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Objective Questions

Latest GSET Mathematical Sciences Exam Question (Objective Questions), MCQ in English

Subjects : Mathematical Science

Question Bank GSET Mathematical Science (Paper-II) Exam - English

Mathematical Science

Question 1Single Answer

The following schematic diagram represents a glaciated terrain with four different points A, B, C and D. The state at different points shown are X-Glacial Ice movement, Y-Glacial plucking, Z-Jointed bedrock, K-Abrasion and polishing. Find the correct match .

A
A - X; B - Z; C - Y; D - K
B
A - K; B - Y; C - Z; D - X
C
A - K; B - Z; C - X; D - Y
D
A - X; B - Y; C - Z; D - K

Question 2Single Answer

The following schematic diagram shows velocities observed by a ground Doppler radar for two atmospheric phenomena. The solid lines indicate wind component towards the radar, the dashed lines indicate wind components away from the radar, and the dotted line indicates the zero isodop. Which of the following options correctly identifies the two phenomena ?

A
(i) Tornadic mesocyclone and (ii) Cold air downburst from a thunderstorm
B
(i) Sea-breeze and (ii) Mountain-valley winds
C
(i) Occluded front and (ii) Extra-tropical cyclone
D
(i) Butterfly effect and (ii) Coriolis effect

Question 3Single Answer

Given profiles are schematic for the surface waters in northern temperate latitudes. Identify A, B, C and D parameters .

A
A – Nutrients, B – Phytoplankton, C –Zooplankton, D – Sunlight
B
A – Sunlight, B – Phytoplankton, C – Nutrients, D – Zooplankton
C
A – Zooplankton, B – Phytoplankton, C – Chlorophyll, D – Sunlight
D
A – Sunlight, B – Phytoplankton, C – Nutrients, D – Chlorophyll

Question 4Single Answer

Find the value of f(0) if f(x+2) = (x + 1)³⁴ – (x + 1)³³ + 5 .

A
5
B
7
C
6
D
72

Question 5Single Answer

The shortest distance between the parallel lines A and B in the following figure is :

Question 6Single Answer

Two varieties A and B of rice cost Rs 30 and Rs 90 per kg, whereas two varieties C and D of pulases, Rs 100 and Rs 120 per kg, respectively. If at least one kg each of A and B and at least half a kg each of C and D have to be purchased, then the minimum and maximum costs of a total of 5 kg of these provisions are respectively .

A
Rs 150 and Rs 600
B
Rs 260 and Rs 530
C
RS 290 and Rs 470
D
Rs 370 and Rs 460

Question 7Single Answer

One of four suspects A, B, C and D has committed a crime. A and D are always truthful, and B and C are always untruthful. C and D are identical twins and the interrogator does not know who is who. If A says "D is innocent", B says, "A is guilty and amoung C and D one say, “A is innocent” and the other says, "B is guilty", then which of the following is FALSE ?

A
D said "A is innocent"
B
D is innocent
C
B is innocent
D
C is innocent

Question 8Single Answer

Which is an appropriate diagram to represent the relations between the following categories : quadruped, mammal, whale, house lizard ?

Question 9Single Answer

A 7 m long tube having inner diameter of 2 cm is filled with water. The water is then poured into a cylindrical bucket having inner base area of 200 cm². What will be the approximate height (in cm) of water in the bucket ?

A
22
B
44
C
9
D
11

Question 10Single Answer

Water is being filled in a cone from the top at a constant volumetric rate. The rate of increase of the height of the water column :

A
is linearly dependent on time
B
depends on the apex angle of the cone
C
increases as cube-root of the volumetric rate
D
increases as square-root of the volumetric rate

Question 11Single Answer

A square board is divided into 9 smaller identical squares by drawing lines. Three bullets are shot at the board randomly. The probability that at least 2 bullets hit the same small square is :

Question 12Single Answer

A and B complete a work in 30 days. B and C complete the same work in 24 days wherea C and A complete the same work in 28 days. Based on these statements which of the following conclusions is correct ?

A
C is the most efficient and B is the least efficient
B
B is the most efficient but the least efficient one cannot be determined
C
C is the most efficient but, the least efficient one cannot be determined
D
C is the most efficient and A is the least efficient

Question 13Single Answer

Clock A loses 4 minutes every hour, clock B always shows the correct time and clock C gains 3 minutes every hour. On a Monday, all the three clock showed the same time, 8pm. On the following Wednesday, when the clock C shows 2 pm, what time will clock A show ?

A
7 : 20 am
B
8 : 40 am
C
9 : 20 am
D
10 : 40 am

Question 14Single Answer

In a class, there is one pencil for every two students, one eraser for every three students, and one ruler for every four students. If the total number of these stationery items required is 65 how many students are present in the class ?

A
55
B
60
C
65
D
70

Question 15Single Answer

The given table shows the numbers of active and recovered case of a certain disease. Assuming that the lienear trend for both continues, on which days will recovered cases be twice that of the active cases ?

Day	0	1	4	7	10
Active cases	990	1000	1030	1060	1090
Recovered cases	760	800	920	1040	1160

A
61
B
62
C
63
D
64

Question 16Single Answer

A boat weights 60kg, and oarsmen A and B weight 80 and 90kg, respectively. Rowing at a constant power, the time required to complete a course is proportional to the total weight. Rowing alone, A and B complete the course in 1 and 1½ hours, respectively. Assuming that their powers add up, how long will they ake to complete the course if the row together ?

A
49.4 min
B
57.5 min
C
62.6 min
D
72.5 min

Question 17Single Answer

The wavelength dependant absorbance of two compounds, A and B, is shown Absorbance of mixture is a linear function of the concentration of the two compounds. R is defined as a ratio of absorbance at 650nm to the abosrbance at 950 nm .

If the mixture contains 95% of compound A then R must be :

A
95
B
5
C
1
D
Less than 1

Question 18Single Answer

An epidemic is spreading in a population of size P. The rate of spread R of the disease at a given time is proportional to both, the number of people affected by the disease (N) and the number of people not yet affected by the disease. Which of the following graphs of R vs N is correct ?

Question 19Single Answer

The figure shows temperature and salinity of four samples of water. Which one of the samples has the highest density ?

Question 20Single Answer

Consider a parallelogram ABCD with centre O and E as the midpoint of side CD. The area of the triangle OAE, is :

Question 21Single Answer

The sum of the first n even numbers is :

A
divisible by n and not by (n + 1)
B
divisible by (n + 1) and not by n
C
divisible by booth n and (n + 1)
D
neither divisible by n nor by (n + 1)

Question 22Single Answer

A, B, C, D and E are the vertices of a regular pentagon as shown in the figure .

The angle ∠ABC is :

A
48°
B
72°
C
54°
D
36°

Question 23Single Answer

On a 200 m long straight road, maximum number of poles are fixed at 200 m interval. How many of the poles should be removed in order to have maximum number of poles at an interval of 40 m on the road ?

Question 24Single Answer

Let (En) be a squence of subsets of

.
Define

Which of the following statements is true ?

A
B
(x : x ∈ E_n for some n)
C
= (x : x ∈ E_n for all but finitely many n)
D
= (x : x ∈ E_n for infinitely many n)

Question 25Single Answer

be a bounded function. Which of the following statements is NOT true ?

Question 26Single Answer

Which of the following statements is true ?

A
There are at most constably many continuous maps from to
B
There are at most finitely many continuous surjective maps from to
C
There are infinitely many continuous injective maps from to
D
There are no continuous bijective maps from to

Question 27Single Answer

The series

converges

A
only for x = 0
B
uniformly only for x ∈ [−π, π]
C
uniformly only for x ∈ (nπ : n ∈ Z)
D
uniformly for all x ∈

Question 28Single Answer

Given (a_n)_n _{≥ 1} a sequence of real numbers, which of the following statements is true :

A
B
There is a subsequence (a_nk)_k≥1 such that converges
C
There is a b such that converges
D
There is a number b and a subsequence (a_nk)_k≥1 such that converges

Question 29Single Answer

Given f, g are continuous function on [0,1] such that f(0) = f(1) = 0 ; = g (1) = 1 and

. Which of the following statements is true ?

A
There is no t ∈ [0,1] such that f(t) = g(t)
B
There is exactly one t ∈ [0,1] such that f(t) = g(t)
C
There are atleast two t ∈ [0,1] such that f(t) = g(t)
D
There are always infinitely many t ∈ [0,1] such that f(t) = g (t)

Question 30Single Answer

Let A be an n x n matrix such that the set of all its non-zero eigenvalues has exactly r elements. Which of the following statements is true ?

A
rank A ≤ r
B
If r = 0, then rank A < n − 1
C
rank A ≥ r
D
A² has r distinct non-zero eigenvalues

Question 31Single Answer

Let A and B be 2 x 2 metries. Then which on the following is true ?

A
det (A+B) + det(A−B) = det A + det B
B
det (A+B) + det(A−B) = 2det A − 2det B
C
det (A+B) + det(A−B) = 2det A + 2det B
D
det (A+ B) − det(A−B) = 2det A − det B

Question 32Single Answer

If A =

, then A²⁰ euqals

Question 33Single Answer

Let A be a 2 x 2 real matrix with det A = 1 and trace A = 3. What is the value of trace A² ?

A
2
B
10
C
9
D
7

Question 34Single Answer

For a, b ∈

, let p (x, y) = a²x₁y₁ + abx₂y₁ + abx₁y₂+b²x₂y₂, x = (x₁, x₂) y = (y₁, y₂) ∈

For what values of a and b does p :

→

define an inner product ?

A
a > 0, b > 0
B
ab > 0
C
a = 0, b = 0
D
For no values of a, b

Question 35Single Answer

Which of the following real quadratic froms on

is positive definite ?

A
Q(X,Y) = XY
B
Q(X,Y) = X²− XY+ Y²
C
Q(X,Y) = X²− 2XY+ Y²
D
Q(X, Y) = X²+ XY

Question 36Single Answer

Let

be the positively oriented circle in the complex plane given by

. Then

equals :

Question 37Single Answer

For a positive integer p, consider the holomorphic function

For which values of p does there exist a holomorphic function

such that f(z) = g' (z) for

A
All even integers
B
All odd integers
C
All multiples of 3
D
All multiples of 4

Question 38Single Answer

Let

be the positively oriented circle in the complex plane given by

. The line integral

equals :

Question 39Single Answer

Let p be a positie integer. Consider the closed curve r(t) = e^it, 0 ≤ t < 2π/. Let f be a function holomorphic in {z : |z| < 2 R} where R > 1. If f has a zero only at z₀, 0 < |z₀| < R, and it is of multiplicity q, then

equals

Question 40Single Answer

Which of the following statements is true ?

A
Every even integer n ≥ 16 divides (n – 1)! +3
B
Every odd integer n ≥ 16 divides (n – 1)!
C
Every even integer n ≥ 16 divides (n – 1)!
D
For every integer n ≥ 16, n² divides n! + 1

Question 41Single Answer

Let X be a non-empty set and P(X) be the set of all subsets of X. On P(X) defines two operations * and Δ as follows :
for A, B

P(X), A * B = A

B ;
A Δ B = (A

B) (A

B) .
Which of the following statements is true ?

A
P(X) is a group under * as well as under Δ
B
P(X) is a group under *, but not under Δ
C
P(X) is a group under Δ, but not under *
D
P(X) is neither a group under * nor under Δ

Question 42Single Answer

Let φ(n) be the cardinality of the set {a|1 ≤ a ≤ n, (a, n ) = 1} where (a, n) denotes the gcd of a and n. Which of the following is NOT true ?

A
There exist infinitely many n such that φ(n) > φ(n+1)
B
There exist infinitely many n such that φ(n) < φ(n+1)
C
There exists N ϵ such that N > 2 and for all n > N, φ(N) < φ(n)
D
The set has finitely many limite points

Question 43Single Answer

For any two metric spaces (X, d_X), (Y, d_Y) a map

is said to be a closed map if whenever F is closed in X, then

(F) is closed in Y. For any subset B of a metric space, B is given the induced metric. The metric on X x Y is given by d((x, y), (x', y')) = max {d_X(x, x'), d_Y(y, y')}. Which of the following are true ?

A
For any subset A X in the inclusion map i : A → X is closed
B
The projection map p₁ : X x Y → X given by p₁ (x, y) = x is closed
C
Suppose that : X → Y, g : Y → Z are continuous maps. If g o : X → Z is a closed map then g|_f(x) : (X) → Z is closed. Here g|_f(x)means the map g restricted to (X)
D
If takes closed balls into closed sets then is closed

Question 44Single Answer

Let k be a positive integer. Consider the differential equation

Which of the following statements is true ?

A
It has a unique solution which is continuously differentiable on (0, ∞)
B
It has at most two solutions which are continuously differentiable on (0, ∞)
C
It has infinitely many solutions which are contiuously differentiable on (0, ∞)
D
It has no continuously differentiable solution on (0, ∞)

Question 45Single Answer

Let y₀ > 0, Z₀ > 0 and α > 1 .
Consider the following tow differential equations :

We say that the solution to a differential equation exists globally if it exists for all t > 0 .
which of the following statements is true ?

A
Both (*) and (**) have global solutions
B
None of (*) and (**) have global solutions
C
There exists a global solution for (*) and there exists a T < ∞ such that
D
There exists a global solution for (*) and there exists a T < ∞ such that

Question 46Single Answer

The genral solution of the surface which are perpendicular to the family of surface
z² = kxy, k ϵ

is :

Question 47Single Answer

The general solution of the equation

Question 48Single Answer

Let

be an infinitely differentiable real valued function on a bounded inerval I. Take n ≥ 1 interpolation points (x₀, x_i, ..., x_n-1}. Take n additional interpolation points

where is such that {x₀, x₁, ............... x_2n-1} are distinct .
Let P_2n-1 be the Lagrange interpolation polynomial of degree 2n - 1 with the interpolation points (x₀, x₁, ....., x_n-1} for the function

.
Let q_2n-1 be the Hermite interpolation polynomial of degree 2n - 1 with the interpolation points {x₀, x₁, ...., x_n-1} for the function

. In the

limit, the quantity

A
does not necessarily converge
B
converges to
C
converges to 0
D
converges to

Question 49Single Answer

The extremal of the function

is :

Question 50Single Answer

The solution of the Fredholm integral equation
y(s) = s + 2

is :

A
y(s) = −(50s + 40s²)
B
y(s) = (30s + 15s²)
C
y(s) = −(30s + 40s²)
D
y(s) = (60s + 50s²)

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Mathematical Science

Question 51Single Answer

Consider the solid S made of a material of constant density in the shape of a hemisphere of unit radius :
S = {x, y, z)|x²+y² +z² ≤ 1 z ≥ 0}
Which of the following statements is true ?

A
The centre of mass of S is at the origin
B
The x-axis is a principal axis for S
C
The moment of inertial tensor of S is not a diagonal matrix
D
The z-axis is a principal axis for S

Question 52Single Answer

In an examination involving multiple choice questions, a student works out the solution in questions the student guesses the answer. However, when the answer is guessed the probbility that is is correct is 0.30. When the student works out the solutions it may be wrong with probability 0.10 .
If the answer to a particular question is correct, what is the probability that the student guessed the answer ?

A
0.25
B
0.50
C
0.90
D
0.30

Question 53Single Answer

Let X₁, X₂, .......... be i.i.d. randomly variables having a x² distribution with 5 degrees of freedom. Let a

be constant.Then the limiting distribution of a

is :

A
Gamma distribution for an appropriate value of a
B
X²- distribution for an appropriate value of a
C
Standard normal distribution for an appropriate value of a
D
A degenerate distribution for an appropirate value of a

Question 54Single Answer

Consider a Markov Chain X₀, X₁, X₂, .......... with state space S. suppose i, j

S are two states which communicate with each other. Whcih of the following statements is NOT correct ?

A
Period of i = period of j
B
i is recurrent if and only if j is recurrent
C
D

Question 55Single Answer

Suppose that X has uniform distribution on the interval [0, 100]. Let Y denote the greatest integer smaller than or equal to X. Which of the following is true ?

Question 56Single Answer

Let X₁, X₂, ......... X_n be i.i.d random variables with common pdf

, for 0 1 is an unknown parameter. Then the statistics

is :

A
sufficient, but not complete
B
sufficient, but not minimal sufficient
C
complete sufficient
D
neither complete, nor sufficient

Question 57Single Answer

Consider the pdf given by

Based on one observation X with the above pdf, a UMP test of a size α for testing H₁ : θ ≤ θ₀ Versus H₁ : θ > θ₀ is :

A
X > k for some k such that α = Pθ₀[X > k]
B
X < k for some k such that α = Pθ₀[X > k]
C
X > k for some k such that α = Pθ₀[X < k]
D
X < k for some k such that α = Pθ₀[X < k]

Question 58Single Answer

Consider 35 i.i.d. observations X₁, X₂, ......... X₁₅ and Y₁, Y₂, .......... Y₂₀, Let R be the Wilicoxon's rank sum statistic based on the ranks of the X's in the combined sample. Then the expected value of R is :

A
270
B
300
C
360.5
D
330.5

Question 59Single Answer

Let I, J > 5. Consider two-way ANOVA, where the observations satisfy the linear model

In this set up

A
β₁ is estimable
B
y₁ is estimable
C
β₁ – β₂ is estimable
D
β₁ + g₂ is estimable

Question 60Single Answer

Let X₁ and X₂ be two i.i.d. p x 1 multivariate normal random vectors with mean µ and positive definite dispersion matrix Σ. Then which of the following random variables always has a central chi-square distribution .

Question 61Single Answer

Consider a Randomized Block Design with b blocks and k treatment. let the observation corrresponding to the ith treatment and the jth block bey_ij, 1≤i ≤k, 1 ≤ j ≤ b, which satisfies the usual linear model. Which of the following is true ?

A
The estimates of any two treatment contrasts are uncorrelated
B
The error sum of squares has bk -1 degrees of freedom
C
The estimate of any treatment contrast is uncorrelated with the estimate of any block contrast
D
The correlation between the estimates of two treatment contrasts is always negative

Question 62Single Answer

The maximum and the minimum values of 5x + 7y, when |x| + |y| ≤ 1 are

A
5 and −5
B
5 and −7
C
7 and −5
D
7 and −7

Question 63Single Answer

Suppose that {f_n} is a sequence of a real valued functions on

. Supose it converges to a continuous function uniformly on each closed and bounded subset of

. Which of the following statements are true ?

A
The sequence {f_n} converges to f uniformly on
B
The sequence {f_n} converges to f pointwise on
C
For all sufficiently large n, the function f_n is bounded
D
For all sufficiently large n, the function f_n is continuous

Question 64Single Answer

Which of the following sets are in bijection with

A
Set of all maps from {0,1} to
B
Set of all maps from to {0,1}
C
Set of all subsets of
D
Set of all subsets of

Question 65Single Answer

Which of the following statements are true ?

A
The series is convergent
B
The series is absolutely convergent
C
The series is convergent
D
The series is convergent c

Question 66Single Answer

Let

be defined by f(x, y) =

Define g(x, y) =

Which of the following statements are true ?

A
The function h(y) = g(c, y) is continuous on for all c
B
g is continuous from ²into
C
g is not a well defined function
D
g is continuous on

Question 67Single Answer

where x

[0, 1] which of the following statements are true ?

A
Both A(x) and B(x) converge pointwise
B
Both A(x) and B(x) converge uniformly
C
A(x) converges uniformly but B(x) does not
D
B(x) converges uniformly but A(x) does not

Question 68Single Answer

For

, consider the improper integral

Which of the following statements are true ?

Question 69Single Answer

Let f(x) = e^−x and g(x) = e^−x². Which of the following statements are true ?

A
Both f and g are uniformly continuous on
B
f is uniformly continuous on every interval of the form [a, +∞), a
C
g is uniformly continuous on
D
f(x) g(x) is uniformly continuous on

Question 70Single Answer

Define f

Which of the following statements are true ?

A
f is continuous at (0, 0)
B
f is continuous at (0, 0)
C
all directional derivatives of f at (0, 0) exist
D
f is not differentiable at (0, 0)

Question 71Single Answer

Define

A
f is continuous at (0, 0)
B
f is bounded in a neighbourhood of (0, 0)
C
f is not bounded in any neighbourhood of (0, 0)
D
f has all directional derivatives at (0, 0)

Question 72Single Answer

Let

be defined by :

Which of the following statements are true ?

A
p(x, y) = 0 if and only if x = y = 0
B
p(x, y) ≥ 0 for all x, y
C
p(αx, αy) = |α|p(x, y) for all and for all x, y
D
p(x₁+ x₂, y₁+ y₂ ) ≤ p(x₁, y₁ ) + p(x₂, y₂) for all (x₁,y₁ ), x₁+ y₂)

Question 73Single Answer

Let P be a square matrix such that P²= P. Which of the following statements are true ?

A
Trace of P is an irrational number
B
Trace of P = rank of P
C
Trace of P is an integer
D
Trace of p is an imaginary complex number

Question 74Single Answer

Let A and B be n x n real matrices and let

.
Which of the following statement are true ?

A
If is an eigenvalue of A + B, then is an eigenvalue of C
B
If is an eigenvalue of A – B, then is an eigenvalue of C
C
If is an eigenvalue of A or B, then is an eigenvalue of C
D
All eigenvalue of C are real

Question 75Single Answer

Let A be an n x n real matrix. Let b be an n x 1 vector. Suppose Ax = b has no solution. Which of the following statements are true ?

A
There exists an n x 1 vector c such that Ax = c has a unique solution
B
There exist infinitely many vector c such that Ax = c has no solution
C
If y is the first solumn of A then Ax = y has a unique solution
D
Det A = 0

Question 76Single Answer

Let A be an n x n matrix such that the first 3 rows of A are linearly independent and the first 5 columns of A are linearly independent. Which of the following statements are true ?

A
A has atleast 5 linearly independent rows
B
3 ≤ rank A ≥ 5
C
Rank A ≥ 5
D
Rank A² ≥ 5

Question 77Single Answer

Let n be a positive integer and F be a nonempty proper subset of {1, 2, .........., n} .
Define

Let

Which of the following statements are true ?
For

Question 78Single Answer

Let

be a non-zero vector. Define a linear transformation

Where x . y denotes the standard inner product in

.
Which of the following statements are true ?

A
The eigenvalues of T are +1, −1
B
The determinant of T is −1
C
The trace of T is +1
D
T is distance preserving

Question 79Single Answer

A quadratic form Q(x, y, z) over

represents 0 non triavially if there exists

such that Q(a, b, c) = 0. Which of the following quadratic forms Q(x, y, z) over

represent 0 non trivially ?

A
Q(x, y, z) = xy + z²
B
Q(x, y, z) = x²+ 3y² −2z²
C
Q(x, y, z) = x²− xy + y² + z²
D
Q(x, y, z) = x² + xy + z²

Question 80Single Answer

Let Q(x, y, z) be a real quadratic form. Which of the following statments are true ?

Question 81Single Answer

For

. Which of the following are true ?

A
has finitely many zeros
B
has a sequence of zeros that converges to a removable singularity of
C
has a sequence of zeros that converges to a pole of
D
has a sequence of zeros that converges to an essential singularity of

Question 82Single Answer

Let f be a holomorphic function on the open unit disc

Which of the following are possible values of

A
− i
B
i
C
1
D
− 1

Question 83Single Answer

Let

be a positive be the open unit disc and let

be a holomorphic function. Suppose that

. Which of the following are possible values of

Question 84Single Answer

Let n be a positive integer. For a real number R > 1, let z(θ) = Re^iθ, 0 ≤ θ < 2π .
The set (θ ∈ [0, 2π) : |z(θ)ⁿ+ 1| = |z(θ)ⁿ-1} contains which of the following sets ?

A
{θ ∈ [0, 2π) : cos nθ = 1}
B
{θ ∈ [0, 2π) : sin nθ = 1}
C
{θ ∈ [0, 2π) : cos nθ = -1}
D
{θ ∈ [0, 2π) : sin nθ = -1}

Question 85Single Answer

Which of the following statement are true ?

A
has countably many subgroups
B
has uncountably many subsets
C
Every finitely generate subgroup of is cyclic
D
is isomorphic to as groups

Question 86Single Answer

A
is a subgroup of
B
is not a normal subgroup of
C
has atleast two elements
D
is uncountable

Question 87Single Answer

Let G be a finite group. Which of the following are true ?

A
If has order m and if n ≥ 1 divids m, then G has a subgroup of order n
B
If for any two subgroups A and B of G, either , then G is cyclic
C
If G is cyclic, then for any two subgroups A and B of G, either
D
If for every positive integer m dividing |G|, G has a subgroup of order m, then G is abelian

Question 88Single Answer

Let R, S be commutative rings with unity, f : R → S be a surjective ring homomorphism

be a non-zero prime ideal. Which of the following statements are true ?

A
f ^-1(Q) is a non-zero prime ideal in R
B
f ^-1(Q) is a maximal ideal in R if R is a PID
C
f ^-1(Q) is a maximal ideal in R if R is a finite commutative ring with unity
D
f ^-1(Q) is a maximal ideal in R if x⁵ = x for all X ∈ R

Question 89Single Answer

Consider the polynomial f(x) = x² + 3x - 1. Which of the following statements are true ?

A
f is ireducible over
B
f is irreducible over
C
f is reducible over
D
is a unique factorization domain

Question 90Single Answer

Let p be an odd prime such that p = 2 (mod 3). Let

be the field with p elements. Consider the subset E of

given by

. Which of the following are true ?

A
E has atleast two elements
B
E has atmost 2p elements
C
E can have p² elements
D
E has atleast 2p elements

Question 91Single Answer

Consider the subset of

defined as follows :

Which of the following statements are true ?

A
A is connected
B
A is compact
C
A is closed
D
A is dense

Question 92Single Answer

Let X be a non-empty set. Suppose that

and

are two topologies over X, such that

.
Which of the following statements imply that

A
is compact and is (Hausdorff)
B
is compact and is (Hausdorff)
C
The connected components of both and are same
D
For any subset the closure of A in is contained in the closure of A in

Question 93Single Answer

The following two-point boundary value problem

has a trivial solution y = 0. It also has a non-trivial solution for :

A
no values of
B
λ = 1
C
λ = n² for all
D
λ ≤ 0

Question 94Single Answer

Let A be an n × n matrix with distinct eigenval ues {λ₁ ...,λ_n} with corresponding linearly in eigenvectors {v₁, .., v_n}. Then, the non-homogeneous differential equation x' (t) = Ax(t) + e^λ₁tv₁

A
does not have a solution of the form e^λ₁ta for any vector
B
has a solution of the form e^λ₁ta for some vector a
C
has a solution of the form e^λ₁ta + te^λ₁tb for some vectors
D
does not have a solution of the form e^λ₁ta + te^λ₁tb for any vectors

Question 95Single Answer

Consider the solutions

to the homogeneous linear system of differential equation

. Which of the following statements are true ?

A
y₁ and y₂ form a basis for the set of all solutions to (*)
B
y₁ and y₂ are linearly independent but do not form a basis for the set of all solutions to (*)
C
There exists another solution y₃ such that {y₁, y₂, y₃) form a basis for the set of all solutions to (*)
D
y, and y₂ are linearly dependent

Question 96Single Answer

Consider the partial differential equations :

Which of the folloss statements are true ?

A
Equation (i) is parabolic for v > 0 and elliptic for y < 0
B
Equation (i) is hyperbolic for y > 0 and elliptic for y > 0
C
Equation (ii) is elliptic in I and III quadrant and hyperbolic in II and IV quadrant
D
Equation (ii) is hyperbolic in I and III quadrant and elliptic in II and IV quadrant

Question 97Single Answer

Consider the Cauch Problem

Which of the following statements are true ?

A
A necessary condition for a solution to exist is that g is an odd function
B
A necessary condition for a solution to exist is that g is an even function
C
The solution (if it exists) is given by
D
The solution (if it exists) is given by

Question 98Single Answer

Fix

Consider the iteration defined by

The above iteration has two distinct fixed points

and

such that

Which of the following statements are true ?

A
The iteration (*) is equivalent to the recurrence relation
B
The iteration (*) is equivalent to the recurrence relation
C
D

Question 99Single Answer

Consider the function

defined by

where β ∈ (0, ∞) is a parameter. Consider the iterations

. Which of the following statements are true about the iteration ?

A
For β = 1, the sequence (x_k) converges to 0 linearly with asymptotic rate of convergence log₁₀2
B
For β > 1, the sequence {x_k} does not converge to 0
C
For β ∈ (0, 1), the sequence {x_k} converges to 0 sublinearly
D
For β ∈ (0, 1), the sequence {x_k} converges to 0 superlinearly