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Latest GATE Mathematics Exam Question (Objective Questions), MCQ in English

Subjects : General Aptitude, Mathematics Engineering

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Mathematics Engineering

Question 1Multiple Answers

The matrix M is defined as

and has eigenvalues 5 and –2. The matrix Q is formed as,

Which of the following is/are the eigenvalue(s) of matrix Q ?

A15
B25
C–20
D–30

Question 2Multiple Answers

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function

â€‹ ?

A
It is continuous on its domain
B
It has a local minimum at x=2
C
It has a local maximum at x=2
DIt is differentiable on its domain

Question 3Multiple Answers

Let y be a non-zero vector of size 2022 × 1. Which of the following statement(s) is/are TRUE ?

A
yy^T is a symmetric matrix
B
y^Ty is an eigenvalue of yy^T
C
yy^T has a rank of 2022
Dyy^T is invertible

Question 4Multiple Answers

P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct ?

A
If P and Q are invertible, then
B
If P and Q are invertible, then â€‹
CIf P and Q are invertible, then â€‹
DIf P and Q are not invertible, then

Question 5Multiple Answers

Which of the following is/are the eigenvector(s) for the matrix given below ?

â€‹

Question 6Multiple Answers

Consider the following wave equation,

â€‹
Which of the given options is/are solution(s) to the given wave equation ?

Question 7Multiple Answers

Consider the following series :

For which of the following combinations of c, d values does this series converge ?

Ac = 1, d = –1
Bc = 2, d = 1
Cc = 0.5, d = –10
Dc = 1, d = – 2

Question 8Multiple Answers

The system of linear equations in real (x, y) given by

â€‹
involves a real parameter a and has infinitely many non-trivial solutions for special value(s) of a. Which one or more among the following options is/are non-trivial solution(s) of (x, y) for such special value(s) of a ?

A
x = 2, y = – 2
Bx = – 1, y = 4
Cx = 1, y = 1
Dx = 4, y = – 2

Question 9Multiple Answers

A is a 3 × 5 real matrix of rank 2 . For the set of homogeneous equations AX = 0, where 0 is a zero vector and X is a vector of unknown variables, which of the following is/are true ?

AThe given set of equations will have a unique solution
BThe given set of equations will be satisfied by a zero vector of appropriate size
CThe given set of equations will have infinitely many solutions
DThe given set of equations will have many but a finite number of solutions

Question 10Multiple Answers

Consider the following partial differential equation (PDE)

where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters

â€‹ and

such that is a solution of thegiven PDE.

Question 11Single Answer

Consider the following expression :

where z, y, and t are variables, and

is a complex number. The partial differential equation derived from the above expression is

Question 12Single Answer

For the equation

the correct description is

A
an ordinary differential equation of order 3 and degree 2
B
an ordinary differential equation of order 3 and degree 3
C
an ordinary differential equation of order 2 and degree 3
Dan ordinary differential equation of order 3 and degree â€‹

Question 13Single Answer

The value of (1 + i)¹², where

â€‹ , is.

A
– 64 i
B
64 i
C64
D– 64

Question 14Single Answer

Given matrix

, the ordered pair (x, y) for which is det (A) = 0 is.

A(1, 1)
B(1, 2)
C(2, 2)
D(2, 1)

Question 15Single Answer

Let

where x is real. The value of

â€‹ at x = –1 is

A
– e
B
e
C1/e
D– 1/e

Question 16Single Answer

The value of the real variable

, which maximizes the function

â€‹is.

A
e
B
0
C1/e
D1

Question 17Single Answer

The partial differential equation

where,

and

, is subjected to the following initial and boundary conditions:

u(x, 0) =

u(0, t) = 0

u(1, t) = 0
The value of t at which

Question 18Single Answer

Consider the polynomial on the domain S given by

. The first and second derivatives are f'(x) and f"(x). Consider the following statements :
I. The given polynomial is zero at the boundary points x = 1 and x = 3.
II. There exists one local maxima of f(x) within the domain S.
III. The second derivative f"(x) > 0 throughout the domain S.
IV. There exists one local minima of f(x) within the domain S.
The correct option is :

A
Only statements I, II and III are correct
B
Only statements I, II and IV are correct
C
Only statements I and IV are correct
DOnly statements II and IV are correct

Question 19Single Answer

is equal to

Question 20Single Answer

The function (x, y) satisfies the Laplace equation

â€‹ on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0 . The value of this function on the circular boundary of this domain is equal to 3. The numerical value of f(0, 0) is :

Question 21Single Answer

Consider a 3 × 3 matrix A whose (i, j)^th element, a_{i, j} = (i – j)³. Then the matrix A will be

A
symmetric
B
skew-symmetric
Cunitary
Dnull

Question 22Single Answer

e^A denotes the exponential of a square matrix A. Suppose 𝞴 is an eigenvalue and v is the corresponding eigen-vector of matrix A.

Consider the following two statements :

Statement 1 : e^𝞴is an eigenvalue of e^A.

Statement 2 : u is an eigen-vector of eA.

Which one of the following options is correct ?

A
Statement 1 is true and statement 2 is false
B
Statement 1 is false and statement 2 is true
C
Both the statements are correct
DBoth the statements are false

Question 23Single Answer

Let

â€‹. Then f(x) decreases in the interval

A
x € (1, 3)
B
x €(2, 3)
Cx €(0, 1)
Dx €(0.5, 1)

Question 24Single Answer

Consider a matrix .

The matrix A satisfies the equation 6A^–1 = A² + cA + d I, where c and d are scalars and I is the identity matrix.
â€‹Then (c + d) is equal to

A
5
B
17
C– 6
D11

Question 25Single Answer

Let, f(x, y, z) = 4x²+ 7xy + 3xz². The direction in which the function f(x, y, z) increases most rapidly at point P = (1, 0, 2) is

Question 26

Consider the following recursive iteration scheme for different values of variable P with the initial guess x₁ = 1 :

For P = 2, x₅ is obtained to be 1.414, rounded-off to three decimal places. For P = 3, x₅ is obtained to be 1.732, rounded-off to three decimal places.If P = 10, the numerical value of x5 is (round off to three decimal places).

Question 27

The Fourier cosine series of a function is given by :

For f(x) = cos⁴x, the numerical value of (f₄ + f₀) is (round off to three decimal places).

Question 28

Consider the differential equation

For the initial condition y = 3 at x = 1, the value of y at x = 1.4 obtained using Euler's method with a step-size of 0.2 is (round off to one decimal place)

Question 29

A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below.
â€‹

x	5	2	4	3
y	16	10	13	12

â€‹Based on the data, the coefficient a of the linear regression model y = a + bx is estimated as 6.1. The coefficient b is ____. (round off to one decimal place)

Question 30

The directional derivative of f(x, y, z) = 4x² + 2y² + z²at the point
â€‹(1, 1, 1) in the direction of the vector

is (rounded off to two decimal places).

Question 31

Consider a sphere of radius 4 , centered at the origin, with outward unit normal

on its surface S. The value of the surface integral

is (rounded off to one decimal place).

Question 32

The

with the initial condition y(x = 0) = 1 is to be solved using a predictor-corrector approach. Use a predictor based on the implicit Euler's method and a corrector based on the trapezoidal rule of integration, each with a full-step size of 0.5. Considering only positive values of y, the value of y at x = 0.5 is (rounded off to three decimal places).

Question 33

A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is (round off to three decimal places).

Question 34

The number of arrangements of six identical balls in three identical bins is

Question 35

The value of the following limit is

Question 36

The radius of convergence of the power series

â€‹ is

Question 37

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

â€‹ subject to y(1.2) = 2. Using the Euler method with the step size h = 0.05, the approximate value of y(1.3), correct to two decimal places, is____.

Question 38

Let a R. If a x is the polynomial which interpolates the function f(x) = sin

x on [–1, 1] at all the zeroes of the polynomial 4x³ – 3x, then a is

Question 39

If u(x, t) is the D' Alembert's solution to the wave equation

, with the condition u(x, 0) = 0 and

, then

â€‹ is

Question 40

The number of non - isomorphic groups of order 10 is____.

Question 41Single Answer

Which of the following is false ?

Af is a continuous linear functional
Bâ€‹
CThere does not exist any y ∈ xc₀₀ such that f(x) = (x, y) for all x ∈ C₀₀
DN1 ≠ N{0}

Question 42Single Answer

Which of the following is false ?

Ac₀₀ ≠ N
BN is closed
CC₀₀ is not a complete inner product space
DC₀₀ = N ⊕ N'

Question 43Single Answer

Which of the following is not an unbiased estimate of u ?

AX₁
Cn[min {X₁, X₂, ..., X_n}]

Question 44Single Answer

Consider the problem of estimating m. the MSE (Mean Square Error) of the estimate

â€‹ is

Question 45Single Answer

The value of no is

Question 46Single Answer

The possible set of eigen values of a 4 × 4 skew-symmetric orthogonal real matrix is

A(± i)
B{± i, ±1}
C(± 1)
D{0, ± i }

Question 47Single Answer

The coeffecient of (z – p)² in the Taylor series expanision of

â€‹ around p is

Question 48Single Answer

Consider R² with the usual topology. Which of the following statements are true for all

â€‹

AP and R
BP and S
CQ and R
DQand S

Question 49Single Answer

Let f : R → R be a continuous function with f(1) = 5 and f(3) = 11. If g(x) =

â€‹ then g'(0) is equal to

AZero
B6
C8
D14

Question 50Single Answer

Let P be a 2 × 2 complex matrix such that tr(P) = 1 & det (P) = 6. Then tr (p⁴ – p³) is equal to

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Mathematics Engineering

Question 51Single Answer

Suppose that R is a unique facersation domain an d that a, b ∈ R are distinct irreducible elements. Which of the following statements is true ?

AThe idal (1 + a) is a prime ideal
BThe ideal (a + b) is a prime ideal
CThe ideal (1 + ab) is a prime ideal
DThe ideal (a) is not necessarily a maximal ideal

Question 52Single Answer

Let X be a compact Housdorft topological space and Y be a topological space. Let f : X → Y be a bijective continuous maping. Which of the following statements is true ?

Af is a closed map but not necessarily an open map
Bf is an open map but not necessarily a closed map
Cf is both an open map and a colsed map
Df need not be an open map or a colsed map

Question 53Single Answer

Consider the linear programming problem Maximise

subject to constraints

and

If S denotes the set of all solutions of the above problem, then

AS is empty
BS is a singleton
CS is a line segment
DS has positive area

Question 54Single Answer

Which of the following groups has a proper subgroup that is not cyclic ?

AZ₁₅ × Z₇₇
BS₃
C(Z, +)
D(Q,+)

Question 55Single Answer

The value of the integral

dydx is

General Aptitude

Question 56Single Answer

Inhaling the smoke from a burning _____ could _____ you quickly.

Atire/tier
Btire/tyre
Ctyre/tire
Dtyre/tier

Question 57Single Answer

Rice, a versatile and inexpensive source of carbohydrate, is a critical component of diet worldwide. Climate change, causing extreme weather, poses a threat to sustained availability of rice. Scientists are working on developing Green Super Rice (GSR), which is resilient under extreme weather conditions yet gives higher yields sustainably.
Which one of the following is the CORRECT logical inference based on the information given in the above passage ?

AGSR is an alternative to regular rice, but it grows only in an extreme weather
BGSR be used in future in response to adverse effects of climate change
CGSR grows in an extreme weather, but the quantity of produce is lesser than regular rice
DRegular rice will continue to provide good yields even in extreme weather

Question 58Single Answer

Given below are three statements and four conclusions drawn based on the statements.
Statement 1: Some engineers are writers.
Statement 2: No writer is an actor.
Statement 3: All actors are engineers.
Conclusion I: Some writers are engineers.
Conclusion II: All engineers are actor.
Conclusion III: No actors is a writer.
Conclusion IV: Some actors are writers.
Which one of the following options can be logically inferred ?

AOnly conclusion I is correct
BOnly conclusion II and conclusion III are correct
COnly conclusion I and conclusion III are correct
DEither conclusion III or conclusion IV is correct

Question 59Single Answer

You should_____ when to say _____ .

Ano/no
Bno/know
Cknow/know
Dknow/no

Question 60Single Answer

Healthy eating is a critical component of healthy aging. When should one start eating healthy ? It turns out that it is never too early. For example, babies who start eating healthy in the first year are more likely to have better overall health as they get older.
Which one of the following is the CORRECT logical inference based on the information in the above passage ?

AHealthy eating is important for those with good health conditions, but not for others
BEating healthy can be started at any age, earlier the better
CEating healthy and better overall health are more correlated at a young age, but not older age
DHealthy eating is more important for adults than kids

Question 61Single Answer

The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is :

Question 62Single Answer

Pipes P and Q can fill a storage tank full of water in 10 and 6 minutes, respectively. Pipe R draws the water out from the storage tank at a rate of 34 litres per minute. P, Q and R operate at a constant rate. If it takes one hour to completely empty a full storage tank with all the pipes operating simultaneously, what is the capacity of the storage tank (in litres) ?

A26.8
B60.0
C120.0
D127.5

Question 63Single Answer

Six persons P, Q, R, S, T and U are sitting around a circular table facing the centre not necessarily in the same order. Consider the following statements:
• P sits next to S and T.
• Q sits diametrically opposite to P
• The shortest distance between S and R is equal to the shortest distance between T and U. Based on the above statements, Q is a neighbour of

AU and S
BR and T
CR and U
DP and S

Question 64Single Answer

Consider the following inequalities.
I 3p – q < 4
II 3q – p < 12
Which one of the following expressions below satisfies the above two inequalities ?

Ap + q < 8
Bp + q = 4
C8 ≤p + q <16
Dp + q ≥ 16

Question 65Single Answer

Two straight lines pass through the origin = (0,0). One of them passes through the point = (1,3) and the other passes through the point = (1, 2).What is the area enclosed between the straight lines in the interval, [0,1] on the -axis ?

A0.5
B1.0
C1.5
D2.0

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Latest GATE Mathematics Exam Question (Objective Questions), MCQ in English

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