| Post Name | LECTURER IN MATHEMATICS - NCA - EZHAVA/THIYYA/BILLAVA - COLLEGIATE EDUCATION |
|---|---|
| Question Paper Code | 088/2016/OL |
| Category Code | 053/2015 |
| Date of Test | 4 January 2017 |
| Medium of Question | ENGLISH |
| Exam Type | Online Examination |
| Total Questions | 100 |
1
Who was the first non-brahmin, who rang the temple bell of Guruvayoor as a part of Guruvayoor Sathyagraha ?
Answer
C. P. Krishna Pillai
2
The weekly 'Vivekodayam' was published under the leadership of :
Answer
A. Kumaranasan and M. Govindan
3
Among the following members who participated in Civil Disobedience Movement holding her two months old daughter in her arms ?
Answer
B. A.V. Kuttimalu Amma
4
The book 'Sivasadakam' written by :
Answer
D. Sreenarayana Guru
5
The book 'Boheemiyan Chithrangal' written by :
Answer
A. S.K. Pottakkad
6
Among the following which is included in the World Heritage Table in 2016 from India ?
Answer
C. Kanjan junga national park
7
Kilkari project related to :
Answer
C. a mobile voice message service related to family health care
8
Mistake in news reading is known as :
Answer
D. fluff
9
The narrow stretch of land that connect Peninsular India with north-eastern states of India :
Answer
D. Siliguri Corridor
10
Which article of Indian Constitution deals about money bill ?
Answer
A. Article - 110
11
'Applying a principle in a new situation' is an example for :
Answer
B. Conceptual objective
12
Hypothetico-deductive thinking is a characteristic in :
Answer
D. Adolescence
13
The correlation between intelligence and achievement :
Answer
A. High correlation
14
Which of the following is an expository method of teaching ?
Answer
C. Lecture method
15
Grading using relative performance of students in a group is :
Answer
D. Norm referenced grading
16
The purpose of pure research is :
Answer
A. To generate new theories and principles
17
Which is the correct sequence in research ?
Answer
C. Observation-Hypothesizing-Verification-Theory making
18
Internal validity in experimental research means :
Answer
A. Genuineness of experimental treatment
19
Impact factor of a research paper is :
Answer
B. Quality indicator of the article
20
The process of getting permission from the subjects before conducting a research is :
Answer
D. Informed consent
21
Which Constitution Amendment Act has substituted, in Clause (4A) of Article 16 of the Constitution of India, for the words "in matters of promotion to any class" the words "in matters of promotion, with consequential seniority, to any class" ?
Answer
B. The Constitution `85^(th)` Amendment Act
22
__________ is not legally enforceable by any court and the state cannot be compelled through the courts to implement them.
Answer
A. Directive principles of State Policy
23
Who said "Preamble is the 'Political Horoscope' of the Constitution" ?
Answer
C. K.M. Munshi
24
The article, which is related with the Residuary Powers of the Indian Parliament is :
Answer
C. Article 248
25
Which article of the Indian Constitution empowers the President of India to declare National Emergency ?
Answer
D. Article 352
26
Who appoints State Chief Information Commissioner and Other Information Commissioners ?
Answer
A. Governor
27
Commission for Protection of Child Rights Act, 2005 provides that Chairman and other members of National Commission for Protection of Child Rights shall hold office as such for a period of ___________ from the date on which he assumes the office.
Answer
B. 3 Years
28
The day, on which The National Food Security Act, 2013 received the assent of the President of India is :
Answer
A. `10^(th)` September, 2013
29
In the year 1986 which of the following Acts was enacted by the Parliament of India for the protection and improvement of environment ?
Answer
B. The Environment (Protection) Act
30
In which year National Rural Employment Guarantee Act is renamed as Mahatma Gandhi National Rural Employment Guarantee Act ?
Answer
C. 2009
31
Find a simpler expression for where S and R are any two sets and U is the Universal Set.
Answer
C. U
32
Fibonacci number fₙ = fₙ₋₁ + fₙ₋₂, where f₁ = 1 and f₂ = 1, is even if n is divisible by:
Answer
B. 3
33
If x, y and z are three elements in a Boolean algebra such that x = yz′ + y′z, then write an expression for x′.
Answer
B. yz + y′z′
34
The domain of the function f(x) = √((1 − |x|)/(|x| − 2)) is:
Answer
C. (−2, −1) ∪ (1, 2)
35
The number of vertices of a 2-regular graph with 16 edges is:
Answer
A. 16
36
Which of the following is not true? K₃,₃ is a:
Answer
B. Complete graph
37
For any vector space V, which of the following sets is a subspace having a single element?
Answer
A. {0}
38
What is the largest possible value for the rank of a 2 × 5 matrix?
Answer
D. None of these
39
If at least two elements a, b other than the identity element are such that 0(a) = finite and 0(b) = ∞, then the group is called:
Answer
C. Mixed group
40
Which of the following pairs of groups are isomorphic?
Answer
A. ℤ₂₄ and ℤ₈ ⊕ ℤ₃
41
The point on the plane curve f(x) = −x³ + 3x + 9x − 12 where the slope of the tangent is maximum is:
Answer
D. None of these
42
The quadratic equation whose roots are the squares of the roots of the equation 2x² − 2x + 5 = 0 is:
Answer
B. 4x² + 16x + 25 = 0
43
If the sum of the distances of a point in the plane from two fixed perpendicular lines is l, then the locus of the point is :
Answer
C. Square
44
The directional derivative of the scalar function 2xy + z² + 1 in the direction of the vector i + 2j − k at the point (1, −1, 2) is:
Answer
Question Cancelled
45
The product of the eigenvalues of the matrix A = [[1, 2], [−1, 3]] is:
Answer
A. 5
46
If f(x) = (ln(sin 2x))², then f′(x) is:
Answer
D. None of these
47
The ratio of the diameter of a sphere to the height of the right circular cone having the greatest volume which can be inscribed in the sphere is:
Answer
A. 3 : 2
48
The dimension of the dual space of the vector space of 2 × 2 matrices with complex entries, over the field of real numbers is:
Answer
B. 8
49
If f is a non-zero linear functional defined on a three dimensional vector space X, then the dimension of the null space of f is :
Answer
B. 2
50
Which of the following is not a Hilbert space?
Answer
A. C[0, 1]
51
Let {xₙ} be a convergent sequence in R, the set of real numbers. Then the number of limit points of {xₙ} is:
Answer
B. Exactly one if R is under usual topology
52
For what values of a is the function f(x) = x² + a for x < 2, and f(x) = 2a − x² for x ≥ 2, continuous at every x?
Answer
C. a = 8
53
If a real valued function f is defined and continuous on a closed and bounded set F of real numbers, then :
Answer
D. f is uniformly continuous on F
54
Let f be a family of measurable real functions. Let G be the set of limit points of sequences of real functions in f. Then G is :
Answer
A. measurable
55
Let f : ℝ → ℝ be differentiable for every x ∈ ℝ, such that lim_{x→∞} f′(x) = ℓ. Then lim_{x→∞} f(x)/x is:
Answer
C. ℓ
56
Let t be a positive integer and define a sequence {xₙ} by xₙ₊₁ = t + xₙ² for all n ≥ 0 with x₀ = 0. If the sequence is convergent, then:
Answer
A. t = 1/4
57
Find the number of continuous onto maps from [0, 1] to (0, 1).
Answer
B. None
58
Let f(x) = a₁ sin x + a₂ sin 2x + … + aₙ sin nx such that |f(x)| ≤ |sin x| for all x ∈ R. Then |a₁ + 2a₂ + … + naₙ| is:
Answer
D. less than or equal to 1
59
Let {Sₙ} be a sequence of real numbers defined by S₁ = √2 and Sₙ₊₁ = √(2 + Sₙ). Then {Sₙ} is:
Answer
A. Monotonically increasing for all n
60
Let f be a function on real line such that |f| is measurable. Then :
Answer
C. f need not be measurable
61
The value of ∫C 1/z² dz, where C is the curve |z − i| = 1/2, is:
Answer
B. 0
62
Which of the following is a bounded complex-valued function?
Answer
D. None of the above (A), (B) and (C)
63
The function g(z) = z/(eᶻ − 1), z ≠ 0, has:
Answer
D. Removable singularity at the origin
64
Which of the following is the value of ∮|z|=1 |z − 1| |dz|?
Answer
C. 8
65
The number of roots of z⁴ − 6z + 3 = 0 with absolute value less than 1 is:
Answer
B. 1
66
If R is the set of real numbers, then which of the following is a metric on R × R? For X = (x₁, y₁) and Y = (x₂, y₂) in R × R:
Answer
D. d(X, Y) = √((x₁ − x₂)² + (y₁ − y₂)²)/(1 + √((x₁ − x₂)² + (y₁ − y₂)²))
67
If (X, d) is an arbitrary metric space and E is a subset of X, choose the false statement from the following.
Answer
A. Every interior point of E is a limit point of E
68
Let R be the set of real numbers and be the semi open interval topology on R. Then, which of the following is true for (R, ).
Answer
C. (R, ) is a separable space.
69
Choose the correct statement from the following:
Answer
D. The torus surface is a quotient space of the unit square.
70
Let X be an uncountable set and 𝒯 be the co-finite topology on X. Then choose the correct one from the following:
Answer
C. (X, 𝒯) is a separable compact space which is not metrizable.
71
From the following given operations on the set of positive integers, choose the one which is a binary operation:
Answer
C. a*b = ab + b
72
Let G = (V, E) be a simple graph. Choose the wrong statement from below :
Answer
A. G is bipartite implies G has a perfect matching
73
Let (X, d) be a metric space and (xₙ) be a sequence in X. Choose the correct statement from the following:
Answer
B. If (xₙ) converges in X, any subsequence converges to the same limit.
74
Let X = R². From the following maps from X to X, choose the one which is a linear homeomorphism:
Answer
D. F(x₁, x₂) = (x₁ − x₂, x₁ + x₂)
75
Let E = {x ∈ R : x is rational, 0 ≤ x ≤ 2}. The Lebesgue measure of E is:
Answer
D. 0
76
Among the following equations, choose the exact differential equation:
Answer
A. 12x²y dx + 4x³ dy = 0
77
Let X = (C², ‖·‖₁). Let A ∈ BL(X) be defined by A(x₁, x₂) = (2x₂, x₁). Then ‖A‖ is:
Answer
C. 2
78
Let μ denote the Möbius function. Then μ(75) is:
Answer
B. 0
79
Consider the vector field defined by X(x₁, x₂) = (x₁, x₂, −x₂, x₁). Choose an integral curve of X from the following:
Answer
Question Cancelled
80
Let X = R², Y = {(x, 0) : x ∈ R}, where ‖·‖₁ is defined on X. Let g : Y → R be defined by g((x, 0)) = x. From the following choose a Hahn-Banach extension of g:
Answer
C. f(x₁, x₂) = x₁ − x₂
81
Which of the following is not a solution of the two-dimensional Laplace equation?
Answer
C. u = x² + y²
82
A partial differential equation of the form auₓₓ + 2buₓᵧ + cuᵧᵧ = F(x, y, u, uₓ, uᵧ) is said to be elliptic if:
Answer
B. ac − b² > 0
83
General integral of yz z_x + xz z_y = xy is:
Answer
A. F(x² − y², z² − y²) = 0
84
Which of the following functions is nowhere analytic?
Answer
B. f(z) = 2xy + i(x² − y²)
85
Let u_{tt} = c²u_{xx} be the one-dimensional wave equation with boundary conditions u(0,t) = 0 and u(L,t) = 0 for all t, and initial conditions u(x,0) = f(x) and u_t(x,0) = g(x) for all x. If u(x,t) = Σ_{n=1}^{∞}(a_n cos λ_n t + b_n sin λ_n t) sin(nπx/L), then a_n =
Answer
C. (2/L)∫₀ᴸ f(x) sin(nπx/L) dx
86
Two-dimensional heat equation is :
Answer
A. u_t = c²(u_{xx} + u_{yy})
87
Let φ(x) = x be an iteration function for solving the equation f(x) = 0 by the fixed point iteration method. The iterations converge to a root if:
Answer
B. |φ′(xₖ)| < 1, for k = 0, 1, 2, …
88
Let x₀ be an initial approximation to the root of an algebraic equation f(x) = 0 and let xₖ be the kth iterate obtained in the Newton-Raphson method. Then for k = 1, 2, 3, …
Answer
D. xₖ₊₁ = xₖ − f(xₖ)/f′(xₖ)
89
If Δ is the forward difference operator, then Δ³f(x_k) =
Answer
D. f(x_{k+3}) − 3f(x_{k+2}) + 3f(x_{k+1}) − f(x_k)
90
Approximate value of f (5) by Lagrange's interpolation polynomial using the following data :
Answer
A. 18
91
A linear transformation T on ℓ²(Zₙ) = {z(1), z(2), …, z(N) / z(i) ∈ C, 1 ≤ i ≤ N} is a Fourier multiplier operator iff the matrix of T in the Fourier basis is __________.
Answer
D. Diagonal
92
Which of the following is an elliptic function?
Answer
A. Weierstrass ρ function
93
Domain of convergence of Riemann Zeta function is __________.
Answer
B. Re z > 1
94
The relative cardinality of the Fuzzy set A={(a, 0.1), (b, 0.5), (c, 0.9)} is __________.
Answer
B. 0.5
95
Tychnoff's theorem states that :
Answer
A. Arbitrary product of compact spaces is compact in the product topology
96
The language generated by the grammar G = ({S}, Σ = {a, b}, S, {S → aSb | λ}) is __________.
Answer
C. {aⁿbⁿ, n ≥ 0}
97
Which of the following is true for the differential equation (1 − x²)y″ − 2xy′ + 2y = 0?
Answer
B. x = ∞ is a regular singular point
98
The integral curve through (1, −1) of the gradient field of f(x, y) = (x² + y²)/2 is __________.
Answer
D. (eᵗ, −eᵗ)
99
The integral equation corresponding to the initial value problem y″ + xy′ + (1 − x²)y = f(x), y(0) = 0, y′(0) = 0 is a __________.
Answer
C. Volterra equation of second kind
100
If a source of strength m is at the origin, the corresponding stream function is ψ=
Answer
A. - mθ
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