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K-2614
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1. D ±Üâo¨Ü ÊæáàÆᤩ¿áÈÉ J¨ÜXst Óܧ٨
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entries, which may disclose your identity, you will render yourself
Answer Sheet, except for the space allotted for the relevant
8. If you write your name or put any mark on any part of the OMR
7. Rough Work is to be done in the end of this booklet.
6. Read the instructions given in OMR carefully.
evaluated.
place other than in the ovals in OMR Answer Sheet, it will not be
in the OMR Sheet kept inside the Booklet. If you mark at any
5. Your responses to the question of Paper III are to be indicated
where (C) is the correct response.
Example :
4.
3.
2.
1.
B
C
D
Instructions for the Candidates
Number of Pages in this Booklet : 16
Time : 2 Hours 30 Minutes
A
correct response against each item.
and (D). You have to darken the oval as indicated below on the
Each item has four alternative responses marked (A), (B), (C)
Booklet will be replaced nor any extra time will be given.
period of 5 minutes. Afterwards, neither the Question
by a correct booklet from the invigilator within the
other discrepancy should be got replaced immediately
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(i) To have access to the Question Booklet, tear off the paper
open the booklet and compulsorily examine it as below :
be given to you. In the first 5 minutes, you are requested to
At the commencement of examination, the question booklet will
This paper consists of seventy five multiple-choice type of questions.
Write your roll number in the space provided on the top of this page.
Number of Questions in this Booklet : 75
Maximum Marks : 150
Subject : MATHEMATICAL SCIENCE
Paper
: III
Name
: ____________________________________
Signature: ____________________________________
Name
: ___________________________________
Signature : ___________________________________
Name & Signature of Invigilator/s
(Figures as per admission card)
Test Subject Code :
K-2614
Test Subject
: MATHEMATICAL SCIENCE
Test Paper
: III
Roll No.
OMR Sheet No. : _________________________________
Test Booklet Serial No. : _______________________
Paper III
K-2614
2
e
locally compact
space and A is open in X, then A is
(D) u (x, t) =
(D) If X is locally compact Hausdorff
compact
space is regular
(B) Every locally compact Hausdorff
(C) u(x, t) =
metrizable space is compact
false ?
3. Which one of the following statements is
(B) u(x, t) =
(D) π log a
e
a
(C)
π log a
π log a
π log a
∫
∞
x2 + a2
log x
(C)
1
(C) ∞
(B) 1
(A) 0
n→∞ n
1. lim
1 + 2 + 3 3 + .... + n n is equal to
1
Note :
(D)
dx is equal to
2
(
∞
n
1
2
+ π
3 12
n =1
∑
∞
4
n
e
1 − 3n2 π 2 t
2 π
+
3 12
n =1
∑
∞
n
(−1)n+1
2 π
+
3 12
n =1
n
∞
( −1)n + 1
∑
ux (0, t) = ux (2, t) = 0, u (x, 0) = 3x, is
ut = 3uxx, 0 < x < 2, t > 0,
5. The solution to the heat equation
2a
0
(D)
∑
4
 2 


cos
− 3n2 π 2 t
 nπx 
(A) u(x, t ) =
2
2. If a > 0 then
n =1
4
 4 
e


cos
− 3n2 π2 t
 nπx 
(A) Every sequentially compact
(A)
2 4
+
3 12
 2 
sin 

 nπx 
(C) Every limit point compact space is
(B)
4
 2 
− 3n2 π 2 t cos  nπx 
)
(B)
(A)
dx 2
d2 y
λ = n2 , n = 1, 2, 3, .....
has non-trivial solutions if
λ = 2n, n = 1, 2, 3, ....
has no non-trivial solutions if
λ = n, n = 1, 2, 3, ....
has a non-trivial solution if
has a non-trivial solution if λ ≤ 0
+ λy = 0 , y (0) = 0, y(π) = 0
4. The sturm-Liouville problem
carries two (2) marks. All questions are compulsory.
This paper contains seventy-five (75) objective type questions. Each question
PAPER – III
MATHEMATICAL SCIENCE
*K2614*
Total Number of Pages : 16
K-2614
Paper III
3
(C) 1.2428
(D) 1.2424
(D) (A × B) ∩ (C × D) = (A ∪ C) × (B ∪ D)
(A) 1.2426
(B) 1.2425
(C) (A × B) ∪ (C × D) ⊂ (A ∪ C) × (B ∪ D)
increment h = 0.2 is
dx
= x + y, y(0) = 1 with
given that
dy
Runge-Kutta method of fourth order,
8. The value of y (0.2) obtained by
(D) φ(x ) = f(x) +
(C) φ(x ) = f(x ) +
λ2 − 1
λ2
0
ex
∫
1
e − y f(y) dy
λ −1 0
∫
λ e x e − y φ(y) dy
1
 λ − 1 0
(B) φ(x ) = f(x) − 
 ∫
 λ + 1 e x e − y f(y ) dy
1
0
λ −1
e x ∫ e − y f(y ) dy
(A) φ(x ) = f(x ) −
λ
1
solution of (1) ?
then which one of the following is the
given real function f(x) (0 ≤ x ≤ 1) . If λ ≠ 1,
(B) (A × B) ∪ (C × D) ⊃ (A ∪ C) × (B ∪ D)
(A) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)
true ?
which one of the following statements is
10. If A, B, C, D are nonempty sets, then
(D) φ (t) ≠ 0 for all t ∈ [a, b] with t ≠ t 0
of t0 ≠ t ∈ [a, b]
(C) φ (t) ≠ 0 and φ′(t ) = 0 for all values
(B) φ (t) = 0 ∀ t ∈ [a, b]
[a, b]
(A) φ (t) ≠ 0 for atleast one value of t in
system such that φ( t 0 ) = 0 . Then
 φn 
 
 ⋅ 
 ⋅ 
φ =   be a solution of the above
 ⋅ 
 φ2 
 1
φ 
Let to be any point of [a, b] and let
0
φ(x) = f(x ) + λ
∫
1
e x − y φ(y ) dy .... (1) for a
7. Consider the Fredholm integral equation
(D)
(C)
(B)
(A)
y (x) = e
y (x) = 0
y (x) = c, where c is any real constant ≠ 0
y (x) = x
a
∫
b
(x + y 2 + 3y′) dx is given by
6. The extremal for the functional
*K2614*
dt
= an1(t)x1 + ... + ann (t)xn.
dx n
•
•
•
•
•
•
•
•
•
•
•
•
dt
= a 21(t ) x1 + ... + a 2n (t )x n
dx 2
dt
= a11(t )x1 + ... + a1n (t )x n
dx1
system
9. Consider the homogeneous linear
Total Number of Pages : 16
Paper III
connected.
(D) Every connected space is path
connected.
under a continuous map is
(C) The image of a connected space
common is connected.
connected sets that have a point in
(B) The union of a collection of
have Bd A ≠ φ .
nonempty proper subset A of X, we
(A) If X is connected, then for every
not true ?
14. Which one of the following statements is
K-2614
4
subfield.
(D) No Q-vector subspace of E is a
E which are subfields is infinite.
(C) Number of Q-vector subspaces of
subspaces of E are subfields of E.
(B) Only finitely many Q-vector
subfield of E.
(A) Every Q - vector subspace of E is a
statements is true ?
[E : Q] > 2. Which one of the following
spaces is compact.
(D) The product of finitely many compact
under a continuous map is compact.
(C) The image of a compact space
topological space is closed.
(B) Every compact subspace of any
compact space is compact.
(A) Every closed subspace of a
false ?
13. Which one of the following statements is
is normal.
(D) Every subspace of a normal space
not be normal.
(C) Product of two normal spaces need
(B) Every regular space is normal.
(A) Every Hausdorff space is regular.
true ?
12. Which one of the following statements is
(D) closed
(C) path connected
(B) bounded
(A) compact
usual topology, then the set IR 2 – A is
11. If A is a countable subset of IR 2 with
*K2614*
field of rational numbers Q. Assume
17. Let E be a finite Galois extension of the
(C) pq – p – q
(D) pq – 1
(A) pq – p – q + 1
(B) pq – p – q – 1
which are coprime to pq is
Then the number of integers a, 1 < a < pq,
16. Let p and q be two distinct prime numbers.
other.
IR are not homeomorphic to each
2
(D) Under the usual topologies, IR and
homeomorphism.
by f(t) = (cos2 π t, sin2 π t) is a
(C) The mapping f : [0, 1) S′ defined
to each other.
(B) (–1, 1) and IR are homeomorphic
IR are homeomorphic to each other.
(A) The subspaces [a, b] and [0, 1] of
not true ?
15. Which one of the following statements is
Total Number of Pages : 16
K-2614
Paper III
5
25 elements
(D) irreducible over any finite field with
elements
5
(C) irreducible over the field IF of five
numbers IR
(B) reducible over the field of real
(A) irreducible over ring of integers 9
(D) does not exist
(C) exists, but not 3
(B) is
3
(A) is 3
A = {p ∈ Q + | p 2 < 3} in Q
24. The supremum of the set
20. The polynomial f(x) = x4 + x3 + x2 + x + 1 is
a=b
(D) For a, b, c ∈ R − {0} , if ac = bc then
has a solution
(C) The equation x = a, a ∈ R always
2
group under multiplication
(B) Non-zero elements can never be a
always a C ∈ R such that a ⋅ c = b
(A) Given any a, b ∈ R − {0} there is
following holds ?
19. In an integral domain R, which one of the
zero ideal of 9.
9[i], the intersection 9 p is a non(D) For any non-zero prime ideal p of
prime ideal.
ideal generated by ‘p’ in 9[i] is a
(C) For any prime number ‘p ’ in 9, the
extension of its prime filed.
9[i]/ p is always a degree 2
(B) If p is a prime ideal of 9[i], then
9[i]/p is a field.
(A) If p is a prime ideal of 9[i], then
statements is true ?
integers. Which one of the following
18. Let 9 [ i ] denote the ring of Gaussian
*K2614*
relation
(D) both transitive and symmetric
(C) a symmetric relation
(B) a transitive relation
(A) an equivalence relation
“a divides b” is
23. In the set of integers, the relation
(D) G cannot be a cyclic group
(C) G is a cyclic group
(B) G cannot be an abelian group
(A) G is an abelian group
center. Assume G Z is cyclic. Then
22. Let G be a finite group and Z ⊂ G be its
(D) an associative ring
identity
(C) a non commutative ring without
(B) a commutative ring with identity
(A) a non-associative ring
21. The ring M2 (IR) of all 2 × 2 real matrices is
Total Number of Pages : 16
Paper III
2
1
(D) is
(C) is 1
(B) is 0
(A) does not exist
f(x) → 0 as x ∞ . The xlim∞ f ′(x)
(0, + ∞ ), f′′ is bounded on (0, + ∞ ) and
27. Suppose f is twice differentiable on
Stielties integrable functions
(D) f is a product of two Riemann-
U(p, f, α) − L(p, f, α ) < ∈
partition p such that
(C) for every ∈ > 0, there exists a
(B) f is monotonic on [a, b]
(A) f is continuous on [a, b]
respect to α if and only if
Riemann-stielties in tegrable with
[a, b]. Then the function f : [a, b] IR is
26. Let α be monotonically increasing on
0
∫
(D)
2
1
0
(C)
∫
∞
3

 
 log  x   dx
 1 

0
∫
1
1 + 2x 2
7e − x − 1
4
x
log x
(D) nlim∞ xn = 1
n
∞
(C) lim n xn = 2
2
(B) nlim∞ nxn = 1
(A) nlim∞ nxn = 0
monotonically. Then
which the terms xn decrease
n =1
30. Let
∑
xn be a convergent series in
∞
(C)
2
1
(D) 1
(A) 0
(B)
n =1
∑
4
1
n4 + n2 + 1
n
∞
dx
dx
integrals diverges ?
25. Which one of the following improper
*K2614*
(D) is either 1 or –1
(C) is always 1
(B) may be any non-zero integer
(A) may not be an integer
Then det (A)
Assume that A–1 also has integer entries.
31. Let A be a 3 × 3 matrix with integer entries.
29. The sum of the series
0
(B) ∫ e
dx
−x
2
∞
(A)
K-2614
6
(D) Diverges for a >
is
3
1
(C) Diverges for all real values of a
(B) Converges for a >
3
1
(A) Converges for all real values of a
n=1
28. The series
∑
∞
n

 − sin 1n 
1

( )
a
Total Number of Pages : 16
K-2614
T is diagonal
with respect to which the matrix of
(D) There may not be any basis of V
digonal
respect to which the matrix of T is
the field of complex numbers with
(C) There is always a basis of V over
diagonal
respect to which the matrix of T is
the field of real numbers with
(B) There is always a basis of V over
(A) The matrix of T is always diagonal
following is true ?
of rational numbers. Which one of the
dimensional vector space V over the field
34. Let T be an endomorphism of a two
values cannot be distinct
(D) If A is invertible then the eigen
values must be distinct
(C) If A is invertible then the eigen
a non-zero real eigen value for A.
(B) If A is invertible then there is always
complex eigen value
(A) A is invertible if A has a non real,
of the following is true ?
33. Let A be a 2 × 2 real matrix. Then which
composition of endomorphisms
EndF(V) always form a group under
(D) Non-zero endomorphisms of
F
(C) End (V) is always a F-vector space
endomorphisms
addition and composition of
commutative ring under usual
F
(B) End (V) can never be a
structure
F
(A) End (V) has no F-vector space
following statements is true ?
F-endomorphisms of V. Which one of the
32. EndF (V) denotes the set of all
*K2614*
Paper III
7
0 0 1


(D) 0 1 0
 1 1 0
0 0 0


(C) 0 − 1 0
 1 0 0
0 2 0


(B)  1 − 1 0
 1 − 1 0
0 2 0


(A)  1 0 0
0 0 0 
respect to the basis {1 + x, x, x2} is
dx
. Then the matrix of T with
T( f ) = f ′ =
df
field F. Define T : F 2[x] F 2 [x] by
polynomials of degree atmost 2 over a
36. Let F 2 [x] be the vector space of all
0
(D) 
1
0 1
1

and 
 1 2
1
0
(C) 
1
0 1
1

and 
 1 0
1
 1 0 0
0 0 0




(B) 0 0 1 and 0 0 1
0 1 0
0 1 0 
1
(A) 
1
0 0
1

and 
3 0
1
over IR are similar ?
35. Which of the following pairs of matrices
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Paper III
K-2614
8
 y d
z c 
(D) 
 and 

a x
b w 
(D) 2πi e4
 y d
 x d
(C) 
 and 

a x
a y
(B) 2πi (e2 − e4 )
z c 
y a

 and 
(B) 
b w 
d x
y b
z d 
 and 

(A) 
a x
c w
matrices can never be similar ?
numbers, which pair of the following
and w, x, y and z represent any real
39. If a < b < c < d are fixed real numbers
(D) 4
(C) 2πi e2
(A) 2πi (e4 − e2 )
|z| = 5, positively oriented, is
C
∫ (z − 1) (z − 2) dz where C is the circle
e2z
42. The value of the integral
(C)
(B)
(C) 3
(B) 2
(A) 1
The dimension of V over IR is
and d = a + b}
( )
2
2 cos (α )
2
2 sin (α )
2
2 sin (α )
(D) 2 cos α 2
(A)
then | z| is
41. Suppose α is real and z = 1 – cosα + i sin α ,
38. Let V = {(a, b, c, d) | a, b, c, d ∈ IR , a = c
(D) e + z
(D) have absolute value 1
z
(C) have multiplicity 2
(C) zeiz
(B) are real
(B) e – iz
(A) are purely imaginary
(A) e + iz
values of A
 i − 3 − 4i
5 


37. If A = 0
2
− 3 + 4i then the eigen
1
0
−i 
*K2614*
z
z
f(z) can be
for all z ∈ " and u(x, y) = y + ex cosy. Then
40. Given that f(z) = u(x, y) + i ν (x, y) is analytic
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K-2614
Paper III
9
(D) Standard exponential
(C) Degenerate at 1
(B) Degenerate at 0
(A) Uniform over [0, 1]
Y = f(U), then what is the distribution of Y ?
uniform random variable over [0, 1], U and
46. If f is the probability density function of
(D) an essential singularity at 1
(C) a pole at 1 of order 2
(B) a pole at 0 of order 2
Then
2 .
f(z) has
1
45. Consider the complex valued function
2
(D) z = nπ + (−1)n π , n ∈ 9
2
(C) z = nπ + π , n ∈ 9
( )
2
z = nπ + ( −1)n (π − 4i), n ∈ 9
(B) z = (−1) ⋅ nπ + π 2 − 4i , n ∈ 9
n
(A)
X+Y+Z
is Cauchy
X
distribution
(C) X +Y +Z has chi-square
2
2
2
(B) ex + ey + ez is exponential
(A) X given Y and Z is normal
correct.
distribution, which one of the following is
49. If (X, Y, Z) has tri-variate normal
(D)
(A) a removable singularity at 1
f(z) = (z – 1)2 ⋅ e ( z −1)
normal random variables
(D) X + Y and X – Y are independent
distribution
(C) (X + Y) (X – Y) has chi-square
distributed
(B) X + Y and X – Y are identically
random variables
(A) X + Y and X – Y are dependent
following is true ?
random variables, which one of the
50. If X and Y are independent normal
z ∈ " are
44. The roots of the equation sinz = cosh4,
(D) Beta distribution
(C) Uniform [0, 1] distribution
(B) Degenerate distribution at 1
(A) Degenerate distribution at 0
distribution over [0, 1] ?
converge to, U having uniform
48. What does the distribution function of U
(D) 0
(C) 4πi
 2
(D) Binomial  l, 
 1
(B) 1
(A) 2πi
 4
(C) Binomial  l, 
 1
2
(B) Geometric  
 1
C
∫
z −1
dz is
f (z )
anticlockwise. Then the value of
radius 2 with center as (1, 0), oriented
43. Let f(z) = z3 – 1 and C denote the circle of
*K2614*
n
1
(A) Poisson (2)
of X given X + Y = l ?
random variables, what is the distribution
47. If X and Y are independent Poisson (2)
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Paper III
K-2614
10
moments finite
(D) Student’s t-distribution has all
(D)
5
1
distribution
generalization of Cauchy
(C) Student’s t-distribution is a
(C)
5
2
(B)
5
3
(B) Student’s t-distribution is symmetric
sampling distribution
(A) Student’s t-distribution is a
53. Which of the following is not true ?
(A) 1
→∞ 11
what is nlim
p( n) ?
states 1, 2, given that p12 =
(D) MLE of θ does not exist
(C) Sample range is the MLE of θ
56. With reference to a Markov chain with
(D) i is non-null
(B) Sample mean is the MLE of θ
(C) i is recurrent null
(A) Sample median is the MLE of θ
(B) i is aperiodic
the following is correct ?
f(x; θ) =
2
e
,
. Which of
1 −|x −θ| x ∈ IR , θ ∈ IR
the probability density function
52. Let {X1, ..., Xn} be a random sample from
6
3
, p 21 = ,
1
2
(A) i is ergodic
sufficient condition for existence of ai ?
in n-steps. Which of the following is a
probability of going to state i from state i
n→∞
ii
ai = lim p(nii) where p(n) is the transition
55. Let i be a state of a Markov chain and
3
(D)
2F + G
4
(C)
F+G
(B) FG
2
(A)
F+G
function ?
of the following is not a distribution
51. If F and G are distribution functions, which
*K2614*
(D) The errors have zero variances
(C) The errors have zero expectations
(B) The error variances are different
(A) The error variances are same
‘heteroscedastic’ mean ?
54. In a linear model, what does
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(D)
6
1
(D) e , t ∈ IR
t2
(C) e , t ∈ IR
3
(C)
2
(B)
–t 2
(B) e–|t|, t ∈ IR
(A) e , t ∈ IR
5
2
–t
n
X1 + ... + X n
5
(A)
1
is idle ?
steady state probability that the system
service rate 2 and no waiting, what is the
59. In an M/M/1 queue with arrival rate 3,
 4

(D) B   , t ≥ 0
 t

(C) {| B(t) |, t ≥ 0}
(B) {eB( t ) , t ≥ 0}
(A) {B(t + 2) − B(t ), t ≥ 0}
is a Brownian motion ?
Brownian motion, which of the following
58. If {B(t ), t ≥ 0} denotes a standard
(D)
n =1
∑
∞
(C)
n =1
∑
∞
converges to 0
p(kk
n)
is divergent
p(kk
n)
n→∞
(B) lim p kk = 0
(n)
n=1
(A)
Paper III
11
∑
∞
kk
p(n) converges
following ?
Markov chain is which one of the
57. A criterion for state k to be recurrent in a
*K2614*
as n → ∞ ?
of the characteristic function of
1 − | t | if | t | ≤ 1;
what is the limit
φ(t ) = 
 0 if | t | > 1,
characteristic function
62. If X1, X2, ... , Xn are independent with
(D) Tn is never unbiased for θ
regularity conditions
(C) Tn is consistent for θ under Cramer’s
(B) Tn is never consistent for θ
(A) Tn is always unbiased for θ
is true ?
distribution F(⋅ ; θ) , which of the following
61. If Tn is the MLE of a parameter θ in a
→∞
(D) tlim
t
µ
= a.s.
N(t ) 1
t→∞
t
= µ a.s
N(t )
t→∞
t
= 0 a.s.
N(t )
t→∞
t
= 1 a.s.
N(t )
(C) lim
(B) lim
(A) lim
µ . Which of the following is true ?
mean finite and mean inter-arrival time
60. Let {N(t ), t ≥ 0} be a renewal process with
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Paper III
(D)
K-2614
12
( )
EY
X 
− Cov. , Y 
Y 
( )
EX
(C)
X 
− Cov. , Y 
Y 
(D)
64e2
(C)
64e2
(B)
64e2
(A)
32e2
( )
EY
(B)
X 
− Cov. , X 
Y 
27
81
11
27
P (X + Y = 1) if X and Y are independent ?
defined for appropriate values of t, what is
( )
EX
X 
(A)
− Cov. , X 
Y 
Mx (t )


 4 
=

 3 + et 
in a simple random sampling ?
65. What is the bias of the ratio estimator
x
y
(D) θ1 = θ2 − θ3
(C) θ3 = θ1 + θ2
(B) θ2 = θ1 + θ3
(A) θ1 = θ2 + θ3
a′θ = a1θ1 + a 2θ2 + a 3θ3 estimable
when is the linear parametric function
0 1 1

,
A =  1 0 1
 1 1 2
64. Given the linear model (Y, Aθ, σ 2 I 3 ) with
(C) 0.6
(D) – 0.3
(A) – 0.4
(B) 0.3
between 1 + X and 1 – 2Y ?
Y is 0.3, what is the correlation coefficient
63. If correlation coefficient between X and
*K2614*
and My(t) = e2(e –1)
t
3
68. Given the moment generating functions
(D)
(C)
(B)
(A)
5
10
0
–5
x ≥ 0, y ≥ 0
minimize x + y subject to x – y = – 5,
at an optimal solution of the LPP.
67. Find the value of the objective function
contrast
(D) a′α is an elementary treatment
of the design matrix
(C) Vector a belongs to the row space
space of the C-matrix of the design
(B) Vector a belongs to the column
space of the design matrix
(A) Vector a belongs to the column
effects alone, in a general block design ?
parametric function a′α of treatment
condition for estimability of a linear
66. What is a necessary and sufficient
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Paper III
13
(D) P [| X | ≤ ∈] ≥
free
n
(D) None of D , Dn+ , Dn− are distribution
not D
(C) P [| X | ≤ ∈] ≤
n
(B) P [| X | ≥ ∈] ≤
(C) Dn+ and Dn− are distribution free but
n
(B) D is distribution free but not Dn+ and Dn−
n
(A) D , Dn+ and Dn− are distribution free
following is true ?
x
D −n = sup (F(x ) − Fn (x )) , which of the
D +n =
x
sup (
Fn ( x ) − F( x )
),
x
sup | Fn (x ) − F(x) |,
=
distribution function. Let
Dn
Fn
be the corresponding empirical
a continuous distribution function F and
70. Let {X1, ..., Xn} be a random sample from
(D) 0
(C)
3
1
(B)
2
1
(A) 1
often) equal to ?
n = 1, 2, ... , what is P({Xn = 0} infinitely
2n
= 1 − P( Xn = 1) ,
69. Given that P( Xn = 0) =
1
*K2614*
(A) P [| X | ≥ ∈] ≤
∈k
E | X |k
∈k
E | X |k
E | X |k
∈k
∈k
E | X |k
Then the Markov inequality states that
73. Let X be a random variable with E |X|k < ∞ .
components
(D) Median survival time of its
components
(C) Mean survival time of its
components
(B) Maximum of the survival times of its
components
(A) Minimum of the survival times of its
system survival time is
72. In a parallel system of k components, the
(D)
θ
σ 2n
(C) σ 2n ⋅θ
(B) σ 2n ⋅θ2
(A)
θ2
σ 2n
variance
σ2n then log Tn is CAN for log θ with
71. If Tn is a CAN estimator of θ with variance
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Paper III
K-2614
14
___________
(D) there is multicollinearity in the model
(D) s p2 =
n −1
p2q
(C) s p =
2
n −1
npq
(B) sp =
2
n
pq
(A) s p =
2
n −1
pq
are linearly independent
(C) the columns of the regression matrix
vector is singular
(B) the dispersion matrix of the error
(A) the errors are autocorrelated
estimator is proposed when
linear regression model, a ridge
74. While estimating the parameters of a
*K2614*
variance of proportion p is
proportion q of category II then the
has proportion p of items of category I and
dichotomous population. If the sample
75. A sample of size n is drawn from a
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K-2614
15
Paper III
Space for Rough Work
bñÜᤠŸÃÜÖÜPÝRX ÓܧÙÜ
*K2614*
Total Number of Pages : 16
Paper III
16
K-2614
Space for Rough Work
bñÜᤠŸÃÜÖÜPÝRX ÓܧÙÜ
*K2614*
Total Number of Pages : 16