y^{2} > 2$
C	elliptic in the region $0 < x^{2} + y^{2} > 2$	D	hyperbolic in the region $0 < x^{2} + y^{2} > 2$

Category: GATE Mathematics 2019

Question 5:

u_{n} = \int_{1}^{n} e^{- t^{2}} d t, n = 1, 2, 3, . . .,

then which one of the following statements is TRUE?

A	Both the sequence ${\{u_{n}\}}^{\infty}_{n = 1}$ and the series $\sum_{n = 1}^{\infty} u_{n}$ are convergent
B	Both the sequence ${\{u_{n}\}}^{\infty}_{n = 1}$ and the series $\sum_{n = 1}^{\infty} u_{n}$ are divergent
C	The sequence ${\{u_{n}\}}^{\infty}_{n = 1}$ is convergent but the series $\sum_{n = 1}^{\infty} u_{n}$ is divergent
D	$\lim_{n \to \infty} u_{n} = \frac{2}{e}$

Category: GATE Mathematics 2019

Question 6:

Let

T = \{(x, y, z) \in R^{3} : - 1 < x < 1, - 1 < y < 1, - 1 < z < 1\}

and

ϕ : T \leftrightarrow R

be a function whose all second-order partial derivatives exist and are continuous. If

ϕ

satisfies the Laplace equation

\nabla^{2} ϕ = 0

for all

(x, y, z) \in T

then which one of the following statements is TRUE in T? ( is the set of all real numbers, and

R^{3} = \{x, y, z\} : x, y, z \subset R})

A	$\vec{\nabla} ϕ$ is solenoidal but not irrotational	B	$\vec{\nabla} ϕ$ is irrotational but not solenoidal
C	$\vec{\nabla} ϕ$ is both solenoidal and irrotational	D	$\vec{\nabla} ϕ$ is neither solenoidal nor irrotational

Category: GATE Mathematics 2019

Question 7:

Let

X = {(x_{1}, x_{2}, . . .) : x_{i} \in R

and only finitely many

x_{i}' s

are non-zero} and

d : X \times X \to R

be a metric on X defined by

d (x, y) = \underset{i e N}{s u p} |x_{i} - y_{i}| f o r x = (x_{1}, x_{2}, . . .), y = (y_{1}, y_{2}, . . .) i n X .

( R is the set of all real numbers and N is the set of all natural numbers) Consider the following statements: P:

(X, d)

is a complete metric space. Q: The set

\{x \in X : d (0, x) \leq 1\}

is compact, where

0

is the zero elements of X

A	Both P and Q	B	P only
C	Q only	D	Neither P nor Q

Category: GATE Mathematics 2019

Question 8:

Consider the following statements: I. The set

Q \times Z

is uncountable. II. The set

{f : f

is a function from to {0, 1}} f f is uncountable. III. The set {

\sqrt{p} : p

is a prime number} is uncountable. IV. For an infinite set, there exists a bijection from the set to one of its proper subsets. ( Q is the set of all rational numbers, Z is the set of all integers, and N is the set of all-natural numbers) Which of the above statements is TRUE?

I and IV only

II and IV only

II and III only

I, II and IV only

Category: GATE Mathematics 2019

Question 9:

Let

f : R^{2} \to R

be defined by

f (x, y) = x^{6} - 2 x^{2} y - x^{4} y + 2 y^{2} .

( is the set of all real numbers and

R^{2} = \{(x, y) : x, y \in R\})

) Which one of the following statements is TRUE?

A	$f$ has a local maximum at the origin	B	$f$ has a local minimum at origin
C	$f$ has a saddle point at the origin	D	The origin is not a critical point of $f$

Category: GATE Mathematics 2019

Question 10:

Let

{\{a_{n}\}}_{n = 0}^{\infty}

be any sequence of real numbers such as that

\sum_{n = 0}^{\infty} {|a_{n}|}^{2} < \infty

. If the radius of convergence of

\sum_{n = 0}^{\infty} a_{n} x^{n}

is r, then which one of the following statements is necessarily TRUE?

A	$r \geq 1$ or r is infinite	B	$r < 1$
C	$r = {(\sum_{n = 0}^{\infty} {\|a_{n}\|}^{2})}^{\frac{1}{2}}$	D	$r = \sum_{n = 0}^{\infty} {\|a_{n}\|}^{2}$

Category: GATE Mathematics 2019

Question 11:

Let

T_{1}

be the co-countable topology on (the set of real numbers) and

T_{2}

be the co-finite topology on R. Consider the following statements: I. In1

(R, T_{1})

the sequence

{\{\frac{1}{n}\}}_{n = 1}^{\infty}

converges to 0. II. In

(R, T_{1})

, T the sequence

{\{\frac{1}{n}\}}_{n = 1}^{\infty}

converges to 0. III. In

(R, T_{1})

there is no sequence of rational numbers which converges to

\sqrt{3}

. IV. In

(R, T_{2})

there is no sequence of rational numbers which converges to

\sqrt{3}

. Which of the above statements is TRUE?

I and II only

II and III only

III and IV only

I and IV only

Category: GATE Mathematics 2019

Question 12:

Let X and Y be normed linear spaces, and let

T : X \to Y

be any bijective linear map with a closed graph. Then which one of the following statements is TRUE?

A	The graph of T is equal to $X \times Y$	B	$T^{- 1}$ is continuous
C	The graph of $T^{- 1}$ is closed	D	T is continuous

Category: GATE Mathematics 2019

Question 13:

Let

g : R^{2} \to R^{2}

be a function defined by

g (x, y) = (e^{x} \cos y, e^{x} \sin y)

and

(a, b)

= g (1, \frac{π}{3})

. (R is the set of all real numbers and

R^{2} = \{(x, y) : x, y \in R\}

) Which one of the following statements is TRUE?

A	g is injective
B	If h is the continuous inverse of g, defined in some neighborhood of $(a, b) \in R^{2}$ , such that $h (a, b) = (1, \frac{π}{3}),$
C	If h is the continuous inverse of g, defined in some neighbourhood of $(a, b) \in R^{2}$ , such that $h (a, b) = (1, \frac{π}{3}),$ then the Jacobian of h at $(a, b) i s e^{- 2}$
D	g is surjective

Category: GATE Mathematics 2019

Question 14:

Let

u_{n} = \frac{n!}{1.3.5 . . . (2 n - 1)}, n \in N

(the set of all-natural numbers). Then

\lim_{n \to \infty} u_{n}

is equal to _____ .

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2019

Question 15:

If the differential equation

\frac{d y}{d x} = \sqrt{x^{2} + y^{2}}, y (1) = 2

is solved using the Euler’s method with step-size h= 0.1, then y(1.2) is equal to _____ (round off to 2 places of decimal).

2.40 to 2.50

2.50 to 2.60

2.60 to 2.70

2.70 to 2.80

Category: GATE Mathematics 2019

Question 16:

Let f be any polynomial function of degree at most 2 over (the set of all real numbers). If the constants a and b are such that

\frac{d f}{d x} = a f (x) + 2 f (x + 1) + 1 b f (x + 2), f o r a l l x \in R,

then 4a + 3b is equal to _____ (round off to 2 places of decimal).

7.5 to 7.5

-7.5 to 7.5

-7.5 to -7.5

None of above

Category: GATE Mathematics 2019

Question 17:

Let L denote the value of the line integral

\oint_{c} (3 x - 4 x^{2} y) d x + (4 x y^{2} + 2 y) d y,

Where C, a circle of radius 2 with centre at the origin of the XY-plane, is traversed once in the anti-clockwise direction. Then

\frac{L}{π}

is equal to _____.

31.60 to 32.10

31.70 to 32.10

31.80 to 32.10

31.90 to 32.10

Category: GATE Mathematics 2019

Question 18:

The temperature

T : R^{3} \ \{(0, 0, 0)\} \to R

at any point

P (x, y, z)

is inversely proportional to the square of the distance of P from the origin. If the value of the temperature T at the point R(0,0,1) is

\sqrt{3}

, then the rate of change of T at the point Q (1,1,2) in the direction of QR is equal to _____ (round off to 2 places of the decimal). ( R is the set of all real numbers,

R^{3} = \{(x, y, z) : x, y, z \in R\}

and

R^{3} \ \{(0, 0, 0)\}

denotes

R^{3}

excluding the origin)

0.20 to 0.22

0.21 to 0.23

0.22 to 0.24

0.23 to 0.25

Category: GATE Mathematics 2019

Question 19:

Let

f

be a continuous function defined on

[0, 2]

such that

f (x) \geq 0

for all

x \in [0, 2]

. If the area bounded by

y = f (x), x = 0, y = 0

and

x = b i s \sqrt{3 + b^{2}} - \sqrt{3},

Where

b \in [0, 2]

, then

f (1)

is equal to _____ (round off to 1 place of decimal).

0.5 to 0.5

1.0 to 1.0

1.5 to 1.5

1.0 to 1.0

Category: GATE Mathematics 2019

Question 20:

If the characteristic polynomial and minimal polynomial of a square matrix A is

(λ - 1) {(λ + 1)}^{4} {(λ - 2)}^{5} a n d (λ - 1) (λ + 1) (λ - 2)

respectively, then the rank of the matrix A+I respectively, then the rank of the matrix of appropriate order.

2 to 2

4 to 4

6 to 6

8 to 8

Category: GATE Mathematics 2019

Question 21:

Let

ω

be a primitive complex cube root of unity and

i = \sqrt{- 1}

. Then the degree of the field extension Q

Q (i, \sqrt{3} ω)

over Q (the field of rational numbers) is _____.

2 to 2

4 to 4

6 to 6

8 to 8

Category: GATE Mathematics 2019

Question 22:

Let

α = \int_{c} \frac{e^{i πz} d z}{2 z^{2} - 5 z + 2}, C : \cos t + i \sin t, 0 \leq t \leq 2 π, i = \sqrt{- 1} .

Then the greatest integer less than or equal to

|α|

is _____.

2 to 2

4 to 4

6 to 6

8 to 8

Category: GATE Mathematics 2019

Question 23:

Consider the system:

3 x_{1} + x_{2} + 2 x_{3} - x_{4} = a, x_{1} + x_{2} + x_{3} - 2 x_{4} = 3 x_{1}, x_{2}, x_{3}, x_{4} \geq 0 .

x_{1} = 1, x_{2} = b, x_{3} = 0, x_{4} = c

is a basic feasible solution of the above system (where a b, and c are real constants), then

a + b + c

is equal to _____.

1 to 1

3 to 3

5 to 5

7 to 7

Category: GATE Mathematics 2019

Question 24:

Let

f : C \to C

be a function defined by

f (z) = z^{6} - 5 z^{4} + 10

. Then the number of zeros of

f i n \{z \in C : |z| < 2\}

is __________________. ( C is the set of all complex numbers)

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2019

Question 25:

Let

e^{2} = \{x = (x_{1}, x_{2}, . . .) : x_{i} \in C, \sum_{i = 1}^{\infty} {|x_{i}|}^{2} < \infty\}

e be a normed linear space with the norm

{||x||}_{2} = {(\sum_{i = 1}^{\infty} {|x_{i}|}^{2})}^{\frac{1}{2}}

Let

g : l^{2} \to C

be the bounded linear functional defined by

g (x) = \sum_{n = 1}^{\infty} \frac{x_{n}}{3^{n}} f o r a l l x = (x_{1}, x_{2}, . . .) \in l^{2} .

Then

{(s u p \{|g (x)| : {||x||}_{2} \leq 1\})}^{2}

is equal to ___________ (round off to 3 places of decimal). (C is the set of all complex numbers).

0.124 to 0.124

0.125 to 0.125

0.126 to 0.126

0.127 to 0.127

Category: GATE Mathematics 2019

Question 26:

For the linear programming problem (LPP):

M a x i m i z e Z = 2 x_{1} + 4 x_{2} s u b j e c t t o - x_{1} + 2 x_{2} \leq 4, 3 x_{1} + β x_{2} \leq 6, x_{1}, x_{2} \geq 0, β \in R,

( R is the set of all real numbers) consider the following statements: I. The LPP always has a finite optimal value for any

β \geq 0

. II. The dual of the LPP may be infeasible for some

β \geq 0

. III. If for some

β

, the point (1,2) is feasible to the dual of the LPP, then

Z \leq 16

, for any feasible solution

(x_{1}, x_{2})

of the LPP. IV. If for some

β

x_{1} a n d x_{2}

are the basic variables in the optimal table of the LPP with

x_{1} = \frac{1}{2},

then the optimal value of dual of the LPP is 10. Then which of the above statements are TRUE?

I and III only

I, III and IV only

III and IV only

II and IV only

Category: GATE Mathematics 2019

Question 27:

Let

f : R^{2} \to R

be defined by

f (x, y) = \{\begin{cases} (x^{2} + y^{2}) \sin (\frac{1}{x^{2} + y^{2}}), i f (x, y) \neq (0, 0) \\ 0, i f (x, y) = (0, 0) \end{cases} .

Consider the following statements: I. The partial derivatives,

\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}

exist at (0, 0) but are unbounded in any neighbourhood of (0, 0). II. f is continuous but not differentiable at (0, 0). III. f is not continuous at (0, 0). IV. f is differentiable at (0, 0). Which of the above statements is/are TRUE?

I and II only

I and IV only

IV only

III only

Category: GATE Mathematics 2019

Question 28:

Let

K = {[k_{i . j}]}_{i, j = 1}^{\infty}

be an infinite matrix over C (the set of all complex numbers) such that (i) for each

i \in N

(the set of all natural numbers), the

i^{t h}

row

(k_{i, 1}, k_{i, 2}, . . .)

of K is in

l^{\infty}

and (ii) for everyone

x = (x_{1}, x_{2}, . . .) \in l^{1}, \sum_{j = 1}^{\infty} k_{i, j} x_{j}

is summable for all

i \in N

, and

(y, y_{2}, . . .) \in l^{1},

where

y_{i} = \sum_{j = 1}^{\infty} k_{i, j} x_{j}

. Let the set of all rows of K be denoted by E . Consider the following statements: P: E is a bounded set in

l^{\infty}

, Q: E is a dense set in

l^{\infty}

(l^{1} = \{(x_{1}, x_{2}, . . .) : x_{i} \in C, \sum_{i = 1}^{\infty}\}) (l^{\infty} = \{(x_{1}, x_{2}, . . .) : x_{i} \in C, \underset{i e N}{s u p} |x_{i}| < \infty\})

Which of the above statements is/are TRUE?

Both P and Q

P only

Q only

Neither P nor Q

Category: GATE Mathematics 2019

Question 29:

Consider the following heat conduction problem for a finite rod

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}} - x e^{t} - 2 t, t > 0, 0 < x < π,

with the boundary conditions

u (0, t) = - t^{2}, u (π, t) = - {πe}^{t} - t^{2}, t > 0

and the initial condition

u (x, 0) = \sin x - \sin^{3} x - x, 0 \leq x \leq π .

v (x, t) = u (x, t) + x e^{t} + t^{2},

then which one of the following is CORRECT?

A	$v (x, t) = \frac{1}{4} (e^{- t} \sin x + e^{- 9 t} \sin 3 x)$	B	$v (x, t) = \frac{1}{4} (7 e^{- t} \sin x - e^{- 9 t} \sin 3 x)$
C	$v (x, t) = \frac{1}{4} (e^{- t} \sin x + e^{- 3 t} \sin 3 x)$	D	$v (x, t) = \frac{1}{4} (3 e^{- t} \sin x - e^{- 3 t} \sin 3 x)$

Category: GATE Mathematics 2019

Question 30:

Let

f : C \to C

be non-zero and analytic at all points in Z. If

F (z) = πf (z) \cot (πz) for z \in \ Z,

then the residue of F at

n \in Z

is ____________. (C is the set of all complex numbers,Z is the set of all integers andC\ Zdenotes the set of all complex numbers excluding integers)

A	$πf (n)$	B	$f (n)$
C	$\frac{f (n)}{π}$	D	${(\frac{df}{d z})}_{z = n}$

Category: GATE Mathematics 2019

Question 31:

Let the general integral of the partial differential equation

(2 xy - 1) \frac{\partial z}{\partial x} + (z - 2 x^{2}) \frac{\partial z}{\partial y} = 2 (x - yz)

be given by

f (u, v) = 0

, where

f : R^{2} \to R

is a continuously differentiable function ( R is the set of all real numbers and

R^{2} = \{(x, y) : x, y \in R\})

Then which one of the following is TRUE?

A	$u = x^{2} + y^{2} + z, v = xz + y$	B	$u = x^{2} + y^{2} - z, v = xz - y$
C	$u = x^{2} - y^{2} + z, v = xz + y$	D	$u = x^{2} + y^{2} - z, v = xz - y$

Category: GATE Mathematics 2019

Question 32:

Consider the following statements: I. If Q denotes the additive group of rational numbers and

f : Q \to Q

is a non-trivial homomorphism, then f is an isomorphism. II. Any quotient group of a cyclic group is cyclic. III. If every subgroup of a group G is a normal subgroup, then G is abelian. IV. Every group of order 33 is cyclic Which of the above statements are TRUE?

II and IV only

II and III only

I, II and IV only

I, III and IV only

Category: GATE Mathematics 2019

Question 33:

A solution to the Dirichlet problem

\nabla^{2} u (r, θ) = 0, 0 < r < 1, - π \leq θ \leq π, u (1, θ) = |θ|, - π \leq θ \leq π,

is given by

A	$u (r, θ) = \frac{π}{2} + \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n} - 1}{n^{2}}] r^{n} \cos (nθ)$	B	$u (r, θ) = \frac{2}{π} + \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n + 1}}{n^{2}}] r^{n} \cos (nθ)$
C	$u (r, θ) = \frac{π}{2} + \frac{2}{π} \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n} - 1}{n^{2}}] r^{n} \cos (nθ)$	D	$u (r, θ) = \frac{π}{2} + \frac{2}{π} \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n} + 1}{n^{2}}] r^{n} \cos (nθ)$

Category: GATE Mathematics 2019

Question 34:

Consider the subspace

Y = \{(x, x) : x \in C\}

of the normed linear space

(C^{2} {||||}_{\infty}) .

ϕ

is a bounded linear functional on Y, defined by

ϕ (x, x) = x,

, then which one of the following sets is equal to

\{ψ (1, 0) : ψ is a norm preserving extension of ϕ to (C^{2}, {||||}_{\infty})\}

( is the set of all complex numbers,

C^{2} = \{(x, y) : x, y \in C\}

and

{||(x_{1}, x \infty_{2})||}_{} = \sup \{|x_{1}|, |x_{2}|\})

A	$\{1\}$	B	$[\frac{1}{2}, \frac{3}{2}]$
C	$[1, \infty]$	D	$[0, 1]$

Category: GATE Mathematics 2019

Question 35:

Consider the following statements: I. The ring

Z [\sqrt{- 1}]

is a unique factorization domain. II. The ring

Z [\sqrt{- 5}]

is a principal ideal domain. III. In the polynomial ring

Z_{2} [x],

, the ideal generated by

x^{3} + x + 1

is a maximal ideal. IV. In the polynomial ring

Z_{3} [x]

, the ideal generated by

x^{6} + 1

is a prime ideal. ( Z denotes the set of all integers,

Z_{n}

denotes the set of all integers modulo

n

, for any positive integer n) Which of the above statements is TRUE?

I, II and III only

I and III only

I, II and IV only

II and III only

Category: GATE Mathematics 2019

Question 36:

Let M be a

3 \times 3

real symmetric matrix with eigenvalues 0, 2, and a with the respective eigenvectors

u = {(4, b, c)}^{τ}, v = {(- 1, 2, 0)}^{τ} and w = {(1, 1, 1)}^{τ} .

Consider the following statements: I.

a + b - c = 10

II. The vector

{(0, \frac{3}{2}, \frac{1}{2})}^{τ}

satisfies

Mx = v + x

. III. For any

d \in

span

\{u, v, w\}, M x = d

has a solution. IV. The trace of the matrix

M^{2} + 2 M is 8 .

Which of the above statements is TRUE?

I, II and III only

I and II only

II and IV only

III and IV only

Category: GATE Mathematics 2019

Question 37:

Consider the region

Ω = \{x + i y : - 1 \leq x \leq 2, \frac{- π}{3} \leq y \leq \frac{π}{3}\}, i = \sqrt{- 1}

in the complex plane. The transformation

x + i y \mapsto e^{x + i y}

maps the region Ω onto the region

S \subset C

(the set of all complex numbers). Then the area of the region S is equal to

A	$\frac{π}{3} (e^{4} - e^{- 2})$	B	$\frac{π}{3} (e^{4} + e^{- 2})$
C	$\frac{2 π}{3} (e^{4} - e^{- 2})$	D	$\frac{π}{6} (e^{4} - e^{- 2})$

Category: GATE Mathematics 2019

Question 38:

Consider the sequence

{\{g_{n}\}}_{n = 1}

of functions, where

g_{n} (x) = \frac{x}{1 + n x^{2}}

x \in R, n \in N

and

g'_{n} (x)

is the derivative of

g_{n} (x)

with respect to x. (R is the set of all real numbers,N is the set of all-natural numbers). Then which one of the following statements is TRUE?

A	${\{g_{n}\}}_{n = 1}^{\infty}$ does NOT converge uniformly on R
B	${\{g'_{n}\}}_{n = 1}^{\infty}$ converges uniformly on any closed interval which does NOT contain 1
C	${\{g'_{n}\}}_{n = 1}^{\infty}$ converges point-wise to a continuous function on R
D	${\{g'_{n}\}}_{n = 1}^{\infty}$ converges uniformly on any closed interval which does NOT contain 0

Category: GATE Mathematics 2019

Question 39:

Consider the boundary value problem (BVP)

\frac{d^{2} y}{{dx}^{2}} + α y (x) = 0, α \in R (the set of all real numbers),

with the boundary conditions

y (0) = 0, y (π) = k (k is a non - zero real number) .

Then which one of the following statements is TRUE?

A	For $α = 1,$ the BVP has infinitely many solutions
B	For $α = 1$ , the BVP has a unique solution
C	For $α = - 1$ , the BVP has a solution y(x) such that $y (x) > 0$ for all $x \in (0, π)$
D	For $α = - 1, k < 0,$ , the BVP has a solution $y (x)$ such that $y (x) > 0$ for all $x \in (0, π)$

Category: GATE Mathematics 2019

Question 40:

Consider the ordered square

I_{0}^{2},

The set

[0, 1] \times [0, 1]

with the dictionary order topology. Let the general element of

I_{0}^{2}

be denoted by

x \times y

, where

x, y \in [0, 1]

. Then the closure of the subset

S = \{x \times \frac{3}{4} : 0 < α < x < b < 1\} in I_{0}^{2}

A	$S \cup ((a, b] \times \{0\} \cup ([a, b) \times \{1\})$	B	$S \cup ([a, b) \times \{0\} \cup (a, b] \times \{1\})$
C	$S \cup ((a, b) \times \{0\} \cup ((a, b) \times \{1\})$	D	$S \cup ((a, b] \times \{0\})$

Category: GATE Mathematics 2019

Question 41:

Let

P_{2}

be the vector space of all polynomials of degree at most 2 over R (the set of real numbers). Let a linear transformation

T : P_{2} \to P_{2}

be defined by

T (a + bx + {cx}^{2}) = (a + b) + (b - c) x + (a + c) x^{2} .

Consider the following statements: I. The null space of T is

\{a (+ x + x^{2}) : a \in R\}

. II. The range space of T is spanned by the set

\{1 + x^{2}, 1 + x\}

. III.

T (T (1 + X)) = 1 + X^{2}

. IV. If M is the matrix representation of T with respect to the standard basis

\{1, x, x^{2}\}

P_{2}

, then the trace of the matrix M is 3. Which of the above statements is TRUE?

I and II only

I, III and IV only

I, II and IV only

II and IV only

Category: GATE Mathematics 2019

Question 42:

Let

T_{1} and T_{2}

be two topologies defined on

N

(the set of all-natural numbers), where

T_{1}

is the topology generated by

B = \{\{2 n - 1, 2 n\} : n \in N\} and T_{2}

is the discrete topology on N. Consider the following statements : I. In

(N, T_{1})

, T every infinite subset has a limit point. II. The function

f : (N, T_{1}) \to (N, T_{2})

defined by

F (n) = \{\begin{cases} \frac{n}{2}, if n is even \\ \frac{n + 1}{2} if n isodd \end{cases}

is a continuous function. Which of the above statements is/are TRUE?

Both I and II

I only

II only

Neither I nor II

Category: GATE Mathematics 2019

Question 43:

Let

1 \leq p < q < \infty .

Consider the following statements: I.

l^{p} \subset l^{q}

II.

L^{p} [0, 1] \subset L^{q} [0, 1],

where

l^{p} = \{(x_{1}, x_{2}, . . .) : x_{i} \in R, \sum_{i = 1}^{\infty} {|x_{i}|}^{p} < \infty\} and

L^{p} [0, 1] = \{f : [0, 1] \to R : f is - measurable \int_{[0, 1]} {|f|}^{p} d μ < \infty, where is the Lebesgue measure\}

( R is the set of all real numbers) Which of the above statements is/are TRUE?

Both I and II

I only

II only

Neither I nor II

Category: GATE Mathematics 2019

Question 44:

Consider the differential equation

t \frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} + t y = 0, t > 0, y (0 +) = 1 {(\frac{d y}{d t})}_{t = 0 +} = 0 .

If Y (s) is the Laplace transform of y(t), then the value of Y(1) is _____ (round off to 2 places of the decimal). (Here, the inverse trigonometric functions assume principal values only)

0.72 to 0.83

0.74 to 0.83

0.76 to 0.83

0.78 to 0.83

Category: GATE Mathematics 2019

Question 45:

Let R be the region in the xy-plane bounded by the curves

y = x^{2}, y = 4 x^{2}, x y = 1 and xy = 5 .

Then the value of the integral

\int \int_{R} \frac{y^{2}}{x} d y d x

is equal to _________.

11 to 11

12 to 12

13 to 13

14 to 14

Category: GATE Mathematics 2019

Question 46:

Let V be the vector space of all

3 \times 3

matrices with complex entries over the real field. If\

W_{1} = \{A \in V : A = A^{T}\} and W_{2} = \{A \in V : trace of A = 0\},

then the dimensions of

W_{1} + W_{2}

is equal to ________. (

A^{T}

denotes the conjugate transpose of A)

15 to 15

16 to 16

17 to 17

18 to 18

Category: GATE Mathematics 2019

Question 47:

The number of elements of order 15 in the additive group

Z_{60} \times Z_{50}

IS __________. (

Z_{n}

denotes the group of integers modulo n, under the operation of addition modulo n, for any positive integer n)

45 to 48

46 to 48

47 to 48

48 to 48

Category: GATE Mathematics 2019

Question 48:

Consider the following cost matrix of assigning four jobs to four persons:

Then the minimum cost of the assignment problem subject to the constraint that job

J_{4}

is assigned to person

P_{2}

, is _____.

25 to 25

26 to 26

27 to 27

28 to 28

Category: GATE Mathematics 2019

Question 49:

Let

y : - [- 1, 1] \to R

with y(1) =1 satisfy the Legendre differential equation

(1 - x^{2}) \frac{d^{2} y}{{dx}^{2}} - 2 x \frac{dy}{dx} + 6 y = 0 for |x| < 1 .

Then the value of

\int_{- 1}^{1} y (x) (x + x^{2}) d x

is equal to _____ (round off to 2 places of the decimal).

0.15 to 0.20

0.25 to 0.30

0.35 to 0.40

0.45 to 0.50

Category: GATE Mathematics 2019

Question 50:

The maximum value of the error term of the composite Trapezoidal rule when it is used to evaluate the definite integral

\int_{0.2}^{1.4} (\sin x - \log_{e} x) dx

with 12 subintervals of equal length, is equal to _____ (round off to 3 places of the decimal).

0.022 to 0.028

0.024 to 0.028

0.026 to 0.028

0.028 to 0.028

Category: GATE Mathematics 2019

Question 51:

Let

Z_{125}

be the ring of integers modulo 125 under the operations of addition modulo 125 and multiplication modulo 125. If m is the number of maximal ideals of

Z_{125}

and n is the number of non-units of

Z_{125}

, then m + n is equal to _____.

222 to 22

24 to 24

26 to 26

28 to 28

Category: GATE Mathematics 2019

Question 52:

By the Simplex method, the optimal table of the linear programming problem:

Maximize Z = {αx}_{1} + 3 x_{2} subject to {βx}_{1} + x_{2} + x_{3} = 8, 2 x_{1} + x_{2} + x_{4} = γ, x_{1} + x_{2} + x_{3} + x_{4} \geq 0,

where

α, β, γ

are real constants, is

Then the value

α + β + γ

is _____.

17.0 to 18.5

14.0 to 16.0

14.0 to 16.0 OR 17.0 to 18.5

None of above

Category: GATE Mathematics 2019

Question 53:

Consider the inner product space

P_{2}

of all polynomials of degree at most 2 over the field of real numbers with the inner product

<f, g> = \int_{0}^{1} f (t) g (t) dt for f, g \in P_{2}

Let

\{f_{0}, f_{1}, f_{2}\}

be an orthogonal set in

P_{2}, where f_{0} = 1, f_{1} = t + c_{1}, f_{2} = t^{2} + c_{2} f_{1} + c_{3} and c_{1}, c_{2}, c_{3}

are real constants. Then the value of

2 c_{1} + c_{2} + 3 c_{3}

2 c_{1} + c_{2} + 3 c_{3}

is equal to _____.

-3 to -3

3 to -3

3 to 3

None of above

Category: GATE Mathematics 2019

Question 54:

Consider the system of linear differential equations

\frac{{dx}_{1}}{dt} = 5 x_{1} - 2 x, \frac{{dx}_{2}}{dt} = 4 x_{1} - x_{2},

with the initial conditions

x_{1} (0) = 0, x_{2} (0) = 1 .

Then

\log_{e} (x_{2} (2) - x_{1} (2))

is equal to _____.

1.95 to 2.01

1.95 to 2.03

1.95 to 2.05

1.95 to 2.07

Category: GATE Mathematics 2019

Question 55:

Consider the differential equation

x (1 + x^{2}) \frac{d^{2} y}{{dx}^{2}} - 9 \frac{dy}{dx} + 7 y = 0 .

The sum of the roots of the indicial equation of the Frobenius series solution for the above differential equation in a neighborhood of x = 0 is equal to _____.

8 to 8

9 to 9

10 to 10

11 to 11

⮜ Back Next ⮞

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Answered	0	Marked for Review
Answered & Marked for Review (will be considered for evaluation)

Question 1:

For a balanced transportation problem with three sources and three destinations where costs, availabilities and demands are all finite and positive, which one of the following statements is FALSE?

Correct Answer : The transportation problem has at most 5 basic feasible solutions

Question 2:

Let

f : [a, b] \to R

(the set of all real numbers) be any function that is twice differentiable in

(a, b)

with only one root

α

(a, b)

. Let

f' (x)

and

f'' (x)

denote the first and second-order derivatives of

f (x)

with respect to x. It

α

is a simple root and is computed by the Newton-Raphson method, then the method converges if

A	$\|f (x) f'' (x)\| < {\|f' (x)\|}^{2}, f o r a l l x \in (a, b)$	B	$\|f (x) f' (x)\| < {\|f'' (x)\|}^{2}, f o r a l l x \in (a, b)$
C	$\|f' (x) f'' (x)\| < {\|f (x)\|}^{2}, f o r a l l x \in (a, b)$	D	$\|f (x) f'' (x)\| < \|f' (x)\|, f o r a l l x \in (a, b)$

Correct Answer : A

Question 3:

Let

f : C \to C

(the set of all complex numbers) be defined by

f (x + i y) = x^{3} + 3 x y^{2} + i (y^{3} + 3 x^{2} y), i = \sqrt{- 1 .}

Let

f' (z)

denote the derivative of f with respect to z. Then which one of the following statements is TRUE?

A	$f' (1 + i) e x i s t s a n d \|f' (1 + i)\| = 3 \sqrt{5}$	B	$f$ is analytic at the origin
C	$f$ is not differentiable at i	D	$f$ is differentiable at 1

Correct Answer : D

Question 4:

The partial differential equation

(x^{2} + y^{2} - 1) \frac{\partial^{2} u}{\partial x^{2}} + \partial \frac{\partial^{2} u}{\partial x \partial y} + (x^{2} + y^{2} - 1) \frac{\partial^{2} u}{\partial y^{2}} = 0

A	parabolic in the region $x^{2} + y^{2} > 2$	B	hyperbolic in the region $x^{2} + y^{2} > 2$
C	elliptic in the region $0 < x^{2} + y^{2} > 2$	D	hyperbolic in the region $0 < x^{2} + y^{2} > 2$

Correct Answer : D

Question 5:

u_{n} = \int_{1}^{n} e^{- t^{2}} d t, n = 1, 2, 3, . . .,

then which one of the following statements is TRUE?

A	Both the sequence ${\{u_{n}\}}^{\infty}_{n = 1}$ and the series $\sum_{n = 1}^{\infty} u_{n}$ are convergent
B	Both the sequence ${\{u_{n}\}}^{\infty}_{n = 1}$ and the series $\sum_{n = 1}^{\infty} u_{n}$ are divergent
C	The sequence ${\{u_{n}\}}^{\infty}_{n = 1}$ is convergent but the series $\sum_{n = 1}^{\infty} u_{n}$ is divergent
D	$\lim_{n \to \infty} u_{n} = \frac{2}{e}$

Correct Answer : C

Question 6:

Let

T = \{(x, y, z) \in R^{3} : - 1 < x < 1, - 1 < y < 1, - 1 < z < 1\}

and

ϕ : T \leftrightarrow R

be a function whose all second-order partial derivatives exist and are continuous. If

ϕ

satisfies the Laplace equation

\nabla^{2} ϕ = 0

for all

(x, y, z) \in T

then which one of the following statements is TRUE in T? ( is the set of all real numbers, and

R^{3} = \{x, y, z\} : x, y, z \subset R})

A	$\vec{\nabla} ϕ$ is solenoidal but not irrotational	B	$\vec{\nabla} ϕ$ is irrotational but not solenoidal
C	$\vec{\nabla} ϕ$ is both solenoidal and irrotational	D	$\vec{\nabla} ϕ$ is neither solenoidal nor irrotational

Correct Answer : C

Question 7:

Let

X = {(x_{1}, x_{2}, . . .) : x_{i} \in R

and only finitely many

x_{i}' s

are non-zero} and

d : X \times X \to R

be a metric on X defined by

d (x, y) = \underset{i e N}{s u p} |x_{i} - y_{i}| f o r x = (x_{1}, x_{2}, . . .), y = (y_{1}, y_{2}, . . .) i n X .

( R is the set of all real numbers and N is the set of all natural numbers) Consider the following statements: P:

(X, d)

is a complete metric space. Q: The set

\{x \in X : d (0, x) \leq 1\}

is compact, where

0

is the zero elements of X

A	Both P and Q	B	P only
C	Q only	D	Neither P nor Q

Correct Answer : D

Question 8:

Consider the following statements: I. The set

Q \times Z

is uncountable. II. The set

{f : f

is a function from to {0, 1}} f f is uncountable. III. The set {

\sqrt{p} : p

Correct Answer : II and IV only

Question 9:

Let

f : R^{2} \to R

be defined by

f (x, y) = x^{6} - 2 x^{2} y - x^{4} y + 2 y^{2} .

( is the set of all real numbers and

R^{2} = \{(x, y) : x, y \in R\})

) Which one of the following statements is TRUE?

A	$f$ has a local maximum at the origin	B	$f$ has a local minimum at origin
C	$f$ has a saddle point at the origin	D	The origin is not a critical point of $f$

Correct Answer : C

Question 10:

Let

{\{a_{n}\}}_{n = 0}^{\infty}

be any sequence of real numbers such as that

\sum_{n = 0}^{\infty} {|a_{n}|}^{2} < \infty

. If the radius of convergence of

\sum_{n = 0}^{\infty} a_{n} x^{n}

is r, then which one of the following statements is necessarily TRUE?

A	$r \geq 1$ or r is infinite	B	$r < 1$
C	$r = {(\sum_{n = 0}^{\infty} {\|a_{n}\|}^{2})}^{\frac{1}{2}}$	D	$r = \sum_{n = 0}^{\infty} {\|a_{n}\|}^{2}$

Correct Answer : A

Question 11:

Let

T_{1}

be the co-countable topology on (the set of real numbers) and

T_{2}

be the co-finite topology on R. Consider the following statements: I. In1

(R, T_{1})

the sequence

{\{\frac{1}{n}\}}_{n = 1}^{\infty}

converges to 0. II. In

(R, T_{1})

, T the sequence

{\{\frac{1}{n}\}}_{n = 1}^{\infty}

converges to 0. III. In

(R, T_{1})

there is no sequence of rational numbers which converges to

\sqrt{3}

. IV. In

(R, T_{2})

there is no sequence of rational numbers which converges to

\sqrt{3}

. Which of the above statements is TRUE?

Correct Answer : II and III only

Question 12:

Let X and Y be normed linear spaces, and let

T : X \to Y

be any bijective linear map with a closed graph. Then which one of the following statements is TRUE?

A	The graph of T is equal to $X \times Y$	B	$T^{- 1}$ is continuous
C	The graph of $T^{- 1}$ is closed	D	T is continuous

Correct Answer : C

Question 13:

Let

g : R^{2} \to R^{2}

be a function defined by

g (x, y) = (e^{x} \cos y, e^{x} \sin y)

and

(a, b)

= g (1, \frac{π}{3})

. (R is the set of all real numbers and

R^{2} = \{(x, y) : x, y \in R\}

) Which one of the following statements is TRUE?

A	g is injective
B	If h is the continuous inverse of g, defined in some neighborhood of $(a, b) \in R^{2}$ , such that $h (a, b) = (1, \frac{π}{3}),$
C	If h is the continuous inverse of g, defined in some neighbourhood of $(a, b) \in R^{2}$ , such that $h (a, b) = (1, \frac{π}{3}),$ then the Jacobian of h at $(a, b) i s e^{- 2}$
D	g is surjective

Correct Answer : C

Question 14:

Let

u_{n} = \frac{n!}{1.3.5 . . . (2 n - 1)}, n \in N

(the set of all-natural numbers). Then

\lim_{n \to \infty} u_{n}

is equal to _____ .

Correct Answer : 0 to 0

Question 15:

If the differential equation

\frac{d y}{d x} = \sqrt{x^{2} + y^{2}}, y (1) = 2

is solved using the Euler’s method with step-size h= 0.1, then y(1.2) is equal to _____ (round off to 2 places of decimal).

Correct Answer : 2.40 to 2.50

Question 16:

Let f be any polynomial function of degree at most 2 over (the set of all real numbers). If the constants a and b are such that

\frac{d f}{d x} = a f (x) + 2 f (x + 1) + 1 b f (x + 2), f o r a l l x \in R,

then 4a + 3b is equal to _____ (round off to 2 places of decimal).

Correct Answer : -7.5 to -7.5

Question 17:

Let L denote the value of the line integral

\oint_{c} (3 x - 4 x^{2} y) d x + (4 x y^{2} + 2 y) d y,

Where C, a circle of radius 2 with centre at the origin of the XY-plane, is traversed once in the anti-clockwise direction. Then

\frac{L}{π}

is equal to _____.

Correct Answer : 31.90 to 32.10

Question 18:

The temperature

T : R^{3} \ \{(0, 0, 0)\} \to R

at any point

P (x, y, z)

is inversely proportional to the square of the distance of P from the origin. If the value of the temperature T at the point R(0,0,1) is

\sqrt{3}

, then the rate of change of T at the point Q (1,1,2) in the direction of QR is equal to _____ (round off to 2 places of the decimal). ( R is the set of all real numbers,

R^{3} = \{(x, y, z) : x, y, z \in R\}

and

R^{3} \ \{(0, 0, 0)\}

denotes

R^{3}

excluding the origin)

Correct Answer : 0.21 to 0.23

Question 19:

Let

f

be a continuous function defined on

[0, 2]

such that

f (x) \geq 0

for all

x \in [0, 2]

. If the area bounded by

y = f (x), x = 0, y = 0

and

x = b i s \sqrt{3 + b^{2}} - \sqrt{3},

Where

b \in [0, 2]

, then

f (1)

is equal to _____ (round off to 1 place of decimal).

Correct Answer : 0.5 to 0.5

Question 20:

If the characteristic polynomial and minimal polynomial of a square matrix A is

(λ - 1) {(λ + 1)}^{4} {(λ - 2)}^{5} a n d (λ - 1) (λ + 1) (λ - 2)

respectively, then the rank of the matrix A+I respectively, then the rank of the matrix of appropriate order.

Correct Answer : 6 to 6

Question 21:

Let

ω

be a primitive complex cube root of unity and

i = \sqrt{- 1}

. Then the degree of the field extension Q

Q (i, \sqrt{3} ω)

over Q (the field of rational numbers) is _____.

Correct Answer : 4 to 4

Question 22:

Let

α = \int_{c} \frac{e^{i πz} d z}{2 z^{2} - 5 z + 2}, C : \cos t + i \sin t, 0 \leq t \leq 2 π, i = \sqrt{- 1} .

Then the greatest integer less than or equal to

|α|

is _____.

Correct Answer : 2 to 2

Question 23:

Consider the system:

3 x_{1} + x_{2} + 2 x_{3} - x_{4} = a, x_{1} + x_{2} + x_{3} - 2 x_{4} = 3 x_{1}, x_{2}, x_{3}, x_{4} \geq 0 .

x_{1} = 1, x_{2} = b, x_{3} = 0, x_{4} = c

is a basic feasible solution of the above system (where a b, and c are real constants), then

a + b + c

is equal to _____.

Correct Answer : 7 to 7

Question 24:

Let

f : C \to C

be a function defined by

f (z) = z^{6} - 5 z^{4} + 10

. Then the number of zeros of

f i n \{z \in C : |z| < 2\}

is __________________. ( C is the set of all complex numbers)

Correct Answer : 4 to 4

Question 25:

Let

e^{2} = \{x = (x_{1}, x_{2}, . . .) : x_{i} \in C, \sum_{i = 1}^{\infty} {|x_{i}|}^{2} < \infty\}

e be a normed linear space with the norm

{||x||}_{2} = {(\sum_{i = 1}^{\infty} {|x_{i}|}^{2})}^{\frac{1}{2}}

Let

g : l^{2} \to C

be the bounded linear functional defined by

g (x) = \sum_{n = 1}^{\infty} \frac{x_{n}}{3^{n}} f o r a l l x = (x_{1}, x_{2}, . . .) \in l^{2} .

Then

{(s u p \{|g (x)| : {||x||}_{2} \leq 1\})}^{2}

is equal to ___________ (round off to 3 places of decimal). (C is the set of all complex numbers).

Correct Answer : 0.125 to 0.125

Question 26:

For the linear programming problem (LPP):

M a x i m i z e Z = 2 x_{1} + 4 x_{2} s u b j e c t t o - x_{1} + 2 x_{2} \leq 4, 3 x_{1} + β x_{2} \leq 6, x_{1}, x_{2} \geq 0, β \in R,

( R is the set of all real numbers) consider the following statements: I. The LPP always has a finite optimal value for any

β \geq 0

. II. The dual of the LPP may be infeasible for some

β \geq 0

. III. If for some

β

, the point (1,2) is feasible to the dual of the LPP, then

Z \leq 16

, for any feasible solution

(x_{1}, x_{2})

of the LPP. IV. If for some

β

x_{1} a n d x_{2}

are the basic variables in the optimal table of the LPP with

x_{1} = \frac{1}{2},

then the optimal value of dual of the LPP is 10. Then which of the above statements are TRUE?

Correct Answer : I, III and IV only

Question 27:

Let

f : R^{2} \to R

be defined by

f (x, y) = \{\begin{cases} (x^{2} + y^{2}) \sin (\frac{1}{x^{2} + y^{2}}), i f (x, y) \neq (0, 0) \\ 0, i f (x, y) = (0, 0) \end{cases} .

Consider the following statements: I. The partial derivatives,

\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}

Correct Answer : I and IV only

Question 28:

Let

K = {[k_{i . j}]}_{i, j = 1}^{\infty}

be an infinite matrix over C (the set of all complex numbers) such that (i) for each

i \in N

(the set of all natural numbers), the

i^{t h}

row

(k_{i, 1}, k_{i, 2}, . . .)

of K is in

l^{\infty}

and (ii) for everyone

x = (x_{1}, x_{2}, . . .) \in l^{1}, \sum_{j = 1}^{\infty} k_{i, j} x_{j}

is summable for all

i \in N

, and

(y, y_{2}, . . .) \in l^{1},

where

y_{i} = \sum_{j = 1}^{\infty} k_{i, j} x_{j}

. Let the set of all rows of K be denoted by E . Consider the following statements: P: E is a bounded set in

l^{\infty}

, Q: E is a dense set in

l^{\infty}

(l^{1} = \{(x_{1}, x_{2}, . . .) : x_{i} \in C, \sum_{i = 1}^{\infty}\}) (l^{\infty} = \{(x_{1}, x_{2}, . . .) : x_{i} \in C, \underset{i e N}{s u p} |x_{i}| < \infty\})

Which of the above statements is/are TRUE?

Correct Answer : P only

Question 29:

Consider the following heat conduction problem for a finite rod

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}} - x e^{t} - 2 t, t > 0, 0 < x < π,

with the boundary conditions

u (0, t) = - t^{2}, u (π, t) = - {πe}^{t} - t^{2}, t > 0

and the initial condition

u (x, 0) = \sin x - \sin^{3} x - x, 0 \leq x \leq π .

v (x, t) = u (x, t) + x e^{t} + t^{2},

then which one of the following is CORRECT?

A	$v (x, t) = \frac{1}{4} (e^{- t} \sin x + e^{- 9 t} \sin 3 x)$	B	$v (x, t) = \frac{1}{4} (7 e^{- t} \sin x - e^{- 9 t} \sin 3 x)$
C	$v (x, t) = \frac{1}{4} (e^{- t} \sin x + e^{- 3 t} \sin 3 x)$	D	$v (x, t) = \frac{1}{4} (3 e^{- t} \sin x - e^{- 3 t} \sin 3 x)$

Correct Answer : A

Question 30:

Let

f : C \to C

be non-zero and analytic at all points in Z. If

F (z) = πf (z) \cot (πz) for z \in \ Z,

then the residue of F at

n \in Z

is ____________. (C is the set of all complex numbers,Z is the set of all integers andC\ Zdenotes the set of all complex numbers excluding integers)

A	$πf (n)$	B	$f (n)$
C	$\frac{f (n)}{π}$	D	${(\frac{df}{d z})}_{z = n}$

Correct Answer : B

Question 31:

Let the general integral of the partial differential equation

(2 xy - 1) \frac{\partial z}{\partial x} + (z - 2 x^{2}) \frac{\partial z}{\partial y} = 2 (x - yz)

be given by

f (u, v) = 0

, where

f : R^{2} \to R

is a continuously differentiable function ( R is the set of all real numbers and

R^{2} = \{(x, y) : x, y \in R\})

Then which one of the following is TRUE?

A	$u = x^{2} + y^{2} + z, v = xz + y$	B	$u = x^{2} + y^{2} - z, v = xz - y$
C	$u = x^{2} - y^{2} + z, v = xz + y$	D	$u = x^{2} + y^{2} - z, v = xz - y$

Correct Answer : A

Question 32:

Consider the following statements: I. If Q denotes the additive group of rational numbers and

f : Q \to Q

Correct Answer : I, II and IV only

Question 33:

A solution to the Dirichlet problem

\nabla^{2} u (r, θ) = 0, 0 < r < 1, - π \leq θ \leq π, u (1, θ) = |θ|, - π \leq θ \leq π,

is given by

A	$u (r, θ) = \frac{π}{2} + \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n} - 1}{n^{2}}] r^{n} \cos (nθ)$	B	$u (r, θ) = \frac{2}{π} + \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n + 1}}{n^{2}}] r^{n} \cos (nθ)$
C	$u (r, θ) = \frac{π}{2} + \frac{2}{π} \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n} - 1}{n^{2}}] r^{n} \cos (nθ)$	D	$u (r, θ) = \frac{π}{2} + \frac{2}{π} \sum_{n = 1}^{\infty} [\frac{{(- 1)}^{n} + 1}{n^{2}}] r^{n} \cos (nθ)$

Correct Answer : C

Question 34:

Consider the subspace

Y = \{(x, x) : x \in C\}

of the normed linear space

(C^{2} {||||}_{\infty}) .

ϕ

is a bounded linear functional on Y, defined by

ϕ (x, x) = x,

, then which one of the following sets is equal to

\{ψ (1, 0) : ψ is a norm preserving extension of ϕ to (C^{2}, {||||}_{\infty})\}

( is the set of all complex numbers,

C^{2} = \{(x, y) : x, y \in C\}

and

{||(x_{1}, x \infty_{2})||}_{} = \sup \{|x_{1}|, |x_{2}|\})

A	$\{1\}$	B	$[\frac{1}{2}, \frac{3}{2}]$
C	$[1, \infty]$	D	$[0, 1]$

Correct Answer : D

Question 35:

Consider the following statements: I. The ring

Z [\sqrt{- 1}]

is a unique factorization domain. II. The ring

Z [\sqrt{- 5}]

is a principal ideal domain. III. In the polynomial ring

Z_{2} [x],

, the ideal generated by

x^{3} + x + 1

is a maximal ideal. IV. In the polynomial ring

Z_{3} [x]

, the ideal generated by

x^{6} + 1

is a prime ideal. ( Z denotes the set of all integers,

Z_{n}

denotes the set of all integers modulo

n

, for any positive integer n) Which of the above statements is TRUE?

Correct Answer : I and III only

Question 36:

Let M be a

3 \times 3

real symmetric matrix with eigenvalues 0, 2, and a with the respective eigenvectors

u = {(4, b, c)}^{τ}, v = {(- 1, 2, 0)}^{τ} and w = {(1, 1, 1)}^{τ} .

Consider the following statements: I.

a + b - c = 10

II. The vector

{(0, \frac{3}{2}, \frac{1}{2})}^{τ}

satisfies

Mx = v + x

. III. For any

d \in

span

\{u, v, w\}, M x = d

has a solution. IV. The trace of the matrix

M^{2} + 2 M is 8 .

Which of the above statements is TRUE?

Correct Answer : I and II only

Question 37:

Consider the region

Ω = \{x + i y : - 1 \leq x \leq 2, \frac{- π}{3} \leq y \leq \frac{π}{3}\}, i = \sqrt{- 1}

in the complex plane. The transformation

x + i y \mapsto e^{x + i y}

maps the region Ω onto the region

S \subset C

(the set of all complex numbers). Then the area of the region S is equal to

A	$\frac{π}{3} (e^{4} - e^{- 2})$	B	$\frac{π}{3} (e^{4} + e^{- 2})$
C	$\frac{2 π}{3} (e^{4} - e^{- 2})$	D	$\frac{π}{6} (e^{4} - e^{- 2})$

Correct Answer : A

Question 38:

Consider the sequence

{\{g_{n}\}}_{n = 1}

of functions, where

g_{n} (x) = \frac{x}{1 + n x^{2}}

x \in R, n \in N

and

g'_{n} (x)

is the derivative of

g_{n} (x)

with respect to x. (R is the set of all real numbers,N is the set of all-natural numbers). Then which one of the following statements is TRUE?

A	${\{g_{n}\}}_{n = 1}^{\infty}$ does NOT converge uniformly on R
B	${\{g'_{n}\}}_{n = 1}^{\infty}$ converges uniformly on any closed interval which does NOT contain 1
C	${\{g'_{n}\}}_{n = 1}^{\infty}$ converges point-wise to a continuous function on R
D	${\{g'_{n}\}}_{n = 1}^{\infty}$ converges uniformly on any closed interval which does NOT contain 0

Correct Answer : D

Question 39:

Consider the boundary value problem (BVP)

\frac{d^{2} y}{{dx}^{2}} + α y (x) = 0, α \in R (the set of all real numbers),

with the boundary conditions

y (0) = 0, y (π) = k (k is a non - zero real number) .

Then which one of the following statements is TRUE?

A	For $α = 1,$ the BVP has infinitely many solutions
B	For $α = 1$ , the BVP has a unique solution
C	For $α = - 1$ , the BVP has a solution y(x) such that $y (x) > 0$ for all $x \in (0, π)$
D	For $α = - 1, k < 0,$ , the BVP has a solution $y (x)$ such that $y (x) > 0$ for all $x \in (0, π)$

Correct Answer : D

Question 40:

Consider the ordered square

I_{0}^{2},

The set

[0, 1] \times [0, 1]

with the dictionary order topology. Let the general element of

I_{0}^{2}

be denoted by

x \times y

, where

x, y \in [0, 1]

. Then the closure of the subset

S = \{x \times \frac{3}{4} : 0 < α < x < b < 1\} in I_{0}^{2}

A	$S \cup ((a, b] \times \{0\} \cup ([a, b) \times \{1\})$	B	$S \cup ([a, b) \times \{0\} \cup (a, b] \times \{1\})$
C	$S \cup ((a, b) \times \{0\} \cup ((a, b) \times \{1\})$	D	$S \cup ((a, b] \times \{0\})$

Correct Answer : A

Question 41:

Let

P_{2}

be the vector space of all polynomials of degree at most 2 over R (the set of real numbers). Let a linear transformation

T : P_{2} \to P_{2}

be defined by

T (a + bx + {cx}^{2}) = (a + b) + (b - c) x + (a + c) x^{2} .

Consider the following statements: I. The null space of T is

\{a (+ x + x^{2}) : a \in R\}

. II. The range space of T is spanned by the set

\{1 + x^{2}, 1 + x\}

. III.

T (T (1 + X)) = 1 + X^{2}

. IV. If M is the matrix representation of T with respect to the standard basis

\{1, x, x^{2}\}

P_{2}

, then the trace of the matrix M is 3. Which of the above statements is TRUE?

Correct Answer : I and II only

Question 42:

Let

T_{1} and T_{2}

be two topologies defined on

N

(the set of all-natural numbers), where

T_{1}

is the topology generated by

B = \{\{2 n - 1, 2 n\} : n \in N\} and T_{2}

is the discrete topology on N. Consider the following statements : I. In

(N, T_{1})

, T every infinite subset has a limit point. II. The function

f : (N, T_{1}) \to (N, T_{2})

defined by

F (n) = \{\begin{cases} \frac{n}{2}, if n is even \\ \frac{n + 1}{2} if n isodd \end{cases}

is a continuous function. Which of the above statements is/are TRUE?

Correct Answer : Both I and II

Question 43:

Let

1 \leq p < q < \infty .

Consider the following statements: I.

l^{p} \subset l^{q}

II.

L^{p} [0, 1] \subset L^{q} [0, 1],

where

l^{p} = \{(x_{1}, x_{2}, . . .) : x_{i} \in R, \sum_{i = 1}^{\infty} {|x_{i}|}^{p} < \infty\} and

L^{p} [0, 1] = \{f : [0, 1] \to R : f is - measurable \int_{[0, 1]} {|f|}^{p} d μ < \infty, where is the Lebesgue measure\}

( R is the set of all real numbers) Which of the above statements is/are TRUE?

Correct Answer : I only

Question 44:

Consider the differential equation

t \frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} + t y = 0, t > 0, y (0 +) = 1 {(\frac{d y}{d t})}_{t = 0 +} = 0 .

If Y (s) is the Laplace transform of y(t), then the value of Y(1) is _____ (round off to 2 places of the decimal). (Here, the inverse trigonometric functions assume principal values only)

Correct Answer : 0.76 to 0.83

Question 45:

Let R be the region in the xy-plane bounded by the curves

y = x^{2}, y = 4 x^{2}, x y = 1 and xy = 5 .

Then the value of the integral

\int \int_{R} \frac{y^{2}}{x} d y d x

is equal to _________.

Correct Answer : 12 to 12

Question 46:

Let V be the vector space of all

3 \times 3

matrices with complex entries over the real field. If\

W_{1} = \{A \in V : A = A^{T}\} and W_{2} = \{A \in V : trace of A = 0\},

then the dimensions of

W_{1} + W_{2}

is equal to ________. (

A^{T}

denotes the conjugate transpose of A)

Correct Answer : 17 to 17

Question 47:

The number of elements of order 15 in the additive group

Z_{60} \times Z_{50}

IS __________. (

Z_{n}

denotes the group of integers modulo n, under the operation of addition modulo n, for any positive integer n)

Correct Answer : 48 to 48

Question 48:

Consider the following cost matrix of assigning four jobs to four persons:

Then the minimum cost of the assignment problem subject to the constraint that job

J_{4}

is assigned to person

P_{2}

, is _____.

Correct Answer : 27 to 27

Question 49:

Let

y : - [- 1, 1] \to R

with y(1) =1 satisfy the Legendre differential equation

(1 - x^{2}) \frac{d^{2} y}{{dx}^{2}} - 2 x \frac{dy}{dx} + 6 y = 0 for |x| < 1 .

Then the value of

\int_{- 1}^{1} y (x) (x + x^{2}) d x

is equal to _____ (round off to 2 places of the decimal).

Correct Answer : 0.25 to 0.30

Question 50:

The maximum value of the error term of the composite Trapezoidal rule when it is used to evaluate the definite integral

\int_{0.2}^{1.4} (\sin x - \log_{e} x) dx

with 12 subintervals of equal length, is equal to _____ (round off to 3 places of the decimal).

Correct Answer : 0.022 to 0.028

Question 51:

Let

Z_{125}

be the ring of integers modulo 125 under the operations of addition modulo 125 and multiplication modulo 125. If m is the number of maximal ideals of

Z_{125}

and n is the number of non-units of

Z_{125}

, then m + n is equal to _____.

Correct Answer : 26 to 26

Question 52:

By the Simplex method, the optimal table of the linear programming problem:

Maximize Z = {αx}_{1} + 3 x_{2} subject to {βx}_{1} + x_{2} + x_{3} = 8, 2 x_{1} + x_{2} + x_{4} = γ, x_{1} + x_{2} + x_{3} + x_{4} \geq 0,

where

α, β, γ

are real constants, is

Then the value

α + β + γ

is _____.

Correct Answer : 14.0 to 16.0 OR 17.0 to 18.5

Question 53:

Consider the inner product space

P_{2}

of all polynomials of degree at most 2 over the field of real numbers with the inner product

<f, g> = \int_{0}^{1} f (t) g (t) dt for f, g \in P_{2}

Let

\{f_{0}, f_{1}, f_{2}\}

be an orthogonal set in

P_{2}, where f_{0} = 1, f_{1} = t + c_{1}, f_{2} = t^{2} + c_{2} f_{1} + c_{3} and c_{1}, c_{2}, c_{3}

are real constants. Then the value of

2 c_{1} + c_{2} + 3 c_{3}

2 c_{1} + c_{2} + 3 c_{3}

is equal to _____.

Correct Answer : -3 to -3

Question 54:

Consider the system of linear differential equations

\frac{{dx}_{1}}{dt} = 5 x_{1} - 2 x, \frac{{dx}_{2}}{dt} = 4 x_{1} - x_{2},

with the initial conditions

x_{1} (0) = 0, x_{2} (0) = 1 .

Then

\log_{e} (x_{2} (2) - x_{1} (2))

is equal to _____.

Correct Answer : 1.95 to 2.05

Question 55:

Consider the differential equation

x (1 + x^{2}) \frac{d^{2} y}{{dx}^{2}} - 9 \frac{dy}{dx} + 7 y = 0 .

The sum of the roots of the indicial equation of the Frobenius series solution for the above differential equation in a neighborhood of x = 0 is equal to _____.

Correct Answer : 10 to 10

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Scrore Card
Total Question		Total Attempted
Correct Answers		Incorrect Answers
Score

A	$\|f (x) f'' (x)\| < {\|f' (x)\|}^{2}, f o r a l l x \in (a, b)$	B	$\|f (x) f' (x)\| < {\|f'' (x)\|}^{2}, f o r a l l x \in (a, b)$
C	$\|f' (x) f'' (x)\| < {\|f (x)\|}^{2}, f o r a l l x \in (a, b)$	D	$\|f (x) f'' (x)\| < \|f' (x)\|, f o r a l l x \in (a, b)$

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