| Post Name | HSST(Junior)/NVT(Senior/Junior)- Mathematics 046/2024,442/2024 |
|---|---|
| Question Paper Code | 074/2025 |
| Category Code | 046/2024, 442/2024, 626/2024 |
| Date of Test | 30 June 2025 |
| Medium of Question | ENGLISH |
| Exam Type | OMR Examination |
| Total Questions | 100 |
1
The number of bases for a nontrivial vector space is :
Answer
D. Infinite
1 + t
2
Let V = 2 − 3t / t ∈ R . Vector addition on V is defined by
3 + 2 t
1 + t1 1 + t2 1 + t1 + t2
2 − 3 t + 2 − 3t = 2 − 3 (t + t ) and Scalar Multiplication by
1 2 1 2
3 + 2 t1 3 + 2 t2 3 + 2 (t1 + t2 )
1 + t 1 + ct
c ∗ 2 − 3 t = 2 − 3 ct . If V is a vector space, find the additive identity :
3 + 2 t 3 + 2 ct
1 1
Answer
B. 2
2 3
1 1
3
The dimension of the Vector Space Pn , where Pn represents the set of all polynomials of
degree n or less under usual addition and multiplication is :
Answer
D. n + 1
1 2 3
4
Let A = [[1, 2, 3], [1, 2, −3], [2, −1, 3]]. Then A · adj A =
Answer
C. [[−30, 0, 0], [0, −30, 0], [0, 0, −30]]
5
Let A =
c d
a b 1 0
Answer
D. 0 0
1 0 0
6
Let T: ℝ³ → ℝ³ be a linear operator and B a basis for ℝ³ given by B = {[[1], [0], [0]], [[0], [1], [0]], [[0], [0], [1]]}. T([[1], [0], [0]]) = [[1], [−1], [0]], T([[0], [1], [0]]) = [[2], [0], [1]], T([[0], [0], [1]]) = [[1], [−1], [1]]. Then find T([[1], [2], [3]]):
Answer
B. [[8], [−4], [5]]
7
Find the angle between the two vectors u = [[3], [0], [0]] and v = [[0], [2], [0]].
Answer
A. θ = π/2
8
Find lim = –––––––.
x → 0 tan 4 x
Answer
A. 1
9
The maximum value of log x is:
2x
1 1
Answer
A. =
10
The Rolle’s theorem is applicable for the function y = tan x in the interval :
Answer
D. None of these
11
Let K ⊂ ℝ consist of 0 and the numbers 1/n for n = 1, 2, 3, …. Then K is:
Answer
B. Compact Set
12
Which of the following is true for the series Σ_{n=1}^∞ (−1)ⁿ(x² + n)/n²?
Answer
B. Converges uniformly in every bounded interval
13
If f: R → R is defined by f(x) = [x], where [·] is the greatest-integer function, choose the correct option.
Answer
C. f (x ) is Riemann Integrable
14
Compute dx dy dz :
0 0 0
Answer
A. 8
15
Which of the following sets have Lebesgue measure not equal to zero?
Answer
D. (2, 5)
x − y+2 x −2 y
16
Evaluate lim as (x,y) → (0,0), x ≠ y, of [x − y + 2√x − 2√y]/[√x − √y].
Answer
C. 2
17
Find the tangent plane to f(x,y,z) = x² + y² + z² − 3 = 0 at (1,1,1).
Answer
D. x + y + z = 3
18
Find the directional derivative of f(x, y) = eˣ cos 2y at the point {0, π/4} in the direction v = 2i + 3j:
Answer
B. −6/√13
19
Which of the following is a critical point of the function f(x, y) = x³ + y² − 2xy + 7x − 8y + 2?
Answer
B. (−1/3, 11/3)
20
Which of the following is not true for a Gamma function (Γ)?
Answer
C. Γ(2) = 2
21
Express the integral 1 − x dx in terms of Beta function (β ) :
0
5
1 3 1 1 3 3
Answer
A. β ,
22
Which of the following group is cyclic?
Answer
C. The group of roots of xⁿ − 1 = 0
23
Every finite group is isomorphic to a subgroup of :
Answer
B. Sₙ, for some n ≥ 0
24
Find the order of the element (3, 6, 12, 16) in Z 4 × Z12 × Z 20 × Z 24 ?
Answer
C. 60
25
Which of the following pairs of groups are isomorphic?
(i) The additive group R of real numbers
(ii) The additive group of all integral multiples of π
(iii) Multiplicative group of positive real numbers
(iv) Multiplicative group of non zero complex numbers
Answer
D. Only (i) and (iii)
26
The number of non-isomorphic group of order 10 :
Answer
A. 2
27
Number of generators of the additive group Z36 is equal to :
Answer
C. 12
28
Let μ = (1, 2, 4, 5) (3, 6 ) be a permutation in S6 . Find the index of cyclic subgroup
generated by μ in S6 :
Answer
C. 180
29
The set of all 2 × 2 matrices with entries in the field Z2 is an example of :
Answer
B. Finite noncommutative ring
30
Which is not a solution of x³ − 2x² − 3x = 0 in Z₁₂?
Answer
D. 2
31
The sylow 2 subgroup of a group of order 12 has order __________.
Answer
D. 4
32
Find the degree of field extension Q ( 2 + 3 ) over Q ( 3 ) :
Answer
A. 2
33
The number of maximal ideals of ring Z6 :
Answer
B. 2
34
Find all c ∈ Z5 such that is a field :
(x + c x + 1)
2
Answer
A. 1, 4
35
Number of units in the ring Z × Z :
Answer
C. 4
36
What is the open ball Bₛ(0,1) in the metric subspace S = [0,1] of Euclidean R?
Answer
B. [0,1)
37
Which of the following statement is true about a continuous function?
Answer
C. The image of a connected space under a continuous map is connected
38
Which of the following statement is correct?
Answer
C. Every compact subspace of a Hausdorff space is closed
39
Which of the following property is a hereditary property of a Topological space?
Answer
A. Regularity
40
Let S = {(x, sin(1/x)) : 0 < x ≤ 1} in the plane. Then which of the following is not true?
Answer
C. S̄ is path connected
41
Which of the following statement is not correct?
Answer
D. A subspace of a Lindelof space is Lindelof
074/2025 8 A
42
Which of the following statement is correct about the closure of subsets of a topological space?
Answer
B. overline(A ∪ B) = Ā ∪ B̄
43
What is the radius of convergence of the power series
1
Answer
A. 4
44
What is the residue of f(z) = z³/(z² − 1) at z = ∞?
Answer
D. −1
45
Determine the number of roots of z⁸ − 4z⁵ + z² − 1 that lie inside the circle |z| = 1.
Answer
B. 5
46
What is the value of ∫₀∞ dx/(1 + x⁴)?
Answer
D. π/(2√2)
47
If f (z ) is an analytic function in a domain D, such that real part of f (z ) is constant in D,
then which one of the following is not true?
Answer
B. f(z) is the identity function
48
Consider the two statements for z ∈ ℂ:
(i) e z is a one to one function
(ii) e z is a bounded entire function
Then
Answer
D. both (i) and (ii) are false
A 9 074/2025
49
Let C be the circle |z| = 3/2 in the complex plane that is oriented in the counter clockwise direction. The value of a for which ∮_C [(z + 1)/(z² − 3z + 2) + a/(z − 1)] dz = 0 is:
Answer
C. 2
50
Let H be a Hilbert space over R and x,y ∈ H satisfy ||x|| = 1, ||y|| = 3 and ||x − y|| = 3. Then ||x + y|| is:
Answer
D. √11
51
Let X be an inner product space. Then the orthogonal complement of {0} is :
Answer
A. X
52
Consider the statement :
(i) Every finite dimensional normed linear space is a Banach Space
(ii) Every Banach space is finite dimensional linear space
Answer
A. Only (i) is true
53
Let X and Y be normed space and F : X → Y be a linear map with :
(i) F is bounded on u
Answer
D. , r ) for some r > 0
(ii) F is continuous at 0
(iii) F is continuous on X
(iv) F is uniformly continuous on X . Which of the following is true?
54
Let H be a Hilbert space :
(i) Let A and B be self adjoint then A + B is self-adjoint.
(ii) Let A and B be unity then AB is unitary
(iii) Let A and B be normal then AB is normal. Which of them is correct?
Answer
A. Only (i) and (ii)
55
An operator A is called normal if :
Answer
B. AA* = A*A
56
Consider f : R → R . Which of the following is not a linear map?
Answer
B. f(x) = 5x + 4
57
Which of the following differential equations are homogeneous?
y+x
(i) y′ =
x
y + x2
(ii) y′ =
x3
x
y
2xye
(iii) y′ =
x
x 2 + y2 sin
y
Answer
B. Only (i) and (iii)
58
Find a particular solution of y′ + y² = 0.
Answer
A. 1/x
59
What is the integrating factor of the equation y′ = 2xy − x ?
Answer
D. e−x
60
What is the solution of the partial differential equation (p² + q²)y = qz?
Answer
B. (x + b)²y = az²
61
Which of the following differential equation are linear?
(i) y′ = (sin x ) y + e x
(ii) y′ = x sin y + e x
(iii) y′ = 5
(iv) y′ = y2 + x
Answer
C. Only (i) and (iii)
62
The general solution of z(xp − yq) = y² − x² is:
Answer
D. F(xy,x² + y² + z²) = 0
63
For the initial-value problem y dx/dy = x, x(0) = 0 and y > 0:
Answer
D. There are infinitely many solutions
64
How many Primes are occurred in the etic sequence?
a, a + b, a + 2b, a + 3b,.... where a and b are relatively prime.
Answer
A. Infinite
65
Find the number of positive integers ≤ 2076 and divisible by neither 4 nor 5 :
Answer
C. 831
66
The remainder of 14 84 + 3791 on division by 11 :
Answer
B. 8
67
The inverse of ‘ a ’ modulo p , if p is prime and p does not divide a is :
1
Answer
C. a p −2
68
The value of x for which the linear congruence 35x ≡ 47 (mod 24 ) :
Answer
A. 13
69
Which are the incongruent solutions of x³ − 1 ≡ 0 (mod 13)?
Answer
B. 1, 3, 9
70
The solution of the linear system x ≡ 1(mod 3), x ≡ 2(mod 4 ), x ≡ 3(mod 5) . Is :
Answer
A. 58
71
Which is not a characteristic of Problem Based Learning?
Answer
D. Specification of a predetermined end product
72
Field dependent learners are :
Answer
C. Social and Co-operative
73
Two statements are given below :
(i) Culturally responsive teaching recognizes the importance of creating an
environment where all students feel valued, respected, and understood.
(ii) Avoiding culturally sensitive assessments helps ensure that all students have an
equal opportunity to succeed.
Select the correct one with respect to these statements
Answer
B. Statement (i) is correct, but Statement (ii) is wrong
74
Learner related factors affecting teaching are :
(i) Previous experience of the learner
(ii) Maturity level of the learner
(iii) Parental education of the learner
Answer
B. (i) and (ii)
75
A mathematics teacher asks students to write a reflective journal to document their
approach, strategies, and thought processes as they work through the mathematics
problem. The assessment in this case is :
Answer
A. Assessment as learning
76
Two statements are given below :
Statement (i) : Internal validity of an experimental research is always ensured
statistically.
Statement (ii) : Internal validity examines the extent to which systematic error is
present in the experimental research.
Which among the following is true with respect to these statements?
Answer
C. Statement (i) is incorrect, but (ii) is correct
77
Which qualitative research approach is primarily concerned with understanding the
lived experience?
Answer
D. Phenomenology
78
Dr. X uses their own published work without indicating that it has been published
earlier. This is a case of :
Answer
B. Auto Plagiarism
79
Formulation of hypothesis helps in :
(i) Making the study focused
(ii) Reporting the results objectively
(iii) Review of related studies
Answer
A. (i) and (ii)
80
Consider the following statements :
(i) Conference and symposium focus on dissemination of knowledge
(ii) Conference is a smaller event compared to symposia
(iii) In conference there will be presentations by researchers, professionals and experts
on their latest research findings, exchange ideas and network.
(iv) A symposium is usually centred on a specific theme or topic.
Which statements are correct?
Answer
C. (i), (iii) and (iv)
81
Consider the following statement(s) in respect of Anti-defection Law in India :
(i) The 10th schedule of the Constitution of India is about Anti -Defection Law.
(ii) The decision on disqualification questions on the ground of defection is referred to
the President who will take a decision on the advice of Election Commission.
(iii) All proceedings in relation to the disqualification under this Schedule are
considered to be proceedings in Parliament or the Legislature of a state as is the
case.
Which of the statement(s) given below is/are correct?
Answer
C. Only (i) and (iii)
82
Which of the following statement(s) is/are correct about GST Council?
(i) The Union Finance Minister is the Chairperson of the GST Council
(ii) The Prime Minister of India is not a member of the GST Council
(iii) The Finance Secretary of the Government of India is a member of the GST Council
Answer
A. Only (i) and (ii)
83
Which of the following statement(s) is /are true with reference to the Joint sitting of the
Parliament?
(i) According to Article 108, the President can make rules for the procedure of the
joint sitting after due consultation with the Chairman of the Council of States and
the Speaker of the House of the People.
(ii) The President of India has the power to call a joint sitting
(iii) In the absence of the Speaker and the Deputy speaker of the Lok Sabha, the joint
sitting of the Parliament is presided by the Chairman of the Rajya Sabha
Answer
C. Only (ii)
84
Which statement(s) regarding the Estimates Committee is/ are True?
(i) The Estimates Committee is the Largest Committee of the Parliament
(ii) The term of the committee is Two years
(iii) Estimates committee consisting of members from both Houses of Parliament
Answer
C. Only (i)
85
Which of the following statements about Emergency provisions under the Constitution of
India is/are not correct?
(i) Proclamation of Emergency must be approved by the two houses of Parliament
with special majority within two months
(ii) Every resolution approving the Proclamation of Emergency or its continuance must
be passed by either house of Parliament by a special majority
(iii) The 44th Amendment Act made the Declaration of National Emergency immune to
Judicial review.
Answer
Question Cancelled
86
Consider the statements given below :
(i) 42nd Amendment Act of 1976 moved Four subjects from the State List to the
Concurrent List.
(ii) Article 248 empowers the Parliament to legislate with respect to a matter in the
State List in the national interest.
(iii) Establishment of standards of weight and measure is included in the Union List.
Which of the following statement(s) is/are correct regarding the 7th Schedule of the
Constitution of India?
Answer
D. Only (iii)
87
Which of the following statement(s) is/are correct about Domestic Violence Act of 2005?
(i) Section 16 deals with Proceedings to be held in Camera
(ii) Section 3 deals with Definition of Domestic violence
(iii) Section 22 deals with the Duties and Functions of the Protection Officers
Answer
A. Only (i) and (ii)
88
Which of the following statement(s) regarding Central Information Commission is /are
correct?
(i) The Central Information Commission is a statutory body, set up under Section 18
of the Right to Information Act in 2005
(ii) It consists of a Chief Information Commissioner and not more than Eight
Information Commissioners
(iii) Section 14 of the Right to Information Act 2005 deals with Removal of the Chief
Information Commissioner or Information Commissioners
Answer
C. Only (iii)
89
With reference to the sections of The Wildlife Protection Act 1972, consider the following
statements :
(i) National Board for Wildlife was constituted under Section 5A.
(ii) National Tiger Conservation Authority was constituted under section 38L.
(iii) Section 6 of the Act deals with Prohibition of hunting.
Which of the above statement(s) is/are not correct about Wildlife Protection Act 1972?
Answer
B. Only (iii)
90
Consider the following statements regarding 8th Schedule of the Constitution of India :
(i) The 71st Amendment to the Indian Constitution included Konkani, Manipuri and
Nepali languages in the 8th Schedule.
(ii) The Sindhi language was added in the 8th Schedule by the 31st Amendment Act.
(iii) Bodo, Dongri, Maithili and Santhali were added by the 92nd Amendment Act.
Which of the above statement(s) is/are correct?
Answer
D. Only (i) and (iii)
91
‘Devagita’ is a well-known Malayalam translation of Jayadev's “Geeta Govinda’’. Name
the poet who translated it into Malayalam :
Answer
D. Changampuzha Krishna Pillai
92
The author of the book entitled ‘Wonder that was India’ :
Answer
A. A.L. Basham
93
Three Language formula for education was proposed by :
Answer
A. Kothari commission
94
The Peasant uprising of Malabar was known as :
Answer
B. Mappila Rebellion
95
The Gayathri Mantra is the most reversed Vedic Mantra in Hinduism. Which Vedic Sage
is considered as the author of Gayathri Mantra?
Answer
A. Sage Viswamitra
96
Tinkathia system is associated with which one of the following movement?
Answer
A. Champaran Satyagraha
97
Who was Saradamani?
Answer
B. Wife of Rama Krishna Paramahamsa
98
Community development programme is implemented at the :
Answer
B. Block level
99
Which one of the following is wrongly listed with regard to Kanpur conspiracy case?
Answer
D. M.N. Roy
100
‘Hinduism dies if untouchability lives and untouchability has to die if Hinduism is to
live’. This was said by :
Answer
C. Gandhi
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