Be Paitent While your exam is being loaded

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 complete 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 2020 Mock tests here online for free

View Solution

Participate to compete with other aspirants at an all India level.

Attempt Mock Test

Participate to compete with other aspirants at an all India level.

General Instructions

Choose Your Default Language

Please read the instructions carefully

General Instructions:

- Total duration of is 180 min.
The clock will be set at the server. The countdown timer in the top right corner of screen will display the remaining time available for you to complete the examination. When the timer reaches zero, the examination will end by itself. You will not be required to end or submit your examination.
The Questions Palette displayed on the right side of screen will show the status of each question using one of the following symbols:
1. You have not visited the question yet.
2. You have not answered the question.
3. You have answered the question.
4. You have NOT answered the question, but have marked the question for review.
5. The question(s) "Answered and Marked for Review" will be considered for evalution.
You can click on the ">" arrow which apperes to the left of question palette to collapse the question palette thereby maximizing the question window. To view the question palette again, you can click on "<" which appears on the right side of question window.
You can click on your "Profile" image on top right corner of your screen to change the language during the exam for entire question paper. On clicking of Profile image you will get a drop-down to change the question content to the desired language.
You can click on to navigate to the bottom and to navigate to top of the question are, without scrolling.

Navigating to a Question:

To answer a question, do the following:
1. Click on the question number in the Question Palette at the right of your screen to go to that numbered question directly. Note that using this option does NOT save your answer to the current question.
2. Click on Save & Next to save your answer for the current question and then go to the next question.
3. Click on Mark for Review & Next to save your answer for the current question, mark it for review, and then go to the next question.

Answering a Question:

Procedure for answering a multiple choice type question:
1. To select you answer, click on the button of one of the options.
2. To deselect your chosen answer, click on the button of the chosen option again or click on the Clear Response button
3. To change your chosen answer, click on the button of another option
4. To save your answer, you MUST click on the Save & Next button.
5. To mark the question for review, click on the Mark for Review & Next button.
To change your answer to a question that has already been answered, first select that question for answering and then follow the procedure for answering that type of question.

Navigating through sections:

Sections in this question paper are displayed on the top bar of the screen. Questions in a section can be viewed by click on the section name. The section you are currently viewing is highlighted.
After click the Save & Next button on the last question for a section, you will automatically be taken to the first question of the next section.
You can shuffle between sections and questions anything during the examination as per your convenience only during the time stipulated.
Candidate can view the corresponding section summery as part of the legend that appears in every section above the question palette.

Please note all questions will appear in your default language. This language can be changed for a particular question later on.

I have read and understood the instructions. All computer hardware allotted to me are in proper working condition. I declare that I am not in possession of / not wearing / not carrying any prohibited gadget like mobile phone, bluetooth devices etc. /any prohibited material with me into the Examination Hall.I agree that in case of not adhering to the instructions, I shall be liable to be debarred from this Test and/or to disciplinary action, which may include ban from future Tests / Examinations

Proceed

Category: GATE Mathematics 2020 General Aptitude

Question 1:

Rajiv Gandhi Khel Ratna Award was conferred _____________ Mary Kom, a six-time world champion in boxing, recently in a ceremony _____________- the Rashtrapati Bhawan (the persident's official residence) in New Delhi.

with, at

on, in

on, at

to, at

Category: GATE Mathematics 2020 General Aptitude

Question 2:

Despite a string of poor performances, the chances of K. L. Rahul's selection in the team are _________.

slim

bright

obvious

uncertain

Category: GATE Mathematics 2020 General Aptitude

Question 3:

Select the word that fits the analogy: COver: Uncover: :Associate: ____________

Unassociate

Inassociate

Misassociate

Dissociate

Category: GATE Mathematics 2020 General Aptitude

Question 4:

Hit by floods, the Kharif (summer-sown) crops in various parts of the country have been affected. Officials believe that the loss of production of the Kharif crops can be recovered in the output of the rabi (winter-sown) crops so that the country can achieve its food-grain production target of 291 million tons in the crop year 2019-20 (July-June). They are hopeful that good rains in July-AUgust will help the soil retain moisture for a longer period, helping winter-sown crops such as wheat and pulses during the November-February period. Which of the following statements can be inferred from the given passage?

officials declared that the food-grain production target will be met due to goods rains.

Officials want the food-grain production target to be met by the November-February perood.

Officials feel that the food-grain production target cannot be met due to floods.

Officials hope that the food-grain production target will be met due to a good rabi produce.

Category: GATE Mathematics 2020 General Aptitude

Question 5:

The difference between the sum of the first 2n natural numbers and the sum of the first odd natural numbers is ________.

A	$n^{2} - n$	B	$n^{2} + n$
C	$2 n^{2} - n$	D	$2 n^{2} + n$

Category: GATE Mathematics 2020 General Aptitude

Question 6:

Repo rate is the rate at which the Reserve Bank of India (RBI) lends commercial banks and the reverse repo rate at which RBI borrows money from commercial banks. Which of the following statements can be inferred from the above passage?

Decrease in repo will increase cost of borrowing and decrease lending by commerical banks.

Increase in repo will decrease cost of borrowing and increase lending by commerical banks.

Increase in repo will decrease cost of borrowing and decrease lending by commerical banks.

Decrease in repo will decrease cost of borrowing and increase lending by commerical banks.

Category: GATE Mathematics 2020 General Aptitude

Question 7:

P, Q, R, S, T, U, V, and W seated around a circular table. I. S is seated opposite W. II. U is seated at the second place to the right of R. III. T is seated at the third place to the left of R. IV. V is a neighbour of S. Which of the following must be true?

P is a neighbour of R.

Q is a neighbour of R.

P is not seated opposite to Q.

R is not seated opposite to Q.

Category: GATE Mathematics 2020 General Aptitude

Question 8:

The distance between Delhi and Agra is 233 km. A car P started traveling from Delhi to Agra and another car Q started from Agra to Delhi along the same road 1 hour after the car P started. The cars crossed each other 75 minutes after the car Q started. Both cars were traveling at a constant speed. The speed of car P was 10 km/hr more than the speed of car Q. How many kilometers the car Q had traveled when the cars crossed each other?

66.6

75.2

88.2

116.5

Category: GATE Mathematics 2020 General Aptitude

Question 9:

For a matrix

M = [m_{ij}]; i, j = 1, 2, 3, 4,

the diagonal elements are all zero and

m_{ij} = - m_{ji} .

The minimum number of elements required to fully specify the matrix is _______.

Category: GATE Mathematics 2020 General Aptitude

Question 10:

The profit shares of two companies P and Q are shown in the figure. If the two companies have invested a fixed and equal amount every year, then the ratio of the total revenue of company P to the total revenue of company Q, during 2013-2018 is _______.

15: 17

16: 17

17: 15

17: 16

Category: GATE Mathematics 2020

Question 11:

Suppose that

I_{1} and I_{2}

are topologies on X induced by metrics

d_{1} and d_{2}

, respectively,

I_{1} \underline{\subset} I_{2}

. Then which of the following is TRUE?

A	If a sequence converges in $(X, d_{2})$ then it convergers in $(X, d_{1})$
B	If a sequence converges in $(X, d_{1})$ then it convergers in $(X, d_{2})$
C	Every open ball in $(X, d_{1})$ is an open ball in $(X, d_{2})$
D	The map $x \mapsto x from (X, d_{1}) to (X, d_{2})$ is continuous

Category: GATE Mathematics 2020

Question 12:

Let

D = [- 1, 1] \times [- 1, 1] .

If the function

f : D \to ℝ

is defined by

f (x, y) = \{\begin{cases} \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases},

A	f is continuous at (0, 0)
B	both the first-order partial derivatives of f exist at (0, 0)
C	$\int \int_{D} {\|f (x, y)\|}^{\frac{1}{2}} dx dy$ is finite
D	$\int \int_{D} \|f (x, y)\| dx dy$ is finite

Category: GATE Mathematics 2020

Question 13:

The initial value problem

y' = y^{\frac{3}{5}}, y (0) = b

has

a unique solution if b=0

no solution if b=1

infinitely many solutions if b=2

a unique solution if b=1

Category: GATE Mathematics 2020

Question 14:

Consider the following statements: I:

\log (|z|)

is harmonic on

ℂ \ \{0\}

II.

\log (|z|)

has a harmonic conjugate on

ℂ \ \{0\}

Then

both I and II are true

I is the true but II is false

I is the false but II is true

both I and II are false

Category: GATE Mathematics 2020

Question 15:

Let G and H be defined by

G = ℂ \ \{z = x + iy \in ℂ : x ⩽ 0, y = 0\}, H = ℂ \ \{z = x + iy \in ℂ : x \in ℤ, x \leq 0, y = 0\} .

Suppose

f : G \to ℂ and g : H \to ℂ

are analytic functions. Consider the following statements: I:

\int_{y} f dz

is independent of paths y in G joining -i and i II:

\int_{y} g dz

is independent of paths y in H joining -i and i Then

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

Category: GATE Mathematics 2020

Question 16:

Let

f (z) = e^{1 / z}, z \in ℂ \ \{0\}

and let, for

n \in ℕ,

R_{n} = \{z = x + iy \in ℂ : |x| < \frac{1}{n}, |y| < \frac{1}{n}\} \ \{0\} .

A	$f (R_{n + 1}) \neq f (R_{n})$	B	$f (R_{n}) \ f (R_{n + 1}) = f (R_{n} \ R_{n + 1})$
C	$f (\cap_{n = 1}^{\infty} R_{n}) = \cap_{n = 1}^{\infty} f (R_{n})$	D	$f (R_{n}) = f (R_{n + 1})$

A and B

B and C

C and D

D and A

Category: GATE Mathematics 2020

Question 17:

Suppose that

U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x, y \in ℚ\}, U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x > 0, y \frac{1}{x}\}

Then with respect to the Euclidean metric on

ℝ^{2},

both U and V are disconnected

u is disconnected but V is connected

u is connected but V is disconnected

both U and V are connected

Category: GATE Mathematics 2020

Question 18:

If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and (P2), respectively, where (P1): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1}, x_{2} \geq 0,

(P2): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1} - x_{2} = 10, x_{1}, x_{2} \geq 0,

then

both (D1) and (D2) are infeasible

(P2) is infeasible and (D2) is feasible

(D1) is infeasible and (D2) is feasible but unbounded

(P1) is feasible but unbounded and (D1) is feasible

Category: GATE Mathematics 2020

Question 19:

(4, 0) and (0, - \frac{1}{2})

are critical points of the function

f (x, y) = 5 - (α - β) x^{2} + {βy}^{2} + (α + 1) y^{3} + x^{3},

where

α, β \in ℝ,

then

A	$(4, - \frac{1}{2})$ is a point of local maxima of f
B	$(4, - \frac{1}{2})$ is a saddle point of f
D	$α = 4, β = 2$
C	$(4, - \frac{1}{2})$ is a point of local maxima of f

Category: GATE Mathematics 2020

Question 20:

Consider the iterative scheme

x_{n} = \frac{x_{n - 1}}{2} + \frac{3}{x_{n - 1}}, n \geq 1 .

with initial point

x_{0} > 0 .

Then the sequence

\{x_{n}\}

A	converges only if $x_{0} > 1$
B	converges only if $x_{0} < 3$
C	converges for any $x_{0}$
D	does not converge for any $x_{0}$

Category: GATE Mathematics 2020

Question 21:

Let C[0, 1] denote the space of all real-valued continuous functions on [0,1] equipped with the supremum norm

| | . | |_{\infty} .

Let

T : C [0, 1] \to C [0, 1]

be the linear operator defined by

T (f) (x) = \int_{0}^{x} e^{- y} f (y) dy .

Then

A	$\| \| T \| \| = 1$	B	I-T is not invertible
C	T is subjective	D	$\| \| I + T \| \| = 1 + \| \| T \| \|$

Category: GATE Mathematics 2020

Question 22:

Suppose that M is a

5 \times 5

matrix with real entries and p(x) = diet (xI-M). Then

A	$p (0) = \det (M)$
B	every eigenvalue of M is real if p(1) + p(2)= 0 = p(2) + p(3)
C	$M^{- 1}$ is necessarily a polynomial in M of degree 4 if M is invertible
D	M is not invertible if $M^{2} - 2 M = 0$

Category: GATE Mathematics 2020

Question 23:

Let C [0, 1] denote the space of all real-valued continuous function on [0, 1] equipped with the supremum norm

| | . | |_{\infty} .

Let

f \in C [0, 1]

be such that

|f (x) - f (y)| \leq M |x - y|, for all x, y \in [0, 1] and for some M > 0 .

For

n \in ℕ, let f_{n} (x) = f (x^{1 + \frac{1}{n}}) . If S = \{f_{n} : n \in ℕ\},

then

the closure of S is compact

S is closed and bounded

S is bounded but not totally bounded

S is compact

Category: GATE Mathematics 2020

Question 24:

Let

K : ℝ \times (0, \infty) \to ℝ

be a function such that the solution of the initial value problem

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}}, u (x, 0) = f (x), x \in ℝ, t > 0,

is given by

u (x, t) = \int_{ℝ} k (x - y, t) f (y) dy

for all bounded continuous functions f. Then the value of

\int_{ℝ} k (x, t) dx

is ____________

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2020

Question 25:

The number of cyclic subgroups of the quaternion group

Q_{8} = < a, b | a^{4} = 1, a^{2} = b^{2}, ba = a^{3} b >

is ______________.

1 to 1

3 to 3

5 to 5

7 to 7

Category: GATE Mathematics 2020

Question 26:

The number of elements of order 3 in the symmetric group

S_{6}

is __________

80 to 80

81 to 81

82 to 82

83 to 83

Category: GATE Mathematics 2020

Question 27:

Let F be the field with 4096 elements. The number of proper subfields of F is ____________

5 to 5

5 to 6

5 to 7

5 to 8

Category: GATE Mathematics 2020

Question 28:

(x_{1}^{*}, x_{2}^{*})

is an optimal solution of the linear programming problem,

minimize x_{1} + 2 x_{2} subject to 4 x_{1} - x_{2} \geq 8 2 x_{1} + x_{2} \geq 10 - x_{1} + x_{2} \geq 0

and

(λ_{1}^{*}, λ_{2}^{*}, λ_{3}^{*})

is an optimal solution of its dual problem, then

\sum_{i = 1}^{2} x_{i}^{*} + \sum_{j = 1}^{3} λ_{j}^{*}

is equal to ____________ (correct up to one decimal place)

3.5 to 3.5

4.5 to 4.5

5.5 to 5.5

6.5 to 6.5

Category: GATE Mathematics 2020

Question 29:

Let a, b, c,

\in ℝ

be such that the quadrature rule

\int_{- 1}^{1} f (x) dx \approx af (- 1) + bf (0) + cf' (1)

is exact for all polynomials of degree less than or equal to 2. Then b is equal to __________ (rounded off to two decimal places)

1.40 to 1.50

1.50 to 1.60

1.60 to 1.70

1.70 to 1.80

Category: GATE Mathematics 2020

Question 30:

Let

f (x) = x^{4}

and let p(x) be the interpolating polynomial of f at nodes 1, 2, and 3. Then p(0) is equal to _______

35 to 35

36 to 36

37 to 37

38 to 38

Category: GATE Mathematics 2020

Question 31:

For

n \geq 2,

define the sequence

\{x_{n}\}

x_{n} = \frac{1}{2 π} \int_{0}^{\frac{π}{2}} \tan^{\frac{1}{n}} t dt .

Then the sequence

\{x_{n}\}

converges to _______________ (correct up to two decimal places)

0.25 to 0.25

0.45 to 0.45

0.65 to 0.65

0.85 to 0.85

Category: GATE Mathematics 2020

Question 32:

Let

L^{2} [0, 10] = \{f : [0, 10] \to ℝ : f is Lebesgue measurable and \int_{0}^{10} f^{2} dx < \infty\}

equipped with the norm

| | f | | = {(\int_{0}^{10} f^{2} dx)}^{\frac{1}{2}}

and let T be the linear functional on

L^{2} [0, 10]

given by

T (f) = \int_{0}^{2} f (x) dx - \int_{3}^{10} f (x) dx .

Then ||T|| is equal to ____________

0 to 0

1 to 1

2 to 2

3 to 3

Category: GATE Mathematics 2020

Question 33:

\{x_{13}, x_{22}, x_{23} = 10, x_{31}, x_{32}, x_{34}\}

is the set of basic variables of a balanced transportation problem seeking to minimize the cost of transportation from origins to destinations, where the cost matrix is,

and

λ, μ \in ℝ,

then

x_{32}

is equal to _________

0 to 5

1 to 5

3 to 5

5 to 5

Category: GATE Mathematics 2020

Question 34:

Let

ℤ_{225}

be the ring of integers modulo 225. If x is the number of prime and y is the number of nontrivial units in

ℤ_{225}

, then x + y is equal to ____

121 to 121

122 to 122

123 to 123

124 to 124

Category: GATE Mathematics 2020

Question 35:

Let u(x, t) be the solution of

\frac{\partial^{2} u}{\partial t} - \frac{\partial^{2} u}{\partial x^{2}} = 0, u (x, 0) = f (x), \frac{\partial u}{\partial t} (x, 0) = 0, x \in ℝ, t > 0,

where is a twice continuously differentiable function. If f(-2) =4, f (0) =0, and u (2, 2) = 8, then value of u (1, 3) is _________

9 to 9

10 to 10

11 to 11

12 to 12

Category: GATE Mathematics 2020

Question 36:

Let

{[e_{n}]}_{n = 1}^{\infty}

be an orthonormal basis for a parable Hilbert space H with the inner product <. ,. > Define

f_{n} = e_{n} - \frac{1}{n + 1} e_{n + 1} for n \in ℕ .

Then

A	the closure of the span $\{f_{n} : n \in ℕ\}$	B	$f = 0 if < f, f_{n} > = < f, e_{n} > for all n \in ℕ$
C	${\{f_{n}\}}_{n = 1}^{\infty}$ is an orthogonal sunset of H	D	there does exist nonzero $f \in H such that < f, e_{2} > = < f, f_{2} >$

Category: GATE Mathematics 2020

Question 37:

Suppose V is a finite-dimensional non-zero vector space over

ℂ and T : V \to V

is a linear transformation such that Range (T)=Nullspace (T). Then when of the following statements is FALSE?

A	The dimension of V is even	B	0 is the only eigenvalue of T
C	Both 0 and 1 are eigenvalues of T	D	$T^{2} = 0$

Category: GATE Mathematics 2020

Question 38:

Let

P \in M_{m \times n} (ℝ) .

Consider the following statement: I: If

XPY = 0 for all X \in M_{1 \times m} (ℝ) and M_{n \times 1} (ℝ), then P = 0 .

II. If

m = n, P is symmetric and P^{2} = 0, then P = 0 .

Then

both I and II are true

I is true but II is false

I is false but ii is true

both I and II are false

Category: GATE Mathematics 2020

Question 39:

For

n \in ℕ, let T_{n} : (l^{1}, | | . | |) \to (l^{\infty}, | | . | |) and T : (l^{\infty}, | | . | |_{1}) \to (l^{\infty}, | | . | |_{\infty})

be that bounded linear operators defined by

T_{n} (x_{1}, x_{2}, . .) = (y_{1}, y_{2} . . .), where y_{j} = \{\begin{cases} x_{j}, j \geq n \\ x_{n}, j > n \end{cases}

and

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

A	$\| \| T \| \| does not converge to \| \| T \| \| as n \to \infty$	B	$\| \| T_{n} - T \| converge to zero n \to \infty$
C	$for all x \in l^{1}, \| \| T_{n} (x) - T (x) \| \| converge to zero as n \to \infty$	D	for each non-zero $x \in l^{1},$ exists a continuous linear functional g on $l^{\infty}$ such that $g (T_{n} (x))$ does not $g (T (x)) ad n \to \infty$

Category: GATE Mathematics 2020

Question 40:

Let

P (ℝ)

denote the power set of

ℝ,

equipped with the metric

d (U, V) = \sup_{x \in ℝ} |x_{U} (x) - X_{V} (x)|,

where

X_{U} and X_{V}

denote the characteristic functions of the subsets U and V, respectively, of

ℝ .

The set

\{\{m\} : m \in ℤ\}

in the metric space

(P (ℝ), d)

bounded but not totally bounded

totally bounded but not compact

compact

not bounded

Category: GATE Mathematics 2020

Question 41:

Let

f : ℝ \to ℝ

be defined by

f (x) = \sum_{n = 0}^{\infty} \frac{1}{2^{n}} X_{(n, n + 1]} (x),

Where

X_{(n, n + 1]}

is the characteristic function of the interval

(n, n + 1] . For α \in ℝ, let S_{α} = \{x \in ℝ : f (x) > α\} . Then

A	$S_{\frac{1}{2}}$ is open	B	$S_{\frac{\sqrt{5}}{2}}$ is not measurable
C	$S_{0}$ is closed	D	$S_{\frac{1}{\sqrt{2}}}$ is measurable

Category: GATE Mathematics 2020

Question 42:

For

n, \in ℕ, let f_{n}, g_{n} : (0, 1) \to ℝ

be, a function defined by

f_{n} (x) = x^{n} and g_{n} (x) = x^{n} (1 - x) .

Then

A	$\{f_{n}\}$ converges uniformly but $\{g_{n}\}$ does not converge uniformly	B	$\{g_{n}\}$ converges uniformly but $\{f_{n}\}$ does not converge uniformly
C	both $\{f_{n}\} and \{g_{n}\}$ converge uniformly	D	neither $\{f_{n}\} and \{g_{n}\}$ converge uniformly

Category: GATE Mathematics 2020

Question 43:

Let U be a solution of the differential equation

y' + xy = 0 and let ϕ = uψ

be a solution of the differential equation

y'' + 2 xy' + (x^{2} + 2) y = 0 satisfying ϕ (0) = 1 and ϕ' (0) = 0 . Then ϕ (x) is

A	$(\cos^{2} x) e^{- \frac{x^{2}}{2}}$	B	$(\cos x) e^{- \frac{x^{2}}{2}}$
C	$(1 + x^{2}) e^{- \frac{x^{2}}{2}}$	D	$(\cos x) e^{- x^{2}}$

Category: GATE Mathematics 2020

Question 44:

For

n \in ℕ \{0\}, let y_{n}

be a solution of the differential equation

xy'' + (1 - x) y' + n y = 0

Satisfying

y_{n} (0) = 1 .

For which of the following functions w(x), the integral

\int_{0}^{\infty} y_{p} (x) y_{q} (x) w (x) dx, (p \neq q)

is equal to zero?

A	$e^{- x^{2}}$	B	$e^{- x}$
C	${xe}^{- x^{2}}$	D	${xe}^{- x}$

Category: GATE Mathematics 2020

Question 45:

Suppose that

X = \{(0, 0)\} U \{(x, \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

and

Y = \{(0, 0)\} U \{(x, x \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

are metric spaces with metrics induced by the Euclidean metric of

ℝ^{2} .

Let

B_{X}

and

B_{Y}

be the open unit balls around (0, 0) in X and Y, respectively. Consider the following statement: I: The closure of

B_{X}

in X is compact II. The closure of

B_{y}

in Y is compact Then

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

Category: GATE Mathematics 2020

Question 46:

f : ℂ \ \{0\} \to ℂ

is a function such that

f (z) = f (\frac{z}{|z|})

and its restriction to the unit circle is continuous, then

A	f is continuous but not necessarily analytic
B	f is continuous but not necessarily a constant function
C	f is a constant function
D	$\lim_{z \to 0} f (z)$ exists

Category: GATE Mathematics 2020

Question 47:

For a subset S of a topological space, let Int (S) and

S

denote the interior and closure of S, respectively. Then which of the following statement is TRUE?

A	If S is open, then S = Int $(S)$	B	If the boundary of S is empty, then S is open
C	If the boundary of S is empty, then S is not closed	D	If $S \ S$ is a proper subset of the boundary of S, then S is open

Category: GATE Mathematics 2020

Question 48:

Suppose

I_{1}, I_{2} and I_{3}

are the smallest topologies on

ℝ containing S_{1}, S_{2} and S_{3}, respectively, where

S_{1} = \{(α, α + \frac{π}{n}) : α \in ℚ, n \in ℕ\}, S_{2} = \{(a, b) : a < b, a, b \in ℚ\}, S_{3} = \{(a, b) : a < b, a, b \in ℝ\} .

A	$I_{3} \underline{\supset} I_{1}$	B	$I_{3} \underline{\supset} I_{2}$
C	$I_{1} = I_{2}$	D	$I_{1} \underline{\supset} I_{2}$

Category: GATE Mathematics 2020

Question 49:

Let

M = [\begin{matrix} α & 3 & 0 \\ β & 3 & 1 \\ 0 & 1 & 2 \end{matrix}]

. Consider the following statements: I: There exists a lower triangular matrix L such that M =

{LL}^{t}, where L^{t}

denotes the transpose of L. II: Gauss-Seidel method for

Mx = b (b \in ℝ^{3})

converges for any initial choice

x_{0} \in ℝ^{3} .

Then

A	I is not true when $α > \frac{9}{2}, β = 3$	B	II is not true when $α > \frac{9}{2}, β = - 1$
C	II is not true when $α = 4, β = \frac{3}{2}$	D	$α = 5, β = 3$

Category: GATE Mathematics 2020

Question 50:

Let I and J be the ideals generated by

\{5 . \sqrt{10}\} and \{4, \sqrt{10}\}

in the ring

ℤ [\sqrt{10}] = \{a + b \sqrt{10} | α, b \in ℤ\},

respectively. Then

both I and J are maximal ideals

I is a maximal ideal but J is not a prime ideal

I is not a maximal ideal but J is not a prime ideal

neither I and J are maximal ideals

Category: GATE Mathematics 2020

Question 51:

Suppose V is finite-dimensional vector space over

ℝ . If W_{1}, W_{2}, and W_{3}

are subspaces of V, then which of the following statements is TRUE?

A	If $W_{1} + W_{2} + W_{3} = V then span (W_{1} \cup W_{2}) \cup span (W_{2} \cup W_{3}) \cup span (W_{3} \cup W_{1}) = V$
B	$If W_{1} \cap W_{2} = \{0\} and W_{1} \cap W_{3} = \{0\}, then W_{1} \cap (W_{2} + W_{3}) = \{0\}$
D	If $W_{1} + W_{2} = W_{1} + W_{3}, then W_{2} = W_{3}$
C	$If W_{1} \neq V, then span (V \ W_{1}) = V$

Category: GATE Mathematics 2020

Question 52:

Let

α, β \in ℝ, α \neq 0 . The system

x_{1} - 2 x_{2} + {αx}_{3} = 8 x_{1} - x_{2} + x_{4} = β x_{1}, x_{2}, x_{3}, x_{4} \geq 0

has NO basic feasible solution if

A	$α < 0, β > 8$	B	$α > 0, 0 < β < 8$
C	$α > 0, β < 0$	D	$α < 0, β < 8$

Category: GATE Mathematics 2020

Question 53:

Let

0 < p < 1 and let

X = \{f : ℝ \to ℝ is continuous and \int_{ℝ} {|f (x)|}^{p} dx < \infty\} .

For

f \in X,

define

{|f|}_{p} = {(\int_{ℝ} {|f (x)|}^{p} dx)}^{\frac{1}{p}}

Then

A	${\|.\|}_{p}$ defines a norm on X	B	${\|f + g\|}_{p} \leq {\|f\|}_{p} + {\|g\|}_{p} for all f, g \in X$
C	${\|f + g\|}_{p}^{p} \leq {\|f\|}_{p}^{p} + {\|g\|}_{p}^{p} for all f, g \in X$	D	$if f_{n} converges to f ponitwise on ℝ, then \lim_{n \to \infty} \|f_{n}\| = {\|f\|}_{p}$

Category: GATE Mathematics 2020

Question 54:

Suppose that

ϕ and ϕ_{2}

are linearly independent solutions of the differential equation

2 x^{2} y'' - (x + x^{2}) y' + (x^{2} - 2) y = 0,

and

ϕ (0) = 0 .

Then the smallest positive integer n such that

\lim_{x \to 0} x^{n} \frac{ϕ_{2} (x)}{ϕ_{1} (x)} = 0

is ___________.

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2020

Question 55:

Suppose that

f (z) = \prod_{n = 1}^{17} (z - \frac{π}{n}), z \in ℂ and y (t) = e^{2 it}, t \in [0, 2 π] . If

\int_{Y} \frac{f' (z)}{f (z)} dz = απi,

then the value

α

is equal to _______

55 to 55

56 to 56

57 to 57

58 to 58

Category: GATE Mathematics 2020

Question 56:

y (t) = \frac{1}{2} e^{3 πi t}, t \in [0, 2] and

\int_{y} \frac{1}{z^{2} (e^{z} - 1)} dz = βπi,

then

β

is equal to __________ (correct up to one decimal place)

0.5 to 0.5

0.6 to 0.6

0.7 to 0.7

0.8 to 0.8

Category: GATE Mathematics 2020

Question 57:

Let

K = ℚ (\sqrt{3 + 2 \sqrt{2}}, ω), where ω

is a primitive cube root of unity. Then the degree of extension of K over

ℚ

is ___________

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Mathematics 2020

Question 58:

Let

α \in ℝ . If (3, 0, 0, β)

is an optimal solution to the linear programming problem

minimize x_{1} + x_{2} + x_{3} - {αx}_{4} subject to 2 x_{1} - x_{2} + x_{3} = 6 - x_{1} + x_{2} + x_{4} = 3 x_{1}, x_{2}, x_{3}, x_{4} \geq 0

then the maximum value of

β - α

is _______________

3 to 3

5 to 5

7 to 7

9 to 9

Category: GATE Mathematics 2020

Question 59:

Suppose that

T : ℝ^{4} \to ℝ [x]

is a linear transformation over

ℝ

satisfying

T (- 1, 1, 1, 1) = x^{2} + 2 x^{4}, T (1, 2, 3, 4) = 1 - x^{2}, T (2, - 1, - 1, 0) = x^{3} - x^{4} .

Then the coefficient of

x^{4} in T (- 3, 5, 6, 6)

is _________________

1 to 1

3 to 3

5 to 5

7 to 7

Category: GATE Mathematics 2020

Question 60:

Let

\vec{F} (x, y, z) = (2 x - 2 y \cos x) \hat{i} + (2 y - y^{2} \sin x) \hat{j} + 4 z \hat{k}

and let S be the surface of the tetrahedron bounded by the planes

x = 0, y = 0, z = 0 and x + y + z = 1 .

\hat{n}

is the unit outward normal to the tetrahedron, then the value of

\int \int_{S} \vec{F} . \hat{n} dS

is ______________ (rounded off to two decimal places)

1.00 to 1.10

1.10 to 1.20

1.20 to 1.30

1.30 to 1.40

Category: GATE Mathematics 2020

Question 61:

Let

\vec{F} = (x + 2 y) e^{z} \hat{i} + ({ye}^{z} + x^{2}) \hat{j} + y^{2} z \hat{k}

and let S be the surface

x^{2} + y^{2} + z = 1, z \geq 0 .

\hat{n}

is a unit normal to S and

|\int \int_{S} (\nabla \times \vec{F}) . \hat{n} dS| = απ

Then

α is equal to_______

2 to 2

4 to 4

6 to 6

8 to 8

Category: GATE Mathematics 2020

Question 62:

Let G be a non-cyclic group of order 57. Then the number of elements of order 3 is G is _______________

37 to 37

38 to 38

39 to 39

40 to 40

Category: GATE Mathematics 2020

Question 63:

The coefficient of

{(x - 1)}^{5}

in the Taylor expansion about x = 1 of the function

$F (x) = \int_{1}^{x} \frac{\log_{e} t}{t - 1} dt, 0 < x < 2$

is ___________ (correct uo to two decimal places)

0.01 to 0.04

0.02 to 0.04

0.03 to 0.04

0.04 to 0.04

Category: GATE Mathematics 2020

Question 64:

Let u(x, y) be the solution to the initial value problem

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

Then the value u(0, 1) is ___________ (rounded off to three decimal places)

1.610 to 1.625

1.620 to 1.635

1.630 to 1.645

1.640 to 1.655

Category: GATE Mathematics 2020

Question 65:

The value of

\lim_{n \to \infty} \int_{0}^{1} {nx}^{n} {ex}^{n} e^{x^{2}} dx

is ____________ (rounded off to three decimal places)

2.700 to 2.715

2.710 to 2.725

2.720 to 2.735

2.730 to 2.745

⮜ Back Next ⮞

Not Visited	0	Not Answered
Answered	0	Marked for Review
Answered & Marked for Review (will be considered for evaluation)

Question 1:

Correct Answer : on, at

Question 2:

Despite a string of poor performances, the chances of K. L. Rahul's selection in the team are _________.

Correct Answer : bright

Question 3:

Select the word that fits the analogy: COver: Uncover: :Associate: ____________

Correct Answer : Dissociate

Question 4:

Correct Answer : Officials hope that the food-grain production target will be met due to a good rabi produce.

Question 5:

The difference between the sum of the first 2n natural numbers and the sum of the first odd natural numbers is ________.

A	$n^{2} - n$	B	$n^{2} + n$
C	$2 n^{2} - n$	D	$2 n^{2} + n$

Correct Answer : B

Question 6:

Correct Answer : Decrease in repo will decrease cost of borrowing and increase lending by commerical banks.

Question 7:

Correct Answer : P is not seated opposite to Q.

Question 8:

Correct Answer : 75.2

Question 9:

For a matrix

M = [m_{ij}]; i, j = 1, 2, 3, 4,

the diagonal elements are all zero and

m_{ij} = - m_{ji} .

The minimum number of elements required to fully specify the matrix is _______.

Correct Answer : 6

Question 10:

Correct Answer : 16: 17

Question 11:

Suppose that

I_{1} and I_{2}

are topologies on X induced by metrics

d_{1} and d_{2}

, respectively,

I_{1} \underline{\subset} I_{2}

. Then which of the following is TRUE?

A	If a sequence converges in $(X, d_{2})$ then it convergers in $(X, d_{1})$
B	If a sequence converges in $(X, d_{1})$ then it convergers in $(X, d_{2})$
C	Every open ball in $(X, d_{1})$ is an open ball in $(X, d_{2})$
D	The map $x \mapsto x from (X, d_{1}) to (X, d_{2})$ is continuous

Correct Answer : A

Question 12:

Let

D = [- 1, 1] \times [- 1, 1] .

If the function

f : D \to ℝ

is defined by

f (x, y) = \{\begin{cases} \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}}, (x, y) \neq (0, 0) \\ 0, (x, y) = (0, 0) \end{cases},

A	f is continuous at (0, 0)
B	both the first-order partial derivatives of f exist at (0, 0)
C	$\int \int_{D} {\|f (x, y)\|}^{\frac{1}{2}} dx dy$ is finite
D	$\int \int_{D} \|f (x, y)\| dx dy$ is finite

Correct Answer : C

Question 13:

The initial value problem

y' = y^{\frac{3}{5}}, y (0) = b

has

Correct Answer : a unique solution if b=1

Question 14:

Consider the following statements: I:

\log (|z|)

is harmonic on

ℂ \ \{0\}

II.

\log (|z|)

has a harmonic conjugate on

ℂ \ \{0\}

Then

Correct Answer : I is the true but II is false

Question 15:

Let G and H be defined by

G = ℂ \ \{z = x + iy \in ℂ : x ⩽ 0, y = 0\}, H = ℂ \ \{z = x + iy \in ℂ : x \in ℤ, x \leq 0, y = 0\} .

Suppose

f : G \to ℂ and g : H \to ℂ

are analytic functions. Consider the following statements: I:

\int_{y} f dz

is independent of paths y in G joining -i and i II:

\int_{y} g dz

is independent of paths y in H joining -i and i Then

Correct Answer : I is true but II is false

Question 16:

Let

f (z) = e^{1 / z}, z \in ℂ \ \{0\}

and let, for

n \in ℕ,

R_{n} = \{z = x + iy \in ℂ : |x| < \frac{1}{n}, |y| < \frac{1}{n}\} \ \{0\} .

A	$f (R_{n + 1}) \neq f (R_{n})$	B	$f (R_{n}) \ f (R_{n + 1}) = f (R_{n} \ R_{n + 1})$
C	$f (\cap_{n = 1}^{\infty} R_{n}) = \cap_{n = 1}^{\infty} f (R_{n})$	D	$f (R_{n}) = f (R_{n + 1})$

Correct Answer : D and A

Question 17:

Suppose that

U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x, y \in ℚ\}, U = ℝ^{2} \ \{(x, y) \in ℝ^{2} : x > 0, y \frac{1}{x}\}

Then with respect to the Euclidean metric on

ℝ^{2},

Correct Answer : u is connected but V is disconnected

Question 18:

If (D1) and (D2) denote the dual problems of the linear programming problems (P1) and (P2), respectively, where (P1): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1}, x_{2} \geq 0,

(P2): minimize

x_{1} - 2 x_{2} subject to - x_{1} + x_{2} = 10, x_{1} - x_{2} = 10, x_{1}, x_{2} \geq 0,

then

Correct Answer : both (D1) and (D2) are infeasible

Question 19:

(4, 0) and (0, - \frac{1}{2})

are critical points of the function

f (x, y) = 5 - (α - β) x^{2} + {βy}^{2} + (α + 1) y^{3} + x^{3},

where

α, β \in ℝ,

then

A	$(4, - \frac{1}{2})$ is a point of local maxima of f
B	$(4, - \frac{1}{2})$ is a saddle point of f
D	$α = 4, β = 2$
C	$(4, - \frac{1}{2})$ is a point of local maxima of f

Correct Answer : B

Question 20:

Consider the iterative scheme

x_{n} = \frac{x_{n - 1}}{2} + \frac{3}{x_{n - 1}}, n \geq 1 .

with initial point

x_{0} > 0 .

Then the sequence

\{x_{n}\}

A	converges only if $x_{0} > 1$
B	converges only if $x_{0} < 3$
C	converges for any $x_{0}$
D	does not converge for any $x_{0}$

Correct Answer : C

Question 21:

Let C[0, 1] denote the space of all real-valued continuous functions on [0,1] equipped with the supremum norm

| | . | |_{\infty} .

Let

T : C [0, 1] \to C [0, 1]

be the linear operator defined by

T (f) (x) = \int_{0}^{x} e^{- y} f (y) dy .

Then

A	$\| \| T \| \| = 1$	B	I-T is not invertible
C	T is subjective	D	$\| \| I + T \| \| = 1 + \| \| T \| \|$

Correct Answer : D

Question 22:

Suppose that M is a

5 \times 5

matrix with real entries and p(x) = diet (xI-M). Then

A	$p (0) = \det (M)$
B	every eigenvalue of M is real if p(1) + p(2)= 0 = p(2) + p(3)
C	$M^{- 1}$ is necessarily a polynomial in M of degree 4 if M is invertible
D	M is not invertible if $M^{2} - 2 M = 0$

Correct Answer : C

Question 23:

Let C [0, 1] denote the space of all real-valued continuous function on [0, 1] equipped with the supremum norm

| | . | |_{\infty} .

Let

f \in C [0, 1]

be such that

|f (x) - f (y)| \leq M |x - y|, for all x, y \in [0, 1] and for some M > 0 .

For

n \in ℕ, let f_{n} (x) = f (x^{1 + \frac{1}{n}}) . If S = \{f_{n} : n \in ℕ\},

then

Correct Answer : the closure of S is compact

Question 24:

Let

K : ℝ \times (0, \infty) \to ℝ

be a function such that the solution of the initial value problem

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}}, u (x, 0) = f (x), x \in ℝ, t > 0,

is given by

u (x, t) = \int_{ℝ} k (x - y, t) f (y) dy

for all bounded continuous functions f. Then the value of

\int_{ℝ} k (x, t) dx

is ____________

Correct Answer : 1 to 1

Question 25:

The number of cyclic subgroups of the quaternion group

Q_{8} = < a, b | a^{4} = 1, a^{2} = b^{2}, ba = a^{3} b >

is ______________.

Correct Answer : 5 to 5

Question 26:

The number of elements of order 3 in the symmetric group

S_{6}

is __________

Correct Answer : 80 to 80

Question 27:

Let F be the field with 4096 elements. The number of proper subfields of F is ____________

Correct Answer : 5 to 5

Question 28:

(x_{1}^{*}, x_{2}^{*})

is an optimal solution of the linear programming problem,

minimize x_{1} + 2 x_{2} subject to 4 x_{1} - x_{2} \geq 8 2 x_{1} + x_{2} \geq 10 - x_{1} + x_{2} \geq 0

and

(λ_{1}^{*}, λ_{2}^{*}, λ_{3}^{*})

is an optimal solution of its dual problem, then

\sum_{i = 1}^{2} x_{i}^{*} + \sum_{j = 1}^{3} λ_{j}^{*}

is equal to ____________ (correct up to one decimal place)

Correct Answer : 5.5 to 5.5

Question 29:

Let a, b, c,

\in ℝ

be such that the quadrature rule

\int_{- 1}^{1} f (x) dx \approx af (- 1) + bf (0) + cf' (1)

is exact for all polynomials of degree less than or equal to 2. Then b is equal to __________ (rounded off to two decimal places)

Correct Answer : 1.70 to 1.80

Question 30:

Let

f (x) = x^{4}

and let p(x) be the interpolating polynomial of f at nodes 1, 2, and 3. Then p(0) is equal to _______

Correct Answer : 36 to 36

Question 31:

For

n \geq 2,

define the sequence

\{x_{n}\}

x_{n} = \frac{1}{2 π} \int_{0}^{\frac{π}{2}} \tan^{\frac{1}{n}} t dt .

Then the sequence

\{x_{n}\}

converges to _______________ (correct up to two decimal places)

Correct Answer : 0.25 to 0.25

Question 32:

Let

L^{2} [0, 10] = \{f : [0, 10] \to ℝ : f is Lebesgue measurable and \int_{0}^{10} f^{2} dx < \infty\}

equipped with the norm

| | f | | = {(\int_{0}^{10} f^{2} dx)}^{\frac{1}{2}}

and let T be the linear functional on

L^{2} [0, 10]

given by

T (f) = \int_{0}^{2} f (x) dx - \int_{3}^{10} f (x) dx .

Then ||T|| is equal to ____________

Correct Answer : 3 to 3

Question 33:

\{x_{13}, x_{22}, x_{23} = 10, x_{31}, x_{32}, x_{34}\}

is the set of basic variables of a balanced transportation problem seeking to minimize the cost of transportation from origins to destinations, where the cost matrix is,

and

λ, μ \in ℝ,

then

x_{32}

is equal to _________

Correct Answer : 5 to 5

Question 34:

Let

ℤ_{225}

be the ring of integers modulo 225. If x is the number of prime and y is the number of nontrivial units in

ℤ_{225}

, then x + y is equal to ____

Correct Answer : 121 to 121

Question 35:

Let u(x, t) be the solution of

\frac{\partial^{2} u}{\partial t} - \frac{\partial^{2} u}{\partial x^{2}} = 0, u (x, 0) = f (x), \frac{\partial u}{\partial t} (x, 0) = 0, x \in ℝ, t > 0,

where is a twice continuously differentiable function. If f(-2) =4, f (0) =0, and u (2, 2) = 8, then value of u (1, 3) is _________

Correct Answer : 10 to 10

Question 36:

Let

{[e_{n}]}_{n = 1}^{\infty}

be an orthonormal basis for a parable Hilbert space H with the inner product <. ,. > Define

f_{n} = e_{n} - \frac{1}{n + 1} e_{n + 1} for n \in ℕ .

Then

A	the closure of the span $\{f_{n} : n \in ℕ\}$	B	$f = 0 if < f, f_{n} > = < f, e_{n} > for all n \in ℕ$
C	${\{f_{n}\}}_{n = 1}^{\infty}$ is an orthogonal sunset of H	D	there does exist nonzero $f \in H such that < f, e_{2} > = < f, f_{2} >$

Correct Answer : A

Question 37:

Suppose V is a finite-dimensional non-zero vector space over

ℂ and T : V \to V

is a linear transformation such that Range (T)=Nullspace (T). Then when of the following statements is FALSE?

A	The dimension of V is even	B	0 is the only eigenvalue of T
C	Both 0 and 1 are eigenvalues of T	D	$T^{2} = 0$

Correct Answer : C

Question 38:

Let

P \in M_{m \times n} (ℝ) .

Consider the following statement: I: If

XPY = 0 for all X \in M_{1 \times m} (ℝ) and M_{n \times 1} (ℝ), then P = 0 .

II. If

m = n, P is symmetric and P^{2} = 0, then P = 0 .

Then

Correct Answer : both I and II are true

Question 39:

For

n \in ℕ, let T_{n} : (l^{1}, | | . | |) \to (l^{\infty}, | | . | |) and T : (l^{\infty}, | | . | |_{1}) \to (l^{\infty}, | | . | |_{\infty})

be that bounded linear operators defined by

T_{n} (x_{1}, x_{2}, . .) = (y_{1}, y_{2} . . .), where y_{j} = \{\begin{cases} x_{j}, j \geq n \\ x_{n}, j > n \end{cases}

and

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

A	$\| \| T \| \| does not converge to \| \| T \| \| as n \to \infty$	B	$\| \| T_{n} - T \| converge to zero n \to \infty$
C	$for all x \in l^{1}, \| \| T_{n} (x) - T (x) \| \| converge to zero as n \to \infty$	D	for each non-zero $x \in l^{1},$ exists a continuous linear functional g on $l^{\infty}$ such that $g (T_{n} (x))$ does not $g (T (x)) ad n \to \infty$

Correct Answer : C

Question 40:

Let

P (ℝ)

denote the power set of

ℝ,

equipped with the metric

d (U, V) = \sup_{x \in ℝ} |x_{U} (x) - X_{V} (x)|,

where

X_{U} and X_{V}

denote the characteristic functions of the subsets U and V, respectively, of

ℝ .

The set

\{\{m\} : m \in ℤ\}

in the metric space

(P (ℝ), d)

Correct Answer : bounded but not totally bounded

Question 41:

Let

f : ℝ \to ℝ

be defined by

f (x) = \sum_{n = 0}^{\infty} \frac{1}{2^{n}} X_{(n, n + 1]} (x),

Where

X_{(n, n + 1]}

is the characteristic function of the interval

(n, n + 1] . For α \in ℝ, let S_{α} = \{x \in ℝ : f (x) > α\} . Then

A	$S_{\frac{1}{2}}$ is open	B	$S_{\frac{\sqrt{5}}{2}}$ is not measurable
C	$S_{0}$ is closed	D	$S_{\frac{1}{\sqrt{2}}}$ is measurable

Correct Answer : D

Question 42:

For

n, \in ℕ, let f_{n}, g_{n} : (0, 1) \to ℝ

be, a function defined by

f_{n} (x) = x^{n} and g_{n} (x) = x^{n} (1 - x) .

Then

A	$\{f_{n}\}$ converges uniformly but $\{g_{n}\}$ does not converge uniformly	B	$\{g_{n}\}$ converges uniformly but $\{f_{n}\}$ does not converge uniformly
C	both $\{f_{n}\} and \{g_{n}\}$ converge uniformly	D	neither $\{f_{n}\} and \{g_{n}\}$ converge uniformly

Correct Answer : B

Question 43:

Let U be a solution of the differential equation

y' + xy = 0 and let ϕ = uψ

be a solution of the differential equation

y'' + 2 xy' + (x^{2} + 2) y = 0 satisfying ϕ (0) = 1 and ϕ' (0) = 0 . Then ϕ (x) is

A	$(\cos^{2} x) e^{- \frac{x^{2}}{2}}$	B	$(\cos x) e^{- \frac{x^{2}}{2}}$
C	$(1 + x^{2}) e^{- \frac{x^{2}}{2}}$	D	$(\cos x) e^{- x^{2}}$

Correct Answer : B

Question 44:

For

n \in ℕ \{0\}, let y_{n}

be a solution of the differential equation

xy'' + (1 - x) y' + n y = 0

Satisfying

y_{n} (0) = 1 .

For which of the following functions w(x), the integral

\int_{0}^{\infty} y_{p} (x) y_{q} (x) w (x) dx, (p \neq q)

is equal to zero?

A	$e^{- x^{2}}$	B	$e^{- x}$
C	${xe}^{- x^{2}}$	D	${xe}^{- x}$

Correct Answer : B

Question 45:

Suppose that

X = \{(0, 0)\} U \{(x, \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

and

Y = \{(0, 0)\} U \{(x, x \sin \frac{1}{2}) : x \in ℝ \ \{0\}\}

are metric spaces with metrics induced by the Euclidean metric of

ℝ^{2} .

Let

B_{X}

and

B_{Y}

be the open unit balls around (0, 0) in X and Y, respectively. Consider the following statement: I: The closure of

B_{X}

in X is compact II. The closure of

B_{y}

in Y is compact Then

Correct Answer : I is false but II is true

Question 46:

f : ℂ \ \{0\} \to ℂ

is a function such that

f (z) = f (\frac{z}{|z|})

and its restriction to the unit circle is continuous, then

A	f is continuous but not necessarily analytic
B	f is continuous but not necessarily a constant function
C	f is a constant function
D	$\lim_{z \to 0} f (z)$ exists

Correct Answer : A

Question 47:

For a subset S of a topological space, let Int (S) and

S

denote the interior and closure of S, respectively. Then which of the following statement is TRUE?

A	If S is open, then S = Int $(S)$	B	If the boundary of S is empty, then S is open
C	If the boundary of S is empty, then S is not closed	D	If $S \ S$ is a proper subset of the boundary of S, then S is open

Correct Answer : B

Question 48:

Suppose

I_{1}, I_{2} and I_{3}

are the smallest topologies on

ℝ containing S_{1}, S_{2} and S_{3}, respectively, where

S_{1} = \{(α, α + \frac{π}{n}) : α \in ℚ, n \in ℕ\}, S_{2} = \{(a, b) : a < b, a, b \in ℚ\}, S_{3} = \{(a, b) : a < b, a, b \in ℝ\} .

A	$I_{3} \underline{\supset} I_{1}$	B	$I_{3} \underline{\supset} I_{2}$
C	$I_{1} = I_{2}$	D	$I_{1} \underline{\supset} I_{2}$

Correct Answer : C

Question 49:

Let

M = [\begin{matrix} α & 3 & 0 \\ β & 3 & 1 \\ 0 & 1 & 2 \end{matrix}]

. Consider the following statements: I: There exists a lower triangular matrix L such that M =

{LL}^{t}, where L^{t}

denotes the transpose of L. II: Gauss-Seidel method for

Mx = b (b \in ℝ^{3})

converges for any initial choice

x_{0} \in ℝ^{3} .

Then

A	I is not true when $α > \frac{9}{2}, β = 3$	B	II is not true when $α > \frac{9}{2}, β = - 1$
C	II is not true when $α = 4, β = \frac{3}{2}$	D	$α = 5, β = 3$

Correct Answer : D

Question 50:

Let I and J be the ideals generated by

\{5 . \sqrt{10}\} and \{4, \sqrt{10}\}

in the ring

ℤ [\sqrt{10}] = \{a + b \sqrt{10} | α, b \in ℤ\},

respectively. Then

Correct Answer : I is a maximal ideal but J is not a prime ideal

Question 51:

Suppose V is finite-dimensional vector space over

ℝ . If W_{1}, W_{2}, and W_{3}

are subspaces of V, then which of the following statements is TRUE?

A	If $W_{1} + W_{2} + W_{3} = V then span (W_{1} \cup W_{2}) \cup span (W_{2} \cup W_{3}) \cup span (W_{3} \cup W_{1}) = V$
B	$If W_{1} \cap W_{2} = \{0\} and W_{1} \cap W_{3} = \{0\}, then W_{1} \cap (W_{2} + W_{3}) = \{0\}$
D	If $W_{1} + W_{2} = W_{1} + W_{3}, then W_{2} = W_{3}$
C	$If W_{1} \neq V, then span (V \ W_{1}) = V$

Correct Answer : D

Question 52:

Let

α, β \in ℝ, α \neq 0 . The system

x_{1} - 2 x_{2} + {αx}_{3} = 8 x_{1} - x_{2} + x_{4} = β x_{1}, x_{2}, x_{3}, x_{4} \geq 0

has NO basic feasible solution if

A	$α < 0, β > 8$	B	$α > 0, 0 < β < 8$
C	$α > 0, β < 0$	D	$α < 0, β < 8$

Correct Answer : D

Question 53:

Let

0 < p < 1 and let

X = \{f : ℝ \to ℝ is continuous and \int_{ℝ} {|f (x)|}^{p} dx < \infty\} .

For

f \in X,

define

{|f|}_{p} = {(\int_{ℝ} {|f (x)|}^{p} dx)}^{\frac{1}{p}}

Then

A	${\|.\|}_{p}$ defines a norm on X	B	${\|f + g\|}_{p} \leq {\|f\|}_{p} + {\|g\|}_{p} for all f, g \in X$
C	${\|f + g\|}_{p}^{p} \leq {\|f\|}_{p}^{p} + {\|g\|}_{p}^{p} for all f, g \in X$	D	$if f_{n} converges to f ponitwise on ℝ, then \lim_{n \to \infty} \|f_{n}\| = {\|f\|}_{p}$

Correct Answer : C

Question 54:

Suppose that

ϕ and ϕ_{2}

are linearly independent solutions of the differential equation

2 x^{2} y'' - (x + x^{2}) y' + (x^{2} - 2) y = 0,

and

ϕ (0) = 0 .

Then the smallest positive integer n such that

\lim_{x \to 0} x^{n} \frac{ϕ_{2} (x)}{ϕ_{1} (x)} = 0

is ___________.

Correct Answer : 3 to 3

Question 55:

Suppose that

f (z) = \prod_{n = 1}^{17} (z - \frac{π}{n}), z \in ℂ and y (t) = e^{2 it}, t \in [0, 2 π] . If

\int_{Y} \frac{f' (z)}{f (z)} dz = απi,

then the value

α

is equal to _______

Correct Answer : 56 to 56

Question 56:

y (t) = \frac{1}{2} e^{3 πi t}, t \in [0, 2] and

\int_{y} \frac{1}{z^{2} (e^{z} - 1)} dz = βπi,

then

β

is equal to __________ (correct up to one decimal place)

Correct Answer : 0.5 to 0.5

Question 57:

Let

K = ℚ (\sqrt{3 + 2 \sqrt{2}}, ω), where ω

is a primitive cube root of unity. Then the degree of extension of K over

ℚ

is ___________

Correct Answer : 4 to 4

Question 58:

Let

α \in ℝ . If (3, 0, 0, β)

is an optimal solution to the linear programming problem

minimize x_{1} + x_{2} + x_{3} - {αx}_{4} subject to 2 x_{1} - x_{2} + x_{3} = 6 - x_{1} + x_{2} + x_{4} = 3 x_{1}, x_{2}, x_{3}, x_{4} \geq 0

then the maximum value of

β - α

is _______________

Correct Answer : 7 to 7

Question 59:

Suppose that

T : ℝ^{4} \to ℝ [x]

is a linear transformation over

ℝ

satisfying

T (- 1, 1, 1, 1) = x^{2} + 2 x^{4}, T (1, 2, 3, 4) = 1 - x^{2}, T (2, - 1, - 1, 0) = x^{3} - x^{4} .

Then the coefficient of

x^{4} in T (- 3, 5, 6, 6)

is _________________

Correct Answer : 5 to 5

Question 60:

Let

\vec{F} (x, y, z) = (2 x - 2 y \cos x) \hat{i} + (2 y - y^{2} \sin x) \hat{j} + 4 z \hat{k}

and let S be the surface of the tetrahedron bounded by the planes

x = 0, y = 0, z = 0 and x + y + z = 1 .

\hat{n}

is the unit outward normal to the tetrahedron, then the value of

\int \int_{S} \vec{F} . \hat{n} dS

is ______________ (rounded off to two decimal places)

Correct Answer : 1.30 to 1.40

Question 61:

Let

\vec{F} = (x + 2 y) e^{z} \hat{i} + ({ye}^{z} + x^{2}) \hat{j} + y^{2} z \hat{k}

and let S be the surface

x^{2} + y^{2} + z = 1, z \geq 0 .

\hat{n}

is a unit normal to S and

|\int \int_{S} (\nabla \times \vec{F}) . \hat{n} dS| = απ

Then

α is equal to_______

Correct Answer : 2 to 2

Question 62:

Let G be a non-cyclic group of order 57. Then the number of elements of order 3 is G is _______________

Correct Answer : 38 to 38

Question 63:

The coefficient of

{(x - 1)}^{5}

in the Taylor expansion about x = 1 of the function

$F (x) = \int_{1}^{x} \frac{\log_{e} t}{t - 1} dt, 0 < x < 2$

is ___________ (correct uo to two decimal places)

Correct Answer : 0.04 to 0.04

Question 64:

Let u(x, y) be the solution to the initial value problem

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

Then the value u(0, 1) is ___________ (rounded off to three decimal places)

Correct Answer : 1.610 to 1.625

Question 65:

The value of

\lim_{n \to \infty} \int_{0}^{1} {nx}^{n} {ex}^{n} e^{x^{2}} dx

is ____________ (rounded off to three decimal places)

Correct Answer : 2.710 to 2.725

Back
Scrore Card
Total Question		Total Attempted
Correct Answers		Incorrect Answers
Score

A	$\| \| T \| \| does not converge to \| \| T \| \| as n \to \infty$	B	$\| \| T_{n} - T \| converge to zero n \to \infty$
C	$for all x \in l^{1}, \| \| T_{n} (x) - T (x) \| \| converge to zero as n \to \infty$	D	for each non-zero $x \in l^{1},$ exists a continuous linear functional g on $l^{\infty}$ such that $g (T_{n} (x))$ does not $g (T (x)) ad n \to \infty$

A	${\|.\|}_{p}$ defines a norm on X	B	${\|f + g\|}_{p} \leq {\|f\|}_{p} + {\|g\|}_{p} for all f, g \in X$
C	${\|f + g\|}_{p}^{p} \leq {\|f\|}_{p}^{p} + {\|g\|}_{p}^{p} for all f, g \in X$	D	$if f_{n} converges to f ponitwise on ℝ, then \lim_{n \to \infty} \|f_{n}\| = {\|f\|}_{p}$

GATE 2020 Mathematics complete exam Practice Online with solution

View Solution

Attempt Mock Test

General Instructions

Please read the instructions carefully

General Instructions:

Navigating to a Question:

Answering a Question:

Navigating through sections:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

Question 7:

Question 8:

Question 9:

Question 10:

Question 11:

Question 12:

Question 13:

Question 14:

Question 15:

Question 16:

Question 17:

Question 18:

Question 19:

Question 20:

Question 21:

Question 22:

Question 23:

Question 24:

Question 25:

Question 26:

Question 27:

Question 28:

Question 29:

Question 30:

Question 31:

Question 32:

Question 33:

Question 34:

Question 35:

Question 36:

Question 37:

Question 38:

Question 39:

Question 40:

Question 41:

Question 42:

Question 43:

Question 44:

Question 45:

Question 46:

Question 47:

Question 48:

Question 49:

Question 50:

Question 51:

Question 52:

Question 53:

Question 54:

Question 55:

Question 56:

Question 57:

Question 58:

Question 59:

Question 60:

Question 61:

Question 62:

Question 63:

Question 64:

Question 65:

Exam Summary

Are you sure you want to submit for final marking? No changes will be allowed after submission.

Thank you, Submitted Successfully.

Question 1:

Question 2:

Question 3:

Are you sure you want to submit for final marking?
No changes will be allowed after submission.