| Post Name | Higher Secondary School Teacher (Mathematics) |
|---|---|
| Department | Kerala Higher Secondary Education |
| Question Paper Code | 078/2026/OL |
| Category Code | 219/2025, 720/2025 |
| Date of Test | 5 May 2026 |
| Exam Type | Online Examination |
| Total Questions | 100 |
1
If A is a square matrix of order ļ¬ve and the dimension of the solution space AX = 0 is at least two. Then
Answer
A. Rank(AĀ²) ā¤ 3
2
Let V be an inner product space consisting of the linear polynomials P: [0,1] ā R such that p(x) = ax + b, a,b ā R, with the inner product āØp,qā© = ā«āĀ¹ p(x)q(x) dx, p,q ā V. Then an orthonormal basis for V is
Answer
D. {1, ā3(2xā1)}
3
The number of non-singular 3 3 matrices over F2, the ļ¬nite ļ¬eld with two elements is
Answer
B. 168
4
Let the minimal polynomial of a linear transformation T : R4 R4 be y2 + y+ 1, then
Answer
B. T has no real Eigen values
5
Let T : V V be a linear operator, then which of the following is/are true ? i. {0} is invariant under T ii. V is invariant under T iii. kernel T is invariant under T iv. image of T is invariant under T
Answer
D. All of the above
6
Let A be a square matrix of order four with dim [N(A-2I)]=2, dim [N(A-4I)] = 1 and Rank (A) = 3, then which of the following is not true ?
Answer
C. A is not diagonalizable
7
The number of all non-singular linear transformations T : R4 R3 is
Answer
C. 0
8
The value of c in the Mean Value Theorem, fā²(c) = [f(b)āf(a)]/(bāa), for f(x)=18xĀ²ā9x+7 on [3,37] is
Answer
B. 20
9
Which of the following is not Riemann-integrable on [0,1]?
Answer
B. f(x)=x+1 if xāā; f(x)=x if xāāāā
10
Let A be a 5 4 matrix and B be a 4 5 matrix. Then which of the following is necessarily an eigen value of AB ?
Answer
A. 0
11
Let V be a ļ¬nite vector space. Then which of the following is a possibility for |V| ?
Answer
D. |V|=8
12
Let f:[0,1]āā and g:[0,1]āā be defined by f(x)=1/n if x=1/n, nāN, and 0 otherwise; g(x)=n if x=1/n, nāN, and 0 otherwise. Then
Answer
B. f is Riemann integrable function but g is not
13
If f(x) is twice differentiable and |f(x)| < Ī±, |fā³(x)| < Ī² for x > a, then
Answer
Question Cancelled
14
Consider the improper integrals Iā = ā«ā^ā dx/[xā(xĀ²+1)] and Iā = ā«ā^ā e^(āxĀ²) dx. Then
Answer
C. both I1 and I2 convergent
15
Let f be a bounded measurable function on a set X of finite measure. Then
Answer
B. |ā«_E f dĪ¼| ā¤ ā«_E |f| dĪ¼
16
Let f be a non negative measurable function on E, then which of the following statement is false ?
Answer
A. ā«_E f dĪ¼ = 0 if and only if f = 0 on E
17
Let f be a bounded function deļ¬ned on the closed bounded interval [a, b], i. If f is Riemann integrable over [a, b] then f is Lebegue integrable over [a, b] and the two integrals are equal ii. If f is Riemann integrable over [a, b] then f is Lebegue integrable over [a, b] and the two integrals may be equal Then which is true ?
Answer
A. only i is true
18
Which of the following is true ?
Answer
C. Ī(n+1) = nĪ(n) for n > 1
19
For f(x,y) = 2xĀ³ ā 3xĀ² + 2yĀ³ + 3yĀ², āf = 0 at
Answer
A. (0, 0), (0, -1), (1, 0), (1, -1)
20
Let f:āĀ²āā be defined by f(xā,xā) = xā + xā + xāĀ². The directional derivative at (1,3) in the direction uā = (1,0) is
Answer
A. 1
21
Which statement is true for f(x,y) = xy/(xĀ²+yĀ²) if (x,y) ā (0,0), and f(x,y) = 0 if (x,y) = (0,0)?
Answer
C. partial derivatives exist but f is not diļ¬erentiable at (0,0)
22
Let G be a group, which is not cyclic. Then which is not necessarily true ?
Answer
Question Cancelled
23
What is the non admissible value of order of a group G if the converse of Lagranges Theorem is true ?
Answer
D. None of these
24
The factor group ā/ā¤ is isomorphic to āØG, Ā·ā©. Then G is
Answer
A. G = {z ā ā : |z| = 1}
25
Let p be a prime and let be a group where G = {x / =1, n }. Now consider the following three statements i. G is an inļ¬nite p-group ii. G is an inļ¬nite group in which every element possess ļ¬nite order iii. G is an inļ¬nite group in which every proper subgroups of G are of ļ¬nite order. Then which is true ?
Answer
C. all i, ii and iii are true
26
Let G be a group of order 54 73. Now consider the statements i. G has a subgroup of order 7 ii. G has atleast 6 elements of order 7 Then which is true ?
Answer
C. both i and ii are true
27
Let A be the 13 Ć 13 anti-identity matrix (ones on the anti-diagonal and zeros elsewhere). Then det(A) is:
Answer
A. 1
28
Which is not true ?
Answer
C. A30 has a normal subgroup of order 11
29
Consider the following two statements, i. as a group and 2 are isomorphic ii. as a ring and 2 are isomorphic
Answer
A. only i is true
30
Let F = E = ā¤ā and consider the evaluation homomorphism Ļā:FāE. Then Ļā(3xĀ¹ā°ā¶ + 5xā¹ā¹ + 2xāµĀ³) is
Answer
B. 0
31
Let F is a ļ¬eld of 23 elements. Then is isomorphic to
Answer
A. 2 2 2
32
Let R be the ring of all functions from ā to ā. Which of the following is an ideal of R?
Answer
B. The set of all functions f:āāā such that f(2)=0
33
Let F be ļ¬eld of 260 elements and K be a subļ¬eld of F. Then which of the following is a possibility for number of elements in K ?
Answer
D. 230
34
The spitting ļ¬eld of over Q is,
Answer
D. Q
35
The set of integers ā¤ with the binary operation * defined as a*b = a+b+1 for a,b ā ā¤ is a group. The identity element is
Answer
C. -1
36
For which of the following topology, the set of all real numbers is disconnected ?
Answer
C. Lower limit topology
37
If f is a continuous function from a topological space (X, ) into a topological space (Y, ), then
Answer
B. For all V , f-1 (V)
38
If (X, ) is a topological space satisfy the T1-axiom and then which of the following is incorrect ?
Answer
C. The co-ļ¬nite topology on X is stronger than the topology
39
Which of the following is a Hausdorļ¬ space ?
Answer
D. Discrete topology on an inļ¬nite set
40
Consider with discrete topology. Let A = (0, 1), then set of all limit points of A is
Answer
A. ā
41
Which one of the following is not true ?
Answer
C. An inļ¬nite discrete space is compact
42
Consider with usual topology, then which of the following is true ?
Answer
B. is a dense subset of
43
Which of the following is a closed subset of ā?
Answer
Question Cancelled
44
The radius of convergence of the series z/2 + (1Ā·3)/(2Ā·5) zĀ² + (1Ā·3Ā·5)/(2Ā·5Ā·8) zĀ³ + ā¦ is
Answer
C. 3/2
45
The value of ā®_{|z|=1} eį¶» zā»āæ dz is
Answer
A. 2Ļi/(nā1)!
46
The function f(z) = zĀ³, where z = x+iy, is analytic
Answer
A. in the entire z-plane
47
The value of cot z dz is
Answer
D. 14i
48
The singularity of f(z) = e^z/(zā1)ā“ at z = 1 is
Answer
B. Pole of order 4
49
The analytic function f(z) of real part ex cosy is
Answer
A. ez
50
Which of the following normed space is not a Banach space ?
Answer
C. C00, the space of scalar sequences having only ļ¬ninely many nonzero terms
51
The property of inner products is known as
Answer
B. Schwarz inequality
52
Let H be a Hilbert space. If Sā„ denotes the set of all vectors orthogonal to S, which statement is true?
Answer
A. {0}ā„ = H
53
The distance between two orthonormal vectors in an inner product space is
Answer
D. ā2
54
An operator A is unitary, then
Answer
B. A*=A-1
55
X and Y are two Banach spaces and F : X Y is a closed linear map. Which of the following is always true for F ?
Answer
C. Continuous
56
Which of the following is always true ?
Answer
B. Every Hilbert space is Banach
57
The partial differential equation u_xx ā xĀ²u_yy = 0 is
Answer
D. None of these
58
Lagrange's auxiliary equation corresponding to the partial differential equation yĀ²zp + xĀ²zq = xyĀ² is
Answer
B. dx/(yĀ²z) = dy/(xĀ²z) = dz/(xyĀ²)
59
If the ends of the rod are insulated in a one-dimensional heat-flow problem, the boundary condition is
Answer
B. āu(L,t)/āx = 0
60
The two ordinary differential equations obtained by reducing u_tt = cĀ²u_xx by separation of variables are
Answer
A. Xā³ ā kX = 0, Yā³ ā cĀ²kY = 0
61
The function f satisfies fā²(x) ā 3f(x) = 0 and f(0) = eĀ². Then f is
Answer
B. e^(3x+2)
62
Find the general solution of yāā² = yā + eĀ³Ė£, yāā² = yā ā 3eĀ³Ė£.
Answer
A. yā=cāeā»Ė£+cāeĖ£, yā=ācāeā»Ė£+cāeĖ£āeĀ³Ė£
63
Harmonic functions are solutions of
Answer
D. Laplace equation
64
What is the remainder when 1! + 2! + . . . +100! is divided by 15 ?
Answer
D. 3
65
If f(n) = , then which of the following is always true ?
Answer
D. n is of the form 2k, where k 2
66
The smallest positive number n satisfying the linear congruences n 1 (mod 3), n 2(mod 5) and n 3(mod 7)
Answer
C. 52
67
24 divides 35x-47. Then which of the following can be the value of x ?
Answer
C. 13
68
Euler's Ļ function has the value Ļ(n) = 18. Then n is
Answer
A. 19
69
Number of primitive roots of 54
Answer
A. 6
70
Number of positive integers not greater than 3000 and are divisible by 3, 5 and 7
Answer
Question Cancelled
71
What does the term "feedback loop" signify in the context of assessment and evaluation ?
Answer
A. A continuous process of providing feedback throughout the learning period
72
What is the primary purpose of formative assessment in an evaluation system ?
Answer
B. Providing feedback during the learning process
73
How does the concept of "Inclusive Education" align with eļ¬ective teaching practices ? i. By excluding students with diverse needs ii. By accommodating and valuing the diversity of all students iii. By focusing solely on advanced learners
Answer
B. only ii
74
What does the acronym "STEM" represent in the context of education ?
Answer
C. Science, Technology, Engineering and Mathematics
75
How does the concept of "Blooms Taxonomy" contribute to instructional design ? i. By promoting a single level of cognitive engagement ii. By categorizing cognitive skills into hierarchical levels iii. By emphasizing rote memorization
Answer
B. Only ii
76
In a research context, what does reliability refer to ?
Answer
A. The consistency and stability of measurement
77
What is plagiarism in the context of research ethics ?
Answer
C. Unauthorized use of others' work without proper attribution
78
What is the primary purpose of obtaining informed consent in research ?
Answer
C. To respect participants' autonomy and rights
79
What is the initial step in the research process ?
Answer
D. Identifying the research problem
80
What characterizes a longitudinal research design ?
Answer
C. Data collected over an extended period from the same subjects
81
Which of the following statement is/are correct about the suspension of fundamental rights ? i. Parliament can suspend fundamental rights by passing a law with a two-third majority ii. During an emergency, the President, by order, can suspend the enforcement of some fundamental rights iii. By the proclamation of emergency issued on the ground of war or external aggression, all fundamental rights are automatically suspended iv. Fundamental Rights cannot be suspended
Answer
C. Only ii
82
Match the following Articles in the Constitution with suitable provisions. a. Article 124 i. Extension of jurisdiction of High Court to Union territories b. Article 136 ii. Special leave to appeal by the Supreme Court c. Article 230 iii. Constitution of Supreme Court d. Article 224 iv. Appointment of additional and acting judges in High Court by the President
Answer
D. a-iii, b-ii, c-i, d-iv
83
Assertion (A) : The Legislature of a state or High Court has no role in the removal of the Governor. Reason (R) : No Governor can be removed from the oļ¬ce till the completion of ļ¬ve year.
Answer
C. (A) is true but (R) is false
84
Which of the following features has been correctly listed about Centre-State relations in India ? i. Constitution ensures equal representation of states in Rajya Sabha. ii. Legislative and executive powers are divided between Central Government and State Governments. iii. Union Parliament can make any law for implementing international treaties and agreements without the state's consent, even if the subjects are covered in the State List. iv. If the Rajya Sabha passes a resolution, supported by one-third of members present and voting, that it is necessary or expedient in the national interest, then parliament can make laws on any subject in the State List.
Answer
C. Only ii and iii
85
The Chairman and the members of the UPSC shall only be removed from oļ¬ce, on the ground of misbehaviour, by an order of
Answer
C. the President on the report of Supreme Court
86
Arrange the formation of following commissions in chronological order. i. National Human Rights Commission ii. National Commission for Protection of Child Rights iii. National Commission for Women iv. National Commission for Scheduled Tribes
Answer
B. iii, i, iv, ii
87
How parliament can amend a list in the Seventh Schedule of the Constitution of India ?
Answer
D. The amendment Bill passed in both Houses of Parliament by two third majority of members present and voting and a majority of total membership. Ratiļ¬cation by not less than half of State Legislatures are also required
88
Which of the following statements about the duration of State Emergency is correct ? i. The proclamation, under Article 356, will cease to operate at the expiry of two months unless both houses of the Parliament have approved it before that period. ii. A proclamation issued under Article 356 can be in force normally for one year at a stretch. iii. Parliament can extend the period of State emergency beyond one year, if the Election Commission certiļ¬es that it is necessary on account of the diļ¬culties in holding the general elections to Legislative Assembly of the concerned state. iv. Parliament can extend the period of State Emergency beyond one year if a proclamation of emergency under Article 352 is in operation.
Answer
Question Cancelled
89
Originally Preamble was not considered as a part of constitution. But the Supreme Court held that it was a part of Constitution. The Supreme Court gave this verdict in which of the following case ?
Answer
A. Kesavananda Bharati vs State of Kerala
90
Which of the following statement is/are not the objective of Right to Information Act ? i. to empower the citizens ii. to promote transparency and accountability in the working of government iii. to make our democracy work for the people in the real sense iv. to have time bound redressal of grievances of citizen
Answer
C. Only iv
91
Swami Vagbhatananda denounced caste barriers and idol worship and exhorted his followers to abjure such practices. He was the founder of
Answer
B. Atma Vidya Sangham
92
The ļ¬rst full ļ¬edged Malayalam work printed in the language using Malayalam types was
Answer
B. Samkshepa Vedartham
93
The famous Kerala reform leader K. Ayyappan is related with i. Popularised the idea of Misrabhojanam. ii. Promoted women education. iii. Encouraged Gender Equality. iv. Founded Sahodara Sangham.
Answer
D. All are correct
94
"Enlighten through education and strengthen by union" who said these words ?
Answer
B. Sri Narayana Guru
95
With regard to the incident Gandhiji said, "A miracle of modern times and a smriti which is the people's charter of spiritual emancipation".
Answer
B. Temple entry proclamation
96
Proclamation of Free Primary Education in 1817 is remarkable in the education sector. The proclamation was during the reign of
Answer
C. Gouri Parvathy Bhai
97
Nalappat Narayana Menon has immortalised himself in Malayalam poetry by his beautiful elegy
Answer
B. Kannunirtulli
98
As a part of the campaign for eradication of untouchability a Satyagraha was organised during the period of December-April 1948 and it evoked the willing participation of princes of Royal families of Cochin. Which Satyagraha was mentioned above ?
Answer
C. Paliyam Satyagraha
99
The news paper Swadesabhimani, was owned by
Answer
B. Abdul Khadir Maulavi
100
Royal proclamation of July 26, 1859 is related to
Answer
C. Shanar Agitation
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