Question Paper Code: 66/2022/OL
Category Code: 484/2019
Exam: Higher Secondary School Teacher Mathematics
Date of Test 13-08-2022
Department Kerala Higher Secondary Education
Alphacode A

A:-The set of all non-invertible real matrices
B:-The set of all matrices A with det A = 0
C:-The set of all matrices with trace A = 0
D:-None of the above

Question2:-Let A be a 5 × 3 matrix and B be a 3 × 7 matrix then the rank of AB can be:
A:-5
B:-3
C:-7
D:-All of the above

Question3:-The number of linearly independent eigen vectors of the matrix

is

A:-1
B:-2
C:-3
D:-4

Question4:-Let A be a 5 × 5 real matrix. Which of the following is true?
A:-If A is an orthogonal matrix, then det A = 1
B:-If A is an nilpotent matrix, then det A = 1
C:-If A is an orthogonal matrix, then det A = -1
D:-If A is a skew-symmetric matrix, then det A = 0

Question5:-Which of the following is the Jordan form of a 3 × 3 matrix? Then which of the following statements is true?
A:-T is one-one
B:-T is both one-one and onto
C:-T is onto
D:-T is neither one-one nor onto

A:-1
B:-2
C:-3
D:-4

A:-f is continuous on Q
B:-f is continuous on R/Q
C:-f is continuous on ℝ
D:-f is not continuous at 0

Question10:-Which of the following functions is uniformly continuous on (0,1)?

A:-f has a fixed point
B:-f is not uniformly continuous
C:-f has a unique fixed point
D:-f is not bounded

Which of the following statements is true?
A:-The Riemann lower sum of f corresponding to any partition  is 1
B:-The upper Riemann integral of f is 1/2
C:-f is Riemann integrable
D:-None of the above

Question15:-Which of the following sets has nonzero Lebesgue measure?

Choose the correct statement from the following:
A:-f is continuous at (0,0)
B:-f is differentiable at (0,0)
C:-f has directional derivatives at (0,0) in all directions
D:-The directional derivatives of f at (0,0) in the direction of (1,0) does not exists

Answer : Option C (Question deleted )

Question20:-Let f: ℝ² → ℝ² be defined by f(x,y)=(x+y, x-y), choose the correct statement from the following:
A:-Df(0,0) does not exist
B:-Df(0,0) exists and is invertible
C:-Df(0,0) exists but not invertible
D:-f does not have a local inverse at (0,0)

A:-f is Lebsegue integrable and its Lebsegue integral is 1
B:-f is Riemann integrable
C:-f is not Lebsegue integrable
D:-f is Lebsegue integrable and its Lebsegue integral is 0

Question22:-Let G be a non-abelian group of order 6. Then the number of subgroups of G of order 2 is
A:-1
B:-2
C:-3
D:-4

Question23:-Which of the following pairs of groups are isomorphic:

Question24:-The number of abelian groups of order 36 (up to isomorphism) is
A:-8
B:-4
C:-2
D:-1

Question25:-Let G be a group of order n. Then G is cyclic if n is:
A:-15
B:-21
C:-39
D:-55

Question26:-Let G be a group of order 28. Which of the following statements is true?
A:-G is simple
B:-G has at least 2 subgroups of order 7
C:-G has a normal subgroup of order 7
D:-Number of elements in G of order 7 is 7

Question27:-Which if the following is a class equation of a group of order 10?
A:-10 = 2 + 2 + 2 + 2 + 2
B:-10 = 1 + 2 + 2 + 5
C:-10 = 1+ 1+ 3 +5
D:-10 = 1 + 1 + 1 + 1 + 1+ 5

Question28:-Choose the correct statement from the following:

Question30:-Let <p(x)> denotes the ideal generated by the polynomial p(x) in ℤ [𝑥]. Choose the correct statement from the following
A:- < 𝑥 > is a maximal ideal
B:- < 𝑥2 > is a prime ideal
C:-Every prime ideal in ℤ [𝑥]is a maximal ideal
D:-The set {𝑝[𝑥] ∈ ℤ [𝑥]:𝑝(0) = 0 = 𝑝(1)} is not a prime ideal

Question31:-Choose the true statement from the following:
A:-ℤ is a UFD but  ℤ [𝑥] is not a UFD
B:-ℤ  and ℤ [𝑥] are both PIDs
C:-ℤ is a PID but  ℤ [𝑥] is not a PID
D:-ℤ and ℤ [𝑥] are both Euclidean domains

Question32:-Which of the following is a field?

Question33:-The Galois group of the splitting field of 𝑥3− 2 over ℚ is isomorphic to:
A:-z3
B:- z4
C:-D4
D:-S3

Question34:-The number of generators of the multiplicative group of a field of order 81 is:
A:-54
B:-32
C:-80
D:-79 for all 𝐴, 𝐵 ∈ 𝑀 where M is the set of all 𝑛 × 𝑛  matrices over ℝ , then
A:-d is a metric on 𝑀
B:-d is a pseudo metric on 𝑀
C:-both (1) and (2) are true
D:-None of these are true

Question37:-If the metric space (𝑋, 𝜏) is both totally bounded and complete, then it is
A:-Connected
B:-Discrete
C:-Compact
D:-Not Compact

A:-Compact and open
B:-Compact but not open
C:- not compact but open

Question39:-Which of the following statement is false?
A:-Every regular Liudeloff space is notmal
B:-Every regular separable space is normal
C:-Product of two housdorff spaces is housdorff
D:-Every regular second countable space is normal

Question40:-Which of the following statement is not correct
A:-Connectedness is a hereditary property
B:-Connectedness  is not a weakly hereditary property
C:-If C is a dense subset of topological space X and if C is connected, then X is connected
D:-Compactness is a weakly hereditary property

Question41:- image
A:-Bounded
B:-Unbounded
C:-Finite
D:-Metrizable

Question42:-If [X,𝜏] is Housdorff and X is finite, then 𝜏 is the
A:-Discrete topology
B:-Co-infinite topology
C:-Finite topology
D:-All of these

A:-1-i
B:-1-4i
C:-1+2i
D:--1-1i

Question44:-If 1, 𝑤, 𝑤² are the complex cube roots of unity, then (𝑥 − 𝑦)(𝑥 − 𝑤𝑦)(𝑥 − 𝑤²y)
A:-𝑥 − 𝑦
B:-𝑥²− 𝑦²
C:-𝑥3− 𝑦3
D:-𝑥3+ 𝑦3

Question45:-Pick the correct statement from the following four statements

Question46:-The function 𝑓(𝑧) =z̅̅ is
A:-analytic at 𝑧 = 0
B:-differentiable at 𝑧 = 0
C:-an entire function
D:-continuous everywhere

Question47:-The harmonic conjugate of 𝑥²− 𝑦²  is
A:-𝑥²+ 𝑦²
B:-2xy
C:- 4xy
D:-𝑦²− 𝑥²

Question48:-
A:-f is an onto entire function
B:-g is a one one entire function on S
C:-g is bounded on ℂ
D:-f is bounded on 𝑆

Question50:-
A:-n
B:-n+1
C:-2(n+1)
D:-2n

Question51:-Which of the following is a Banach space
A:-Space of all polynomial functions on [a,b] with the supremum norm
B:-Space of all continuous functions on [a,b] with the supremum norm
C:-Space of all polynomial functions on [a,b] with the p-norm
D:-Space of all continuous functions on [a,b] with the p-norm

Question52:-The term Hilbert space stands for a
A:-Complete inner product space
B:-Compact linear space
C:-Complete normed space
D:-Complete metric space

Question53:-Let H be a Hilbert space and L be a subspace of H. Then which of the following is false.

Question54:-If 𝑝 ≥ 𝑞 ≥ 1 , then which of the following is/are true.

Question55:-Consider the following statements about a Hilbert space
P: H is separable if it has a countable dense subset
Q: H is separable if it has a complete orthonormal system
A:-Only P is true
B:-Only Q is true
C:-Both are true
D:-Neither P nor Q are true

Question56:-If X and Y are normed spaces, and if 𝑇:𝑋 → 𝑌 is a linear operator, then T is bounded if and only if
A:-T maps bounded subsets of X into bounded subsets of Y
B:-T maps open subsets of X into open subsets of Y
C:-T maps closed subsets of X into closed subsets of Y
D:-T is invertible

Question57:-The complete integral of 𝑧²= 𝑝𝑞𝑥𝑦  is

A:-heat equation
B:-wave equation
C:-Laplace Equation
D:-Poisson Equation

Question61:-The solution of the differential equation 𝑑𝑦/𝑑𝑥  + 𝑦 tan 𝑥 = sec 𝑥 is
A:- sin 𝑥 − 𝑐 cos 𝑥
B:- sin 𝑥 + 𝑐 cos 𝑥
C:-(𝑦 − sin 𝑥)sin 𝑥 = 𝑐
D:-None of these

Question62:-The integrating factor of the differential equation 𝑥 log 𝑥 𝑑𝑦/𝑑𝑥 + 𝑦 = 2 log 𝑥  is

Question63:-The solution of the Differential Equation

Question64:-Any prime of the form 3n+1 is also of the form
A:-6m
B:-6m+1
C:-6m+2
D:-6m+3

Question65:-
A:-0
B:-1
C:-2
D:-3

Question66:-Solution of the congruence equation 𝑥 ≡ 1(mod 3), 𝑥 ≡ (mod 5), 𝑥 ≡ ( mod 7)is
A:-32(mod 105)
B:-42(mod 105)
C:-52(mod 105)
D:-62(mod 105)

Question67:-𝜑 is the Euler Phi function, 𝜑(3𝑛) = 2𝜑(𝑛)  if and only if
A:-n is an even number
B:-n is a multiple of 3
C:-n is odd
D:-n is not a multiple of 3

Question68:-128129 mod 17  is
A:-5
B:-7
C:-9
D:-11

Question69:-Find the remainder when 97! is divided by 101
A:-7
B:-13
C:-17
D:-19

Question70:-Which of the following is a primitive root of 19?
A:-5
B:-7
C:-9
D:-10

Question71:-The importance of immediate reinforcement is more emphasised in:
A:-Classical conditioning
B:-Operant conditioning
C:-Insightful learning
D:-Mastery learning

Question72:-Learning outcome is mainly desirable changes in the:
A:-Content
B:-Learning strategy
C:-Evaluation technique
D:-Behaviour of the learner

Question73:-Consistency of a test is referred as:
A:-Validity
B:-Reliability
C:-Objectivity
D:-Practicability

Question74:-Reasoning from general to particular is
A:-deductive
B:-inductive
C:-dialectical
D:-algorithmic

Question75:-Summative assessment is mostly done through
A:-Project work
B:-lab/.lib activities
C:-written tests
D:-home assignments

Question76:-Which among the following is an informal experimental design?
A:-Factorial design
B:-Randomized Block design
C:-Before and after with control design
D:-None of these

Question77:-A teacher conducts a study in her/his classroom situation to correct the errors committed by students in computation. This type of study comes under:
A:-Case study
B:-Action research
C:-Fundamental research
D:-Applied Research

Question78:-Which among the following measurement possesses a true zero point?
A:-Nominal scales
B:-Ratio scales
C:-Interval scales
D:-Ordinal scales

Question79:-Which is NOT related to Empirical research?
A:-It is data based
B:-Findings are verified by observation
C:-It relies on experience or observation
D:-It often considers abstract ideas or theories

Question80:-Which is the first step in writing a research report?
A:-Preparation of draft
B:-Preparation of lay out of the report
C:-Logical analysis of the content
D:-Preparation of Bibliography

Question81:-Consider the following statements:
(i)   The constituent assembly was set up on 6th December 1946
(ii)   The first meeting of the constituent assembly was bold on 10th December 1946
(iii)   The first person who addressed the constituent assembly was Dr. Sachidananda Sinha
(iv)   Total Strength of the constituent assembly fixed by the cabinet mission, was 389
Which of these statements given above is/are correct?
A:-Only (i) and (ii) are correct
B:-Only (ii) and (iii) are correct
C:-Only (iii) and (iv) are correct
D:-Only (i) and (iv) are correct

Question82:-Match List I with List II and choose the correct answer using the options given below:
List I                                            List II
(i)   First Estate                           (a)   legislature
(ii)   Second Estate                      (b)   Executive
(iii)  Third Estate                         (c)    Judiciary
(iv)   Fourth Estate                       (d   Press
A:-(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
B:-(i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
C:-(i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
D:-(i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)

Question83:-Choose the chronological order of Election Commissioners of India from the options given below:
A:-Sukumar Sen, Kalyan Sundaram, Nagendra Singh, S.P. Sen Verma
B:-Sukumar Sen, Kalyan Sundaram, S.P. Sen Verma,  Nagendra Singh
C:-Sukumar Sen, Kalyan Sundaram, T.Swaminathan, Nagendra Singh
D:-Sukumar Sen, Kalyan Sundaram,T.Swaminathan, S.P. Sen Verma

Question84:-Which statement of the following is correct about comptroller and Auditor General (CAG) of India.
A:-CAG is appointed by the prime minister
B:-CAG holds office for a term of 5 year or 65 years of age whichever is earlier
C:-CAG is an officer of the parliament and is called 'Ears and Eyes of the public Accounts committee
D:-Sashikat sharma is the first CAG of India

Question85:-Which of the following statements are correct about Article - 352 of Indian constitution:
(i)   President can proclaim National Emergency under article -352
(ii)   President can proclaim National Emergency only after receiving a written recommendation from the cabinet
(iii)   As per Article - 352, national Emergency is also known as President's Rule
(iv)   During National Emergency our federal constitution will be converted in to a unitary one
A:-Only (i), (ii) and (iii) are correct
B:-Only (ii), (iii) and (iv) are correct
C:-Only (i), (ii) and (iv) are correct
D:-All the above (i), (ii), (iii) and (iv) are correct

Question86:-With reference to National Health Mission, which of the following are the jobs of ASHA (Accredited Social Heath Activist)?
(i)   Encouraging  Family planning
(ii)   A companying women to the health facility for the antenatal care check up
(iii)  Conducting the  delivery of baby
(iv)  Bringing children to immunization
A:-all the above (i), (ii), (iii) and (iv)
B:-Only (i), (ii) and (iii) are correct
C:-Only (ii), (iii) and (iv)  are correct
D:-Only (i), (ii) ad (iv) are correct

Question87:-Match List I and List II and choose the correct answer using the options given below:
List  I                                      List II
(i)     Jan Shree Bima Yojana            (a)   April 8, 2015
(ii)    Antyodaya  Anna Yojana          (b)   April12, 2005
(iii)   Janani Suraksha Yojana           (c)   December  25,200
(iv)   Mudra Bank Yojana                 (d)   August 10,2000
A:-(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
B:-(i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
C:-(i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
D:-(i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)

Question88:-Among the following who are eligible to benefit from the "Mahatma Gandhi National Rural Employment Guarantee Act"?
A:-Adult members of only the scheduled caste and scheduled tribe households
B:-Adult members of Below Poverty line (BPL) households
C:-Adult members of household of all Backward Communities

Question89:-With reference to "Aam Admi Bima Yojana" consider the following statements:
(i)  The member insured under the scheme must be the head of the family in a rural landless  household
(ii)   The member insured must be in the age group of 30 to 65 years
(iii) There is a provision for the free scholarship for upto two children of the insured who are studying between classes 9 and 12
Which of the statements given above is/are correct.
A:-Only (i)
B:-Only (ii) and (iii)
C:-Only (i) and (iii)
D:-All the above (i), (ii) and (iii)

Question90:-Which of the following is a women empowerment and livelyhood programme started in 2008?
A:-Rashtriya Mahila Koshi
C:-Mahila Swayam Sidha Yojana
D:-Indira Mahila Yojana

Question91:-Which state is the official sponsor of Indian Hockey team?
A:-Gujarat
B:-Odisha
D:-Maharashtra

Question92:-Lokayuktha is an authority constituted to prevent
A:-Corruption at administrative, bureaucratic and political levels
B:-Social evils
C:-Pollution of the atmosphere
D:-The intervention of private parties in the functioning of a company

Question93:-Who proposed the Wardha Education plan?
A:-Raja Ram Mohan Roy
B:-D.K. Karve
C:-Dr. Zakir Husain
D:-Mahatma Gandhi

Question94:-Who wrote the fanous patriotic song 'Sare Jahan Se Accha, Hindustan Hamara'?
A:-Allama Muhammed Iqbal
B:-Bankim Chandra Chatterji
C:-Dinabandhu Mitra
D:-Altaf Husain Hali

Question95:-The Mars orbiter Mission (MOM) of India is known as
A:-Chandrayaan
B:-Mangalyaan
C:-Prithviyaan
D:-Gaganyaan

Question96:-Population density refers to
A:-The number of people in 100 meter
B:-the number of people in a kilometer
C:-the number of people in an acre
D:-the number of people per square kilometer

Question97:-Who is the chairman of GST council?
A:-Prime Minister
B:-Finance Secretary
C:-Union Finance Minister
D:-RBI Governor

Question98:-In India, the financial year is from
A:-June 1 to May 31
B:-April 1 to March 31
C:-January 1 to December 31
D:-May 1 to April 30

Question99:-Which of the following is the penalty imposed on an employee per day if he/she gives wrong and unsatisfactory information on an application under Right to Information Act?
A:-Rs. 250
B:-Rs. 300
C:-Rs. 50
D:-Rs. 150

Question100:-Who is the author of the famous work 'Jathikummi'  exposing evils of caste system to bring in social reformation?
A:-Vagbhatananda
B:-Kumaranasan
C:-Pandit Karuppan
D:-Sree Narayana Guru