TS SET 2018: Paper II, Mathematical Sciences
Text of page 1
Subject Code Subject
Question Booklet Number
A-15-18 MATHEMATICAL SCIENCEs
hALL tICKET number PaPER
ii
OMR Sheet Number nUMBER OF qUESTIONS
100
dURATION mAXIMUM mARKS nUMBER OF pAGES
2 hOUrs 200 24
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RE
HE
ITE
WR
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NO
DO
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MATHEMATICAL SCIENCEs
Paper − II
1. Which one of the following sets is 4. Let I = [0, 1] be the closed unit interval.
countable ? Suppose f is a continuous mapping of
(A) [0, 1] I into I. Then
(B) IR (A) f(x) ≠ x for all x in I
(C) The set of all polynomials with integer (B) f(x) > x for all x in I
coefficients (C) f(x) = x for at least one x in I
(D) Cantor set
(D) f(x) < x for all x in I
2. If sn = 1+ cosnπ then lim sup sn =
1
n n→∞
5. Which one of the following statements is
not true ?
(A) −1
(A) Every convex function is continuous
(B) 0
(C) 1 (B) If f is a continuous mapping of
(D) ∞ a compact metric space X into a
metric space Y then f is uniformly
3. Which one of the following sets is a
continuous
compact subspace of the set of all real
numbers with the usual metric ? (C) Every continuous function is
(A) (− 2, 1] bounded
(B) [4, ∞) (D) If f is a continuous real function on a
(C) [4, 5) compact metric space X then f attains
(D) [− 4, 0] its bounds
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6. Define f : [1, 2] → IR by f(x) = x2, x∈[1, 2]. 8. Which one of the following is false ?
5 3 7 1 if x is rational
Let P = 1, , , , 2 be a partition of (A) If f(x) =
4 2 4 0 if x is irrational
then f has a discontinuity of the second
[1, 2]. Then U(P, f) = kind at every point of x
x if x is rational
50 (B) If f(x) =
(A) 0 if x is irrational
32
then f has discontinuity of the second
kind at every point other than 0
41
(B)
16 x 2 + 1 if − 2 < x < −1
(C) If f(x) = 3 + x if −1 ≤ x < 0
− 4 + x 2 if 0 ≤ x < 2
85
(C)
32
then f has a discontinuity of the second
kind at x = 0
87
(D)
32 1
sin if x ≠ 0
(D) If f(x) = x
0 if x = 0
1
, 0< x ≤1
7. f(x) = 3 x then f has a discontinuity of the second
0 , x=0
kind at x = 0
1
Then ∫ f(x) dx =
0 9. Define f : [0, 2] → IR by f(x) = 1 + ex, x∈[0, 2].
Then the total variation of f on [0, 2] is
(A) 1
2 (A) 2
(B) 1
(B) e2−1
3
(C) (C) e2 + 1
2
(D) 2 (D) e − 1
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10. If A : IRn→ IRm is a linear transformation 13. Let w 1 , w 2 be subspaces of a finite
dimensional vector space V over a
on IRnand if x ∈ IRn then A′( x ) =
field F with dimensions 8, 6 respectively.
(A) A If w1∩w2≠{0} and m, n are the least and
greatest possible values of dim (w1 + w2),
(B) 0, the zero transformation then 3 m − n =
(A) 7
(C) 2 A
(B) 9
1
(D) A
2 (C) 11
(D) 13
11. Let IR be the set of all real numbers and
d(x, y) = x − y for x, y∈IR. In the metric 14. Let φ : IR4 → IR3 be a linear transformation
space (IR, d) the derived set of the set Z defined by
of all integers is φ ((x1, x2, x3, x4)) = (x1 + x2 + x3, x2 + x3, x3 − x4)
for all (x1, x2, x3, x4)∈ IR4. Then the nullity of
(A) Z
φ=
(B) IR (A) 0
(C) the set of all rational numbers (B) 1
(D) ∅ the empty set (C) 2
(D) 3
{ }
∞
12. Let (X, d) be a metric space and K n
n=1
15. Let A be a 4 × 4 matrix over the field IR
be a sequence of non-expanding
of all real numbers such that detA = 12.
non-empty compact subsets of X. Then
∞ If φ is the Euler-totient function, then
Kn is
n=1
φ(det(6A)).
(A) equal to X (A) 4836
(B) equal to ∅, the empty set (B) 8154
(C) a non-empty compact set (C) 5184
(D) a compact set (D) 3468
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18. If T: IR3 → IR 3 is a linear transformation
16. If the system of equations
and B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is the
3x + 4y + 7z = 8
standard ordered basis of the vector space
IR3 over the field IR such that
2x + 3y + λz = 10
−1 0 2
x + 2y + 4z = 6 [ T ]B = 7 3 1 , then T((6, 5, 7)) =
4 −6 5
has no solution, then 4λ − 8 =
(A) (18, 29, 9)
(A) 14 (B) (8, 64, 29)
(B) 12 (C) (15, 34, 89)
(C) 8 (D) (12, 85, 72)
19. An orthonormal basis for the inner product
(D) 0
space IR3 ( IR) with respect to the standard
3 4 −1 inner product, among the following is
17. If the inverse of the matrix A = 4 1 2
5 0 8 2 −1 2 2 1 −1 2
2
is lA + mA + nI, then l + m + n = (A) , , , 0, , , 0, ,
3 3 3 5 5 5 5
13
(A) 3 4 − 4 3 5 − 3
59
(B) , , 0 , , , 1 , 0, ,
5 5 5 5 34 34
− 23
(B)
59
2 1 2 3 −4 −1 4 2 − 3
23 (C) , , , , , , , ,
(C) 3 3 3 5 5 5 5 5 5
59
− 13 −4 3 3 4
(D) (D) , 0, , ( 0, 1, 0 ) , , 0,
59 5 5 5 5
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20. Let f be the symmetric bi-linear form 23. The function f(z) = cosz maps the line
defined on the vector space IR2(IR) by segment {(x, y) : − π < x ≤ π, y = 1} onto
f((x1, x2), (y1, y2)) = (x1 + x2) (y1 + y2) for (A) a parabola
all (x1, x2), (y1, y2) in IR2 and q be the (B) an ellipse
quadratic form associated with f. Then for (C) a circle
α= (1, 5), β = (2, 6), q(2α + 3β) = (D) a hyperbola
xy(x 2 − y 2 )
(A) 36 (B) 216 for x 2 + y 2 ≠ 0
24. If f(x, y ) = x 2 + y 2
0 for x 2 + y 2 = 0
(C) 1296 (D) 108
∞
1
21. For z < 1 we know ∑ zn = , using ∂2 f ∂2 f
1− z then − =
∂y∂x ( 0, 0 ) ∂x∂y ( 0, 0 )
n=0
n2
∞
this result, the sum ∑ n =
n=1 3 (A) 0 (B) 1
1 (C) − 2 (D) 2
(A)
2
2 for n = 0
3 25. If an = n
(B) 2 for n = 1,2,3,...
2
(C) 2 then the radius of convergence of the
∞
power series ∑ an . Z is
n
(D) 3 n=0
22. The Mobius transformation that maps (A) 1
i, 2, − 2 onto i, 1, − 1 respectively is
(B) 1
3 + iz 2
3z − 2i
(A) (B)
6z − 4i iz − 6 (C) 2
3z + 2i 3z + 2i
(C)
iz + 6
(D)
6z + i (D) 1
3
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1 z − 3 cosz dz =
26. The Laurent’s series of f(z) =
(z − 1)(z − 2)
28.
∫
z =2 π
2
z − 2
which is valid for 1< z < 2 is
(A) 2πi
−1
1 (B) 4πi
(A) ∑ 2
n= − ∞
n+1
− 1 zn
(C) 6πi
(D) 8πi
−1 ∞
1
(B) − ∑ z − ∑
n
n+1
zn 29. If is a simple closed contour for which
n= − ∞ n= 0 2
z = 0 lies inside and z = 3 lies outside it
dz
∞
1 then ∫ =
(C) ∑ 1− n+1 zn
z( z − 3 )
n= 0 2
(A) 2πi
∞
1 3
(D) ∑ 2 n+1
− 1 zn
(B) − 2πi
n= − ∞
3
πi
(C)
27. If is given by the parametric equation 3
πi
z(t) = t2 + it for 0 ≤ t ≤1 then ∫ z dz =
2
(D) −
3
∞
30. For 0 < λ < 1, x λ−1
2 ∫0 1+ x dx =
(A) (i − 1)
3
πλ
(A)
sin(πλ)
2
(B) (i + 1)
3 λ
(B)
sin(πλ)
2
(C) − (i − 1)
3 π
(C)
sin(πλ)
2
(D) − (i + 1) 1
3 (D)
sin(πλ)
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31. The number of bijections on the set 35. In the symmetric group (S 8 , o ), the
{1, 2, 3, ..., 8} that are strictly increasing
product
is
(A) 0 (1 4 6 8) o (3 4 2 1 6) o (4 8 6 3)−1 =
(B) 1
8! (A) (1 8 2 4 6)
(C)
2
(D) 8! (B) (1 8 3 2 4)
32. The sum of all odd positive divisors of 1800 (C) (1 8 4 3 2)
of
(A) 1331 (D) (1 8 6 2 4)
(B) 304 36. If Z 11∗ = Z 11 − {0} , then in the group
(C) 3113 ( )
Z 11, X11 , the product (11 ) X11 ( 7 ) X11(5) =
∗ −1
(D) 403 (where X11 is the multiplication modulo 11 on
Z 11 ) is
33. If φ is the Euler-totient function and if φ
(12 4 × 15 3 ) is 2 α × 3 β × 5 γ , then (A) 2
α+β+γ=
(B) 4
(A) 16
(B) 17 (C) 6
(C) 18
(D) 8
(D) 19
37. If the ideal n Z is a maximal ideal in
34. The number of binary operations that can
the ring ( Z , +, .) of all integers , then a
be defined on a set A with 6 elements is
possible value of n is
(A) 66
(A) 57
2
(B) 6 (B) 67
36
(C) 6 (C) 77
6
(D) 36 (D) 87
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38. The number of idempotent elements in 41. Let (IR , τ) be the topological space of all
the ring (Z 6, +6, ×6) of all residue classes real numbers with usual topology τ. Let
Y = [0, 1] ∪ [2, 3) be equipped with
of integers modulo 6 is
subspace topology τy= {G∩Y : G∈τ).
(A) 1
Then the correct statement, among the
(B) 2 following is
(C) 3 (A) [0, 1] ∈ τy
(B) [2, 3) ∉ τy
(D) 4
(C) [2, 3) is not closed in Y
39. A polynomial over the field of rational
(D) [0, 1] is not closed in Y
numbers which is not irreducible among
the following is 42. Let X = {a, b, c} and τ1= {∅, {a}, {a, b}, X}.
In the topological space (X, τ), the interior
(A) x3 − 5x + 10
of {b, c} is
(B) 5x4 − 2x3 + 6x2 − 10x +14
(A) ∅
(C) x4 − 3x3 + 6x2 − 9x + 5 (B) {a}
(D) x3 − 9x2 + 3x + 12 (C) {a, b}
(D) X
40. If K is the splitting field of the polynomial
x4 − 3x2 + 4 ∈ [x], then the order of the 43. An example, among the following, a
Galois group G(K/ ) is connected subset of the space IR of real
numbers with usual topology is
(A) 1
(A) (1, 2) ∪ (2, 3)
(B) 2
(B) [1, 2) ∪ (2, 3]
(C) 4
(C) (1, 2] ∪ [2, 3)
(D) 8
(D) (−1, 2) ∪ (2, 4)
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44. The Wronskian of the functions eatsinbt 47. The set of zeros of any non-trivial solution
and eatcosbt is of the equation y″ + (sin2x + 1)y = 0 has
(A) e2at (A) one element
(B) be2at (B) two elements
(C) −be2at (C) finite number of elements
(D) beat (D) infinitely many elements
45. The solution of the initial value problem 48. The eigen values λn for the sturm-Liouville
d2 y dy π
problem y″ + λy = 0, y(0) = 0, y = 0
2 + 2 + 4y = 0, y(0) = 1, y ′(0) = 1 is 2
dt dt
are
t 2
(A) y(t) = e cos 3 t + sin 3 t
3
n2
(A) , n = 1,2,...
−t 2 4
(B) y(t) = e cos 3 t + sin 3 t
3
n2
−t 2 (B) , n = 1,2,...
(C) y(t) = e cos 3 t − sin 3 t 2
3
2
(C) 4n , n = 1,2,...
− 2t 2
(D) y(t) = e cos 3 t + sin 3 t
3
46. The solution of the system (D) 8n2 , n = 1,2,...
dx dy
= x, = − x + 2y is 49. The integral surface of the equation
dt dt
(2xy −1) p + (z − 2x2)q = 2(x − yz) which
(A) x = c1et passes through the line x (s) = 0, y (s) = s
0 0
t 3t
y = c1e + c2e and z0(s) = 1 is
(B) x = c1et
t 2t
(A) x2 + y2 − y + z (1 − x) − 1 = 0
y = c1e + c2e
(C) x = c1et (B) x2 (1 − z2) + (1 − 2xy)z − 1 = 0
y = c1et + c2et
t
(C) x2 + y2 + z = y + 2x
(D) y = c1e
y = c1et + c2e−t (D) x2 (1 − z2) + z(1 − 2xy − x) − y = 0
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50. The general integral of 53. The maximum value for step size h that
can be used in the tabulation of f(x) = sinx in
2x(y + z2) p + y(2y + z2) q = z3 is the interval [1, 3] so that linear interpolation
will be correct to four decimal places after
x z 2 − 2y rounding is
(A) ,
F =0
yz yz (A) 0.2 (B) 0.25
(C) 0.02 (D) 0.5
x
(B) F , (x − y ) (x + y + z) = 0
2
yz 54. Consider the function f (x) = ln x and its
tabular values given below :
x 1 1 1 x : 2.0 2.2 2.6
(C) F , + + = 0
yz x y z f (x) : 0.69315 0.78846 0.95551
y y2 z The derivative of f (x) at 2.0 is approximated
2
(D) F , − − x = 0
z x x by using linear interpolation. The value
f′ (2.0) is
αx + x n − e + 1
51. The iteration formula xn+1 = n
α +1 (A) 0.47655 (B) 0.49619
(α any constant) is to compute a root of
(C) 0.69182 (D) 0.73961
(A) x3 − x2 − α = 0
(B) α x3 − x2 − 1 = 0 55. Consider the variational problem
x1
(C) x3 − x2 − 1 = 0 V [ y (x )] = ∫ (y + xy ′) dx, y (x 0 ) = y 0 , y (x1 ) = y1
x0
(D) x3 − αx2 + 1 = 0
and the statements
52. Consider the system of equations Ax = b
1 −K P : Euler’s equation reduces to the identity
where A = .
− K 1 1≡1
For which values of K, the Jacobi method
Q : Since the integral does not depend on
converges ?
the path of integration, the variational
(A) K > 1 problem is meaningless.
(B) K = 1, − 1 Which one of the following is true ?
(C) K > 2 (A) Both P and Q (B) Only P
(D) K < 1 (C) Only Q (D) Neither P nor Q
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56. Consider the functional 58. R e s o l v e n t k e r n e l o f t h e i n t e g r a l
∂ z 2 ∂ z 2 equation
V [ z (x, y )] = ∫∫ + dxdy ,
∂x ∂y x
D φ (x ) = x + ∫ ( ξ − x ) φ ( ξ ) dξ is
where D is the domain of z (x, y). In this 0
problem, extremal satisfies
(A) cos (ξ– x)
(A) two dimensional wave equations
(B) sin (ξ– x)
∂ 2z ∂ 2z
(B) Poisson equation + = xy
∂ x 2 ∂y 2 (C) e ξ – x
(D) ex – ξ
∂ 2z ∂ 2z
(C) Laplace equation + =0
∂ x 2 ∂y 2
59. A particle of mass m is moving on the inner
∂ 2z ∂ 2z surface of the cone x2 + y2 = c2z2 is
(D) + =1
∂ x 2 ∂y 2 (A) not a mechanical system
57. Consider the integral equation (B) holonomic system
x
F(x) = λ ∫ K(x, ξ) φ (ξ)d ξ (C) non holonomic system
a
(D) neither holonomic nor non holonomic
and the statements
60. Consider motion of a particle with mass m
P : Above is Volterra’s integral equation
in polar coordinate system (r, θ). The force
of first kind
components are Fr and Fθ. The equation
Q : Above is Volterra’s integral equation
of motion is
of second kind if F(x) = φ (x)
(A) Fr = mr − mθ 2r, Fθ = 2mrθ + rm
θ
Which one of the following is correct ?
(A) Only P (B) Fr = mr, rFθ = 2mrθ + rm
θ
(B) Only Q
(C) Fr = mθ 2r, Fθ = 2mrθ + rm
θ
(C) Both P and Q
(D) Neither P nor Q (D) Fr = mr − mθ 2r, rFθ = 2rm
θ
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61. A certain treatment is used in 2 different 63. 3 candidates run for an election as a mayor
centres A and B; patients in centre A were in a city. According to a public opinion poll
25 and were an average 54 years old,
their chances to win are 0.25, 0.35 and
patients treated in centre B were 62 and
0.40. The chances that they build a bridge
had mean age equal to 58 years. what is
the overall mean among all patients who after they have been elected are 0.60,
got the treatment ? 0.90 and 0.80. What is the probability that
(A) 55.85 the bridge will be built after the election ?
(B) 54.85 (A) 0.786
(C) 56.85 (B) 0.781
(D) 56.15 (C) 0.785
62. Pregnant women (within month 4) who (D) 0.782
are being followed up by nutrionist had
weight (kg) equal to 64.3; 65.2; 70; 54.5; 64. In an oral exam you have to solve exactly
54.5; 58.8; 81.5; 61; 62. what is the mean one problem. Which might be one of
and the median ? Do this data suggest a
3 types A, B & C. Which will come up
strong skewness of the distribution of the
with probabilities 30 %, 20 % and 50 %
weight ?
respectively. During your preparation
(A) Mean = 62.66, Median = 62.15, does
you have solved 9 of 10 problems of type
not suggest that weight has a strongly
skewed distribution A, 2 of 10 problems of type B and 6 of
10 problems of type C. What is the probability
(B) Mean = 63.15, Median = 64.66, suggest
a strongly skewed distribution that you will solve the problem of the
exam ?
(C) Mean = 64.66, Median = 63.15, the data
don’t suggest that the variable weight (A) 0.60
has a strongly skewed distribution
(B) 0.61
(D) Mean = 64.66, Median = 63.15, the data
don’t suggest that the variable weight (C) 0.63
has no relevance of symmetricity (D) 0.65
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65. Let X be a random variable that follows 68. The Cauchy random variable is defined
to be the ratio of 2 independent standard
a Poisson distribution with parameter
Normal variables X & Y with probability
λ = 7. What is the lower bound for
density function
P(|X – µ| ≤ 4) ?
(A) 0.4375 1
(A) fv (u) = ;− ∞ < ν < ∞
π(ν + 1)
2
(B) 0.4344
π
(B) fv (u) = ;− ∞ < ν < ∞
(C) 0.4295 (ν + 1)
2
(D) 0.4338 XY
(C) fv (u) = ;− ∞ < ν < ∞
π(ν 2 + 1)
66. Let X 1 , X 2 , ..., X n be a sequence of
π(ν 2 + 1)
i.i.d. random variables each with mean (D) fv (u) = ;− ∞ < ν < ∞
XY
E(Xi) = µ and standard deviation σ, then
SLLN is
69. The sample paths of a Markov chain are
(A) P( lim X n = µ) = 1
n→∞ completely characterised by the
(B) P( lim X n = σ) = 1
n→∞ (A) Normal Transition Probabilities
(C) P( lim X n = x) = 1
n→∞ (B) One step Transition Probabilities
Xn
(D) P = x =1 (C) Two step Transition Probabilities
n
(D) n-step Transition Probabilities
67. Suppose the age of a student graduated
from a state is normally distributed. If
70. An irreducible Markov Process is a Markov
the mean age is 23.1 years and s.d. is
3.1 years. What is the probability that 6 Process for which the embedded Markov
randomly selected students had a mean chain is
age at graduation that was greater than
27 ? (Given the table value = 0.9990) (A) Transition Probability Matrix
(A) 0.0001 (B) Birth and Death Process
(B) 0.0011
(C) Irreducible
(C) 0.0005
(D) Poisson Process
(D) 0.0100
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71. Let m and n be positive Integers, then 74. Let X1, X2, X3 be a random sample from a
K m n
∑ =? distribution of the continuous type having
r= 0 r K − r
probability density function
f(x ) = 2x; 0 < x < 1
m + n
(A) = 0; O.W.
K
m What is the probability that the smallest
(B)
K of X1, X2, X3 exceeds the Median of the
m − n distribution ?
(C)
K
m − 1 (A) 18
(D)
K − 1
1
(B) 7
72. Let X ∼ Exp (λ) using Markov inequality.
what is the upper bound for P(X > a) ?
(C) 16
1
(A)
λa 2
(D) 8
1
(B)
λ
75. Let Xn is an i.i.d. exp (λ), then derive the
1
(C) 2
λ
p.d.f. of X(n)
1
(D)
( λa )
2
(A) n{1 – e–λx}n – 1 λe–λx; x > 0
73. If X ∼ Exp (λ) using Chebyshev’s Inequality,
what is the upper bound for (B) n{1 – e–λx)n – 1; x > 0
P(|X – EX| ≥ b) ?
1 1 (C) e–λx, λe–λx; x > 0
(A) (B) 2 2
λb λ b
2 1 2
(C) λ b (D) 2 (D) e–λx/λe–λ ; x > 0
λb
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76. If H0 : x ~ Poisson ( λ 0) ; 79. If (P, Q) has a Bivariate normal distribution
with parameters µ1, µ2, σ1, σ2 and p then
H1 : x ~ Poisson ( λ 1) ; λ 1 > λ 0
the variance of the conditional distribution
∧ (x) = e ( λ 1 – λ 0) ( λ / λ )x is a of (y/x = x) is
1 0
(A) Simple hypothesis for any λ 1 > λ 0, (A) σ12 (1 – p2)
λ 0 pair
2
(B) Non decreasing function in x for (B) σ 2 p2
any λ 1 > λ 0, λ 0 pair
2
(C) Does not exist for any λ 1 > λ 0, (C) σ 2 (1 – p2)
λ 0 pair
(D) σ1 p2
2
(D) Composite hypothesis for any
λ 1 > λ 0, λ 0 pair
80. Given the J.P.d.f. of x and y as
77. Suppose X1, X2,..., Xn is a random sample f(x, y ) = 4xy; 0 ≤ x ≤ 1; 0 ≤ y ≤ 1
from a normal distribution with mean =0 ; Elsewhere,
‘µ’ and unknown variance ‘σ2’. The null then what is the value of
hypothesis µ = 12 is a
(A) Simple hypothesis
{
p 0 < x < 1 ; 1 ≤ y ≤1
2 2 }
3
(B) Both simple and composite hypothesis (A) 16
(C) Composite hypothesis 5
(B) 16
(D) Not a hypothesis 3
(C) 8
78. Let X1, X2, ..., Xn be a Bernouli random 1
(D) 4
variables with parameter ‘p’. what is the
method of moments estimation of p ? 81. If T1 and T2 are independent and unbiased
2
n
estimators of parameter ‘θ’ with V(T1) = σ1 and
1 n 2 2
(A) ∑ Xi (B) n ∑ Xi V(T2) = 2 σ1 , then unbiased estimate of θ
i =1 i =1 given by T = λ1T1 + (1 – λ)T2, minimum
variance if λ equals
1 n 1 n
(C) n ∑ (Xi − X)
2
(D) n ∑ Xi (A) 12 (B) 13
i =1 i =1
(C) 14 (D) 2 3
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82. You have conducted a study comparing 85. What do you report in a maximum
Army, Navy and RAF cadets on a measure likelihood ratio to say whether your model
of leadership skills. These are unequal was significant or not ?
group sizes and the data is skewed so you
(A) ANOVA
need to use a non parametric test, then
which test you choose ? (B) Correlation
(A) Friedman test (C) R-squared
(B) Mann Whitney test
(D) Beta
(C) Kruskal Wallis test
86. The basic assumption behind regression
(D) Wilcoxon Rank Sum test
analysis is
83. One or two tail tests will determine
(A) To estimate a line that goes through
(A) If the two extreme values (minimum most values in the observed data
or maximum) of the sample need to
(B) To minimise the sum of the squared
be rejected
residuals
(B) If the hypothesis has one or possible
two conclusions (C) Estimate a line that maximises the
n
difference between the ∑ (Yi − Yi )
(C) If the region of rejection is located in ˆ
i=1
one or two tails of the distribution
(D) We accept the test (D) All the above
84. If the ties occur in the Kruskal Wallis test
87. What assumptions does ANCOVA have
with usual notations, the correction ‘C’ for
that ANOVA does not have ?
ties is
(A) Homogeneity of variance
(A) ΣT/K(n2 – 1)
(B) ΣT/(n2 – 1) (B) Homoscedasticity
(C) ΣT/n(n – 1) (C) Homogeneity of sample size
(D) ΣT/(n2 – 1) n (D) Homogeneity of regression slopes
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88. A logistic regression model was used to 90. Match the following :
assess the association between C.V.D. 1. Rotated factor a. This is the number
and obesity. p is defined to be the loading of factors plotted
probability that the people have CVD, against variance
obesity was coded as o = non obese, log 2. The scree plot b. Carried out simply
(p/1 – p) = – 2 + 0.7 (obesity). what is the reduce a larger
log odds ratio for CVD in persons who are data set to a
obese as compared to not obese ? smaller one
(A) 0.7 3. Principles c. You look at this
component to name a factor
(B) –2
analysis
(C) 2.7
4. Correlation d. The first step
(D) exp (0.7) matrix taken to perform
a factor analysis
89. In simple logistic regression the traditional
(A) 1 – c, 2 – a, 3 – b, 4 – d
goodness of fit measure, – 2 (log likelihood
(B) 1 – a, 2 – d, 3 – b, 4 – c
of current model – log likelihood of
previous model) is (C) 1 – c, 2 – d, 3 – b, 4 – a
(A) a statistic that does not follow a χ 2 (D) 1 – b, 2 – a, 3 – d, 4 – c
p.d.f. 91. Which of the following is finally produced
(B) Indicates the spread of answers to a by hierarchical clustering ?
question (A) Final estimate of cluster centroids
(C) An index of how closely the analysis (B) Tree showing how close things are to
reaches statistical significance each other
(D) How close the predicted findings are (C) Assignment of each point to cluster
to actual finding (D) All the mentioned above
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92. In sampling without replacement, the 94. Let N be number of units in a population.
After the selection of one unit from a
standard error of the sample mean
population, every k th (k < n) unit is
vanishes if selected to obtain a sample of size n. Let
P be the inter class correlation between
(A) n = N the units of the same systematic sample. If P =
1, then the relative precision of systematic
(B) n > N
sample with simple random sampling is
(C) n < N (A) a function of N only
(B) a function of N and k only
(D) n ≠ N
(C) a function of N and n only
(D) a function of N, n and k
93. Let ‘µ’ be the population mean and
95. For a BIBD with parameters v = b = 7,
2
‘σ ’ be the population variance and a r = k = 4, λ = 2, the number of treatments
common between any two blocks is
sample of size ‘n’ is drawn by simple
(A) 3
random sampling with replacement from
n
(B) 4
a population of size N. Let Y = ∑ λi xi be
(C) 1
i =1
a best linear unbiased estimator, then the (D) 2
variance of Y is equal to
2 1 n 2 σ2 n 2 96. In a 23 factorial experiment the principal
(A) σ 1 + ∑ i N − 1 ∑ λi
λ +
n − 1 i =1 i =1 blocks of replicates 1 and 2 respectively
consist of {(1), a, BC, ABC} and
σ 2 n
1 n
2 {ABC, AC, B, (1)}. What are the confounded
(B) ∑
N − 1 i =1
λi2 + σ2 1 + ∑ λi
N − 1 i =1 interaction effects in the 2 replicates
respectively ?
2 n n
σ 1
(C) ∑
n − 1 i =1
λi2 − σ2 1 +
N − 1 ∑ λi
i =1
2
(A) BC and AC
(B) ABC and AB
2 1 n 2 σ2 n 2 (C) BC and AB
(D) σ 1 + ∑ i (N − 1) ∑ λi
λ −
N − 1 i =1 i =1
(D) AB and BC
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97. Let R(t) and h(t) respectively denote the 99. If X(t) is the number of customers in an
reliability and hazard rate of a unit at time t,
M/M/1 queuing system with arrival rate
then which of the following relation is
λ > 0 and service rate µ > 0, then what is
correct ?
t the process {X (t), t > 0} is
(A) log R(t) = ∫ r (x ) dx
0
(A) Birth process with (λ – µ)
t
(B) log R(t) = − ∫ r (x ) dx
0 (B) Poisson process with rate (λ – µ)
t
(C) log R(t) = 1 − ∫ r (x ) dx (C) Markov process
0
t (D) Birth and death process
(D) log R(t) = exp − ∫ r (x ) dx
0
100. In a linear programming problem with
98. Consider the L.P.P.
4 constraints and 6 variables which of the
Max Q = x + 3y + 6z – w
following is equal to the maximum number
Subject to 5x + y + 6z + 7w < 20;
6x + 2y + 2z + 9w < 40 of basic solutions ?
x, y, z, w > 0
(A) 4
(A) 55
(B) 10
(B) 58
(C) 60 (C) 24
(D) 62 (D) 15
_______________
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II 24 A-15-18Official key not published for this paper.
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Source: Telangana State Eligibility Test. Question paper published by Telangana State Eligibility Test.