| Post Name | NON VOCATIONAL TEACHER MATHEMATICS (Junior & Senior) - KERALA VHSE |
|---|---|
| Question Paper Code | 083/2023 |
| Category Code | 073/2022, 074/2022, 402/2022, 453/2022, 572/2022, 573/2022, 609/2022 |
| Date of Test | 22 June 2023 |
| Medium of Question | ENGLISH |
| Exam Type | OMR Examination |
| Total Questions | 100 |
1
Let V=ℝ³. Which is not a subspace?
Answer
B. W2 = {( x1 , x 2 , x3 ) : x1 = 1, x 2 = 1}
2
Let {e₁,…,e_n} be the standard basis of ℝⁿ. The triangular family f₁=a₁₁e₁, …, f_n=a₁ₙe₁+⋯+a_nne_n is a basis iff:
Answer
B. a₁₁a₂₂⋯a_nn≠0
3
T:ℝ³→ℝ⁴ has T(1,0,0)=(1,2,0,4), T(0,1,0)=(2,0,−1,−3), T(0,0,1)=(0,0,0). Which expression is T(y₁,y₂,y₃)?
Answer
B. (y₁+2y₂, 2y₁, −y₂, 4y₁−3y₂)
4
Let T:ℝ³→ℝ³ be T(x₁,x₂,x₃)=(x₁,x₂,0). If r=dim(range T) and n=dim(null T), find 2ʳ+3ⁿ.
Answer
B. 7
5
Let λ be an eigen value of a non-singular matrix A. Then eigen value of adjoint of A, adj A is :
1
Answer
D. λ
A 3
6
Minimal polynomial of a matrix M = 0 4 − 2 is :
0 3 − 1
Answer
C. ( λ − 1) ( λ − 2)
7
Let A = − sin φ cos φ 0 be a matrix of orthogonal transformation, then its inverse is given by :
0 0 1
1 0 0 0 0 1
Answer
C. sin φ cos φ 0
8
Which of the following statement is not true?
n2
1
Answer
D. Limit of the sequence 1 + is e
n
9
Which of the following function is not uniformly continuous on
Answer
C. —
10
Let f(x)=|x−1|+|x+1| for x∈ℝ. Which statement is true?
Answer
D. Not differentiable at x=−1 and x=1
11
By Lagranges Mean value theorem the value of 3 65 lies in the interval :
1 1
Answer
C. 4, 4
12
Let {f_n} be defined by: (i) f_n(x)=nxe^(−nx²), x≥0, is not uniformly convergent on [0,1]; (ii) f_n(x)=x/(1+nx²), x∈ℝ, is uniformly convergent on [0,1].
Answer
A. Both (i) and (ii) are true
13
Let g be a bounded function on
Answer
D. all of the above
14
Which of the following improper integrals are convergent?
Answer
D. All of these
15
Let K be a positive integer and let J K = { x ∈
Answer
B. 10
K
4
16
Consider f ( x , y ) = x 2 + y2 . Then which of the following is true?
0; if ( x , y ) =
Answer
B. f is bounded in a neighbourhood of
17
Let f , g :
Answer
B. f is Riemann Integrable, g is Lebesgue Integrable
18
Let f: ℝ³ → ℝ³ be given by f(y₁, y₂, y₃) = (eʸ² cos y₁, eʸ² sin y₁, 2y₁ − cos y₃), and let E = {(y₁, y₂, y₃) ∈ ℝ³ : there exists an open set V around (y₁, y₂, y₃) such that the restriction f|_V is an open map}. Then which of the following is true?
Answer
C. {(y₁, y₂, π/2) ∈ ℝ³ : y₁, y₂ ∈ ℝ} is a proper subset of E
19
The value of dx is :
0
Answer
C. 1
20
The value of
0
1 1
Answer
A. β (2, 5)
21
The Fourier series corresponding to the function f ( x ) = contains :
3, 0
Answer
B. sine terms only
22
Let G have a*b=a+2b−3. The inverse of c∈G is:
Answer
B. (9−2c)/4
23
In the finite group G = {2, 4, 6, 8 } with respect to multiplication modulo 10, the identity
element is :
Answer
C. 6
24
Which of the following is true?
(i) Every group of order 42 has a normal subgroup of order 7
(ii) Every group of order 42 has a normal subgroup of order 8
Answer
C. only (i) is true
25
Every group of even permutations in Sn can be written as a product of —————— number
of transpositions:
Answer
D. 2 n + 8
26
Consider the statements :
(i) Number of homomorphisms from 10 to 25 is 5
(ii) Number of homomorphisms from 100 to 92 is 4
(iii) Number of homomorphisms from 16 to 19 is 2
(iv) Number of non-trivial homomorphisms from 13 to 19 is 1
Answer
B. only (i) and (ii) are true
27
The largest possible order of any element in Aut ( 200) is :
Answer
C. 20
28
Consider the statements :
(i) No group of order 96 is simple
(ii) No group of order 160 is simple
Then
Answer
B. Both (i) and (ii) are true
29
Which of the following is true? (i) S = {[[a, 0], [b, 0]] : a, b ∈ ℤ} is a subring of the ring M₂ of 2 × 2 matrices over integers. (ii) An element x ∈ R is called idempotent if x² = x. (iii) Let R be a ring such that x³ = x for all x ∈ R; then R is commutative. (iv) Let R be the ring of 2 × 2 matrices over reals; then S = {[[x, x], [x, x]] : x is a real number} is not a subring of R.
Answer
B. Only (i), (ii) and (iii) are true
30
Consider the statements: (i) R = {a + b√3 : a, b ∈ ℚ} is a field under usual addition and multiplication. (ii) S = {[[x, y], [−x̄, ȳ]] : x, y ∈ ℂ} is a division ring. (iii) ℂ = {a + ib : a, b ∈ ℝ} is a field under usual addition and multiplication of complex numbers. (iv) ℤ₄ is an integral domain. Then:
Answer
B. Only (i), (ii) and (iii) are true
31
Let R be the ring of all real valued continuous functions of the closed unit interval. Then
which of the following are maximal ideals of R :
1
(i) M 1 = f ∈ R : f = 0
5
2
(ii) M 2 = f ∈ R : f = 0
3
1
(iii) M 3 = f ∈ R : f = 0
3
Then
Answer
A. M 1 , M 2 and M 3 are maximal ideals
32
Let PID, ED and UFD denote the sets of principal ideal domains, Euclidean domains and unique factorization domains respectively. Then:
Answer
C. ED ⊂ PID ⊂ UFD
33
Which of the following is not true?
Answer
C. (1 − i ) is an irreducible element of [i]
34
The degree of the splitting field x 5 − 3x 3 + x 2 − 3 over is :
Answer
C. 4
35
Let G be the Galois group of the splitting field of x 5 − 2 over . Then :
(i) G is cyclic,
(ii) G is non-abelian
(iii) G has order 20
(iv) G has an element of order 4
Answer
C. (ii), (iii) and (iv) are true
36
Let τ be the topology on (natural numbers) consists of ϕ and all subsets of of the form
An = {n, n + 1, n + 2,...}, n ∈ . The set of all accumulation points of the set A =
Answer
C. {1, 2, 3, ...54}
37
Let X be a non-empty set. Under which of the following topologies τ₁ and τ₂ on X is the identity function I = (X, τ₁) → (X, τ₂) continuous? (i) τ₁ and τ₂ are the only two topologies on X. (ii) τ₁ is indiscrete and τ₂ is discrete. (iii) τ₁ is any topology and τ₂ is indiscrete. (iv) τ₁ is discrete and τ₂ is any topology.
Answer
D. Only (iii) and (iv)
38
In the set with infinite topology, the sequence , n ∈ converges to :
n
Answer
B. all real numbers
1 1
39
Which set is dense in ℝ under the topology consisting of ℝ, φ, and all open intervals (x, ∞), x ∈ ℝ?
Answer
A. {−10, −5, 0, 5, 10, …}
40
Which of the following is not metrizable? (i) ℝ with cofinite topology; (ii) ℝ with topology consisting of ℝ, φ and all intervals (x, ∞), x ∈ ℝ; (iii) ℝ with indiscrete topology.
Answer
D. all (i), (ii) and (iii)
41
Let X be a countable set in ℝ. The subspace topology on X induced by the usual topology in ℝ is:
Answer
D. Normal
42
Consider the metrics on ℝ² given by d₁(x,y) = 1 if x ≠ y and 0 if x = y; d₂(x,y) = |x₁ − y₁| + |x₂ − y₂|; and d₃(x,y) = max{|x₁ − y₁|, |x₂ − y₂|}, for x = (x₁,x₂) and y = (y₁,y₂) in ℝ². Then:
Answer
B. Only d₁ and d₃ are equivalent
43
The positive integer n for which (1+i)ⁿ=512 is:
Answer
Question Cancelled
44
The value of the complex number i −2 i is :
Answer
A. e ( 4 n +
45
The function f(z)=|z|² for z∈ℂ is:
Answer
B. differentiable at 0
46
If f and f̄ are both analytic on a region D in the complex plane, then:
Answer
B. f(z)=c for some constant c
47
If C is the circle |z| = 1, then ∮_C (z + 1)/(z⁴ + 2iz³) dz is:
Answer
A. −π/4 + iπ/2
48
If C is the circle |z|=1, then ∮_C e^(3/z) dz is:
Answer
D. 6πi
49
Let G⊂ℂ be open and connected and contain a real interval X. Let f,g:G→ℂ be analytic with f(z)=g(z) on X. Then:
Answer
D. g(z)=f(z)
50
F:ℝ⁵→ℝ is F(x₁,…,x₅)=x₁+2x₂+7x₃+3x₄+x₅. With the 2-norm on ℝ⁵, ||F|| is:
Answer
A. 8
51
Which is a Banach space? (i) c₀₀ as a subspace of ℓ∞; (ii) C[a,b] as a subspace of Lᵖ[a,b], 1 ≤ p < ∞; (iii) C[a,b] with the supremum norm; (iv) C′[a,b] with the supremum norm.
Answer
B. (iii) only
52
The adjoint of T:ℝ²→ℝ², T(x,y)=(2x−y,3x−y), is:
Answer
A. T*(x,y)=(2x+3y,−x−y)
53
Which is a continuous linear functional on an inner-product space X over ℂ? (i) f(x)=||x||; (ii) f(x)=⟨x,y⟩ for some y∈X; (iii) f(x)=⟨y,x⟩ for some y∈X.
Answer
B. (ii)
54
Which of the following metrics in ℝ² is not induced by a norm in ℝ², for x = (x₁,x₂) and y = (y₁,y₂) in ℝ²?
Answer
A. d₁(x,y) = 1 if x ≠ y and 0 if x = y
55
Let M be an orthonormal set. For distinct x,y∈M, d(x,y) is:
Answer
D. √2
56
The Hahn-Banach extension of h(x,y)=2x on Y={(x,y)∈ℝ²:x=y} to ℝ² with the 2-norm is:
Answer
B. f(x,y)=x+y
57
Solution of (tan⁻¹y−x)dy=(1+y²)dx is:
Answer
D. None of the above
58
Solve (1+x)²y″+(1+x)y′+y=2sin[log(1+x)].
Answer
B. y=c₁cos[log(1+x)]+c₂sin[log(1+x)]−log(1+x)cos[log(1+x)]
59
If p(x) is a continuous function, the general solution of y″ − p(x)y′ + [p(x) − 1]y = 0 is:
Answer
D. y = c₁eˣ + c₂eˣ∫e^(−2x + ∫p(x)dx) dx
60
Consider y′=y^(1/3), y(0)=0 on X={(x,y)∈ℝ²:|x|≤1, |y|≤2}. Which is true?
Answer
D. The differential equation does not have a unique solution
61
The equation dy/dx=2y^(1/2), y(0)=0 has:
Answer
B. Infinitely many solutions
62
Solutions of 2xz−px²−2qxy+pq=0, where p=∂z/∂x and q=∂z/∂y, are:
Answer
C. z=c₁y+c₂(x²−c₁)
63
Solve ∂²z/∂x²+4∂²z/(∂x∂y)−5∂²z/∂y²=sin(2x+3y).
Answer
A. z=φ₁(y+x)+φ₂(y−5x)+(1/17)sin(2x+3y)
64
The remainder when 3181 is divided by 17 is :
Answer
D. 5
65
The last two digits of 31000 is :
Answer
B. 01
66
If 1 + 7 + 7² + … + 7¹⁷ ≡ a (mod 19), then a is:
Answer
C. 0
67
The number of positive integers less than or equal to 2076 and divisible by neither 4 nor 5 is :
Answer
B. 1245
68
The ones digit in the decimal value of 176666 is :
Answer
C. 9
69
Which of the following number is divisible by 7?
Answer
Question Cancelled
70
The number of incongruent solutions of 155x ≡ 75 (mod 65) is:
Answer
A. 5
71
The main objective/s of learning is/are :
(i) To gather information about student weaknesses.
(ii) Modification of behaviour.
(iii) To adopt innovative methods of teaching.
(iv) To identify the areas of further improvement in teaching learning process.
Answer
D. (ii) only
72
The evaluation to assess the overall effectiveness of a program and grade the pupil is :
Answer
C. Summative evaluation
73
Backward exploration is a shortcut that makes the process of problem solving quick and
efficient. What does backward exploration mean?
Answer
A. It is a process in which we analyse the previous solutions of a problem, apply
them to the problem and check which is relevant one.
74
What is the sequence of steps of lesson planning?
(i) Followup activity
(ii) Developmental activity
(iii) Setting of specific objectives
(iv) Introductory activity
(v) Recapitulation
Answer
C. (iii), (iv), (ii), (v), (i)
75
At which of the following stages of Piaget’s theory of cognitive development is a child
confronted with confusing ideas?
Answer
D. Preoperational stage
76
Match the statement given in the List A with the List
Answer
C. (3) (1) (2)
77
Which one of the following refers to the authenticity or genuineness of the document?
Answer
D. Internal Criticism
78
Which of the following is the correct sequence of steps in historical research?
(i) Identify primary and secondary data sources.
(ii) Analyze the data and develop a narrative exposition of the findings.
(iii) Evaluate the authenticity and accuracy of source materials.
(iv) Collection of data
Answer
A. (i), (iv), (iii), (ii)
79
Which of the following options most appropriately explains ‘Research Ethics’?
Answer
D. It provides a common set of do's and don’ts of conducting ethical research
80
An academic lecture or presentation to an audience on certain topics of educational nature, it
is called :
Answer
C. Seminar
81
The Constitution of India exhibits federal characteristics, eventhough it does not explicitly
define India as a federal state. Which of the following statements are TRUE about the federal
characteristics of the Indian Constitution?
(i) The Constitution provides for an independent and impartial judiciary.
(ii) The Constitution establishes institutional checks and balances to limit the power of
both states and the centre.
(iii) The central government possesses more powers than state governments, making the
constitution more unitary in nature.
Answer
D. All the above (i), (ii) and (iii)
82
The Indian Constitution includes a comprehensive chapter on fundamental rights. Which of
the following statements regarding fundamental rights in the Indian Constitution is/are
TRUE?
(i) The right to life and liberty prohibits discrimination based on religion, race, caste, sex,
or birthplace.
(ii) Individuals can approach the judiciary in India to enforce their fundamental rights
under the right to constitutional remedy.
(iii) Cultural and educational rights protect the interests of minorities by preserving their
language, script and culture.
Answer
B. Only (ii) and (iii)
83
Which of the following is NOT true about the Indian judiciary?
Answer
D. The doctrine of “due process of law” gives the judiciary more power than the
legislature.
84
Find the appropriate match between the colunms of constitutional Amendments in India and
the Goals of Amendment in the following table. Then, select the correct answer group :
Constitutional Amendments Goals of Amendment
(i) 102nd Amendment (1) The Bill aims to give State governments the power to
identify socially and educationally backward OBCs
(ii) 103rd Amendment (2) To extend the reservation of seats for Scheduled
Castes and Scheduled Tribes in the Lok Sabha
and State Assemblies until 2030
(iii) 104th Amendment (3) To reserve seats for economically weaker sections of
society in government educational institutions
and public employment.
(iv) 105th Amendment (4) Create a National Commission for Backward
Classes and grant the President the authority
to identify socially and educationally disadvantaged groups.
(i) (ii) (iii) (iv)
Answer
D. (4) (3) (2) (1)
85
Though instituted much later, the fundamental duties are an inevitable part of the Indian
Constitution. Which of the following statement is/are TRUE about the fundamental duties
enshrined in the Constitution?
(i) They are a set of moral and ethical obligations that every citizen of India is expected to
fulfil.
(ii) These are not legally enforceable but promote responsibility and patriotism among
citizens.
(iii) They cover a specific range of areas and are limited to duties related to the
Constitution, the national flag and anthem.
(iv) They are complementary to the fundamental rights enshrined in the Indian
Constitution.
Answer
C. Only (i), (ii) and (iv)
86
Which of the following is NOT one of the effects of the emergency provisions mentioned in the
Indian Constitution?
Answer
B. The life of the Rajya Sabha can be extended by one year at a time.
87
The Constitution of India contains an exclusive provision for amendment. Which of the
following statements is TRUE about the amendment procedure in the Constitution of India?
(i) Amendments related to the admission or establishment of new states can be made by a
simple majority of the two houses of Parliament.
(ii) The amendment procedure balances the need for flexibility with the need for stability
and continuity.
(iii) The parliament has initiated the “basic structure” doctrine of the Constitution, which
cannot be amended.
Answer
A. Only (i) and (ii)
88
Which of the following statements about the Directive Principles of State Policy
Answer
C. The DPSPs are enforceable by any court in India.
89
Which among the following are the functions of the NHRC?
(i) To enquire complaints of violation of human rights or negligence in the prevention of
violation by a public servant
(ii) To study human rights treaties and international instruments, and recommend their
effective implementation to the government.
(iii) To spread human rights awareness and encourages efforts in human rights literacy at
national and international levels.
(iv) To hear complaints related to service matters and matters that are sub-judice.
Answer
A. Only (i), (ii) and (iii)
90
Which among the following are the functions of the Union Public Service Commission?
(i) Conducting interviews and assessments for various military positions
(ii) Conducting competitive examinations for various civil services positions.
(iii) Selecting candidates for various civil services positions based on merit and performance
(iv) Advising the government on matters related to personnel management and recruitment
Answer
B. Only (ii), (iii) and (iv)
91
Which statement is/are incorrect regarding the Ramon MagSaySay Award?
(i) It is known as the “Nobel Prize of Asia”.
(ii) Dalailama is the first recipient of this Award as spiritual leader.
(iii) Blessed Mother Teresa is a recipient of this Award in 1986.
(iv) Dr. M.S. Swaminathan is not a recipient of this Award.
Answer
D. (iii) and (iv) only
92
ICC Men’s T20 world cup held in November 2022. Which statement is/are correct regarding
the final match?
(i) It was played at Melbourne Cricket Ground on 13th November 2022.
(ii) England won the match by 5 wickets.
(iii) Kumar Dharmasena was an on-field umpire in the match.
(iv) Joe Root of England was the ‘Player of the Match’.
Answer
B. (i), (ii) and (iii) only
93
Write the chronological order of the formation of the following socio-religious reform
movements of Kerala :
(i) Islam Dharma Paripalana Sangham.
(ii) Vaala Samudaya Parishkarani Sabha.
(iii) Samathwa Samajam
(iv) Sadhu Jana Paripalana Sangham.
Answer
C. (iii), (i), (iv), (ii)
94
Which statement is/are incorrect regarding the Paliyam Sathyagraha?
(i) It was officially inaugurated by Panampilly Govinda Menon.
(ii) It was the first post independent Satyagraha organized in Kerala.
(iii) It was supported by women from all parts of Kerala.
(iv) A.G. Velayudham was killed in police lathi charge.
Answer
A. (i) only
95
The Abstention Movement was a historic event in Travancore history. From the following
statements, find out the wrong statement/statements :
(i) It was a joint venture of the Ezhava, Muslim and Christian Communities.
(ii) C. Kesavan, delivered the famous ‘Kozhencherry Speech’ on May 11, 1934.
(iii) Franchise was widened by reducing the property qualification.
(iv) It brought the issue of responsible government in Travancore.
Answer
B. (ii) only
96
Match the following :
(i) Rajyasamacharam (1) Moorkoth Kumaran
(ii) Vivekodayam (2) Swami Guruprasad
(iii) Gajakesari (3) Basel Mission Society
(iv) Mithavadi (4) Ezhava Gazette
(i) (ii) (iii) (iv)
Answer
D. (3) (4) (2) (1)
97
Find out the incorrect statement/statements from the following :
(i) Bhaskara Menon is regarded as the first detective novel in Malayalam.
(ii) Kerala Sangeetha Nataka Academy was inaugurated by Pandit Jawaharlal Nehru, the
then Prime Minister of India, on 26 April 1958.
(iii) M.K. Sanu was the President of Purogamana Kala Sahithya Sangham during 1988-90.
(iv) Lilathilakam is a 16th Century treatise on Malayalam grammar and poetics.
Answer
D. (iv) only
A 21 083/2023
98
Match the following :
(i) Parvathy Nenmenimangalam (1) Agnipushpangal
(ii) Lalithambika Antharjanam (2) KPCC President in 1944
(iii) A.V. Kuttimalu Amma (3) Founder and editor of the journal “Shreemati”
(iv) Justice Anna Chandy (4) Antharjana Samajam
(i) (ii) (iii) (iv)
Answer
A. (4) (1) (2) (3)
99
Find out the wrong pair from the following table :
(i) Dr. Ayyathan Gopalan – Rao Sahib
(ii) K.P. Vallon – ‘Adhakrithan’ magazine
(iii) P.K. Chathan Master – Member of Legislative Assembly from
Chelakkara
(iv) T.K. Krishnaswami Iyer – Called ‘Untouchable Brahmin’
Answer
C. (iii) only
100
Match the following :
(i) P.Kesavadev (1) Ente Vazhiyambalangal
(ii) Ponkunnam Varkey (2) Ethirppu
(iii) S.K. Pottekkatt (3) Kazhinjakalam
(iv) K.P. Kesava Menon (4) Ente Vazhithirivu
(i) (ii) (iii) (iv)
Answer
B. (2) (4) (1) (3)
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