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KPSC Mock Test: Higher Secondary School Teacher Mathematics - Kerala Higher Secondary Education

KPSC Mock Test: Higher Secondary School Teacher Mathematics - Kerala Higher Secondary Education

Official Kerala Public Service Commission ONLINE Exam Question Paper & Answer Key. • Name of Post: Higher Secondary School Teacher Mathematics - Kerala Higher Secondary Education • Department / Institution: Kerala Higher Secondary Education • Question Paper Code: 66/2022/OL • Category Number: 484/2019 • Date of Test: 13-08-2022 • Answer Key: Provisional Answer Key Question Paper PDF: https://www.keralapsc.gov.in/sites/default/files/2022-10/fky-66-22.pdf Answer Key PDF: https://www.keralapsc.gov.in/sites/default/files/2022-08/provisional_answer_key__hsst_mathematics.pdf

100 Questions
Progress: 0 / 100 Answered
1
The conjugate of 1+i1+i with respect to the circle ∣z−1∣=2|z-1| = 2 is
2
Which of the following is a subspace of Mn(R)M_n(\mathbb{R}), the vector space of all n×nn \times n real matrices
3
Let A be a 5×35 \times 3 matrix and B be a 3×73 \times 7 matrix then the rank of AB can be:
4
The number of linearly independent eigen vectors of the matrix (2200210000300014)\begin{pmatrix} 2 & 2 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 4 \end{pmatrix} is
5
Let A be a 5×55 \times 5 real matrix. Which of the following is true?
6
Which of the following is the Jordan form of a 3×33 \times 3 matrix?
7
Let T:P3(R)→M2×2(R)T : P_3(\mathbb{R}) \to M_{2 \times 2}(\mathbb{R}) be defined by T(f(x))=(f(1)f(−1)f(2)f(−2))T(f(x)) = \begin{pmatrix} f(1) & f(-1) \\ f(2) & f(-2) \end{pmatrix}. Then which of the following statements is true?
8
Let V=M2(R)V = M_2(\mathbb{R}), ATA^T denotes the transpose of the matrix A and I be the identity matrix in V. Define an inner product on V as ⟨A,B⟩=trace(BTA)\langle A, B \rangle = \text{trace}(B^T A). Then the dimension of the orthogonal complement of the set \{I\} is:
9
The series 1−12+14⋅2!+18⋅3!+116⋅4!+…1 - \frac{1}{2} + \frac{1}{4 \cdot 2!} + \frac{1}{8 \cdot 3!} + \frac{1}{16 \cdot 4!} + \dots converges to:
10
Let f:R→Rf : \mathbb{R} \to \mathbb{R} be defined as :
11
Which of the following functions is uniformly continuous on (0,1)?
12
Let f:[0,1]→[0,1]f : [0,1] \to [0,1] be a continuous function. Then which of the following statements is true?
13
Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a thrice differentiable function such that f(a)=f′(a)=f(b)=f′(b)=0f(a) = f'(a) = f(b) = f'(b) = 0 where a,b∈R,a<ba, b \in \mathbb{R}, a < b. Then choose the correct statement.
14
Consider the sequences of functions (fn)(f_n) and (gn)(g_n) defined by fn(x)=11+nx,x∈(0,1)f_n(x) = \frac{1}{1+nx}, x \in (0,1), gn(x)=x1+nx,x∈(0,1)g_n(x) = \frac{x}{1+nx}, x \in (0,1). Choose the correct statement from the following:
15
Let f:[0,1]→[0,1]f : [0,1] \to [0,1] be defined as: f(x)={0if x∈Qxif x∈R∖Qf(x) = \begin{cases} 0 & \text{if } x \in \mathbb{Q} \\ x & \text{if } x \in \mathbb{R} \setminus \mathbb{Q} \end{cases}. Which of the following statements is true?
16
Which of the following sets has nonzero Lebesgue measure?
17
Consider the function f:R2→Rf:\mathbb{R}^2 \to \mathbb{R} defined by f(x)=x2y2x2+y4f(x) = \frac{x^2 y^2}{x^2 + y^4} if x≠0x \neq 0, 00 if x=0x = 0. Choose the correct statement from the following:
18
The value of the integral ∫01t−1/2(1−t)−1/2dt\int_0^1 t^{-1/2}(1-t)^{-1/2} dt
19
Let f(x)=x,0<x<πf(x) = x, 0 < x < \pi. Which of the following is true?
20
Let f:R→Rf:\mathbb{R} \to \mathbb{R} be a Lebesgue measurable function and E⊂RE \subset \mathbb{R} be a measurable set. Choose the correct statement from the following:
21
Let f:R2→R2f:\mathbb{R}^2 \to \mathbb{R}^2 be defined by f(x,y)=(x+y,x−y)f(x,y) = (x+y, x-y), choose the correct statement from the following:
22
Consider the function f:[0,1]→Rf:[0,1] \to \mathbb{R} defined by f(x)=1f(x) = 1 if xx is rational and 11 if xx is irrational. Which of the following statements is true?
23
Let G be a non-abelian group of order 6. Then the number of subgroups of G of order 2 is
24
Which of the following pairs of groups are isomorphic:
25
The number of abelian groups of order 36 (up to isomorphism) is
26
Let G be a group of order n. Then G is cyclic if n is:
27
Let G be a group of order 28. Which of the following statements is true?
28
Which of the following is a class equation of a group of order 10?
29
Choose the correct statement from the following:
30
Let R[x]\mathbb{R}[x] be the polynomial ring over the field of real numbers. Which of the following set I is an ideal in R[x]\mathbb{R}[x]
31
Let ⟨p(x)⟩\langle p(x) \rangle denotes the ideal generated by the polynomial p(x) in Z[x]\mathbb{Z}[x]. Choose the correct statement from the following
32
Choose the true statement from the following:
33
Which of the following is a field?
34
The Galois group of the splitting field of x3−2x^3 - 2 over Q\mathbb{Q} is isomorphic to:
35
The number of generators of the multiplicative group of a field of order 81 is:
36
Let p(x)∈R[x]p(x) \in \mathbb{R}[x]. If p(x) is irreducible then p(x) can be:
37
If d:M×M→[0,∞)d: M \times M \to [0, \infty) defined by d(A,B)=∣det⁡(A)−det⁡(B)∣d(A,B) = |\det(A) - \det(B)| for all A,B∈MA, B \in M where M is the set of all n×nn \times n matrices over R\mathbb{R}, then
38
If the metric space (X,τ)(X, \tau) is both totally bounded and complete, then it is
39
The set {1m+1n;m,n∈N}∪{0}\{\frac{1}{m} + \frac{1}{n} ; m, n \in \mathbb{N}\} \cup \{0\}
40
Which of the following statement is false?
41
Which of the following statement is not correct
42
The space [0,1]J[0,1]^J with the product topology is
43
If [X,τ][X, \tau] is Hausdorff and X is finite, then τ\tau is the
44
If 1,w,w21, w, w^2 are the complex cube roots of unity, then (x−y)(x−wy)(x−w2y)(x-y)(x-wy)(x-w^2y)
45
Pick the correct statement from the following four statements
46
The function f(z)=zˉf(z) = \bar{z} is
47
The harmonic conjugate of x2−y2x^2 - y^2 is
48
Consider the functions f(z)=ez2f(z) = e^{z^2} and g(z)=ezg(z) = e^z and S={z∈C:Re z∈(−π,π)}S = \{z \in \mathbb{C} : \text{Re } z \in (-\pi, \pi)\}. Then
49
The power series ∑n=0∞2−nz2n\sum_{n=0}^{\infty} 2^{-n} z^{2^n} converges is
50
Dimensions of Cn\mathbb{C}^n over R\mathbb{R} is
51
Which of the following is a Banach space
52
The term Hilbert space stands for a
53
Let H be a Hilbert space and L be a subspace of H. Then which of the following is false.
54
If p≥q≥1p \ge q \ge 1, then which of the following is/are true.
55
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
56
If X and Y are normed spaces, and if T:X→YT:X \to Y is a linear operator, then T is bounded if and only if
57
The complete integral of z2=pqxyz^2 = pqxy is
58
A complete solution of the partial differential equation ∂z∂x−3x2=(∂z∂y)2−y\frac{\partial z}{\partial x} - 3x^2 = \left(\frac{\partial z}{\partial y}\right)^2 - y is
59
The partial differential equation ut=c2uxxu_t = c^2 u_{xx} where c≠0c \neq 0 is known as
60
The general integral of the PDE is y2p−xyq=x(z−2y)y^2 p - xy q = x(z - 2y) is
61
The solution of the differential equation dydx+ytan⁡x=sec⁡x\frac{dy}{dx} + y \tan x = \sec x is
62
The integrating factor of the differential equation xlog⁡xdydx+y=2log⁡xx \log x \frac{dy}{dx} + y = 2 \log x is
63
The solution of the Differential Equation dydx=yx+y2x2−1\frac{dy}{dx} = \frac{y}{x} + \sqrt{\frac{y^2}{x^2}-1} is
64
Any prime of the form 3n+13n+1 is also of the form
65
What is the remainder when the sum 15+25+35+⋯+995+10051^5 + 2^5 + 3^5 + \dots + 99^5 + 100^5 is divided by 4
66
Solution of the congruence equation x≡1(mod3),x≡2(mod5),x≡3(mod7)x \equiv 1 \pmod 3, x \equiv 2 \pmod 5, x \equiv 3 \pmod 7 is
67
φ\varphi is the Euler Phi function, φ(3n)=2φ(n)\varphi(3n) = 2\varphi(n) if and only if
68
128129(mod17)128^{129} \pmod{17} is
69
Find the remainder when 97!97! is divided by 101
70
Which of the following is a primitive root of 19?
71
The importance of immediate reinforcement is more emphasised in:
72
Learning outcome is mainly desirable changes in the:
73
Consistency of a test is referred as:
74
Reasoning from general to particular is
75
Summative assessment is mostly done through
76
Which among the following is an informal experimental design?
77
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:
78
Which among the following measurement possesses a true zero point?
79
Which is NOT related to Empirical research?
80
Which is the first step in writing a research report?
81
Consider the following statements:\n(i) The constituent assembly was set up on 6th December 1946\n(ii) The first meeting of the constituent assembly was bold on 10th December 1946\n(iii) The first person who addressed the constituent assembly was Dr. Sachidananda Sinha\n(iv) Total Strength of the constituent assembly fixed by the cabinet mission, was 389\nWhich of these statements given above is/are correct?
82
Match List I with List II and choose the correct answer using the options given below:\nList I\n(i) First Estate\n(ii) Second Estate\n(iii) Third Estate\n(iv) Fourth Estate\nList II\n(a) legislature\n(b) Executive\n(c) Judiciary\n(d) Press
83
Choose the chronological order of Election Commissioners of India from the options given below:
84
Which statement of the following is correct about comptroller and Auditor General (CAG) of India.
85
Which of the following statements are correct about Article - 352 of Indian constitution:\n(i) President can proclaim National Emergency under article -352\n(ii) President can proclaim National Emergency only after receiving a written recommendation from the cabinet\n(iii) As per Article - 352, national Emergency is also known as President's Rule\n(iv) During National Emergency our federal constitution will be converted in to a unitary one
86
With reference to National Health Mission, which of the following are the jobs of ASHA (Accredited Social Heath Activist)?\n(i) Encouraging Family planning\n(ii) A companying women to the health facility for the antenatal care check up\n(iii) Conducting the delivery of baby\n(iv) Bringing children to immunization
87
Match List I and List II and choose the correct answer using the options given below:\nList I\n(i) Jan Shree Bima Yojana\n(ii) Antyodaya Anna Yojana\n(iii) Janani Suraksha Yojana\n(iv) Mudra Bank Yojana\nList II\n(a) April 8, 2015\n(b) April 12, 2005\n(c) December 25, 2000\n(d) August 10, 2000
88
Among the following who are eligible to benefit from the "Mahatma Gandhi National Rural Employment Guarantee Act"?
89
With reference to "Aam Admi Bima Yojana" consider the following statements:\n(i) The member insured under the scheme must be the head of the family in a rural landless household\n(ii) The member insured must be in the age group of 30 to 65 years\n(iii) There is a provision for the free scholarship for upto two children of the insured who are studying between classes 9 and 12
90
Which of the following is a women empowerment and livelyhood programme started in 2008?
91
Which state is the official sponsor of Indian Hockey team?
92
Lokayuktha is an authority constituted to prevent
93
Who proposed the Wardha Education plan?
94
Who wrote the famous patriotic song 'Sare Jahan Se Accha, Hindustan Hamara'?
95
The Mars Orbiter Mission (MOM) of India is known as
96
Population density refers to
97
Who is the chairman of GST council?
98
In India, the financial year is from
99
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?
100
Who is the author of the famous work 'Jathikummi' exposing evils of caste system to bring in social reformation?

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