MH SET 2013: 17 February, Paper III, Mathematical Sciences
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Mathematical Science
Paper III
Time Allowed : 2½ Hours] [Maximum Marks : 150
Note : This Paper contains Seventy Five (75) multiple choice questions, each
question carrying Two (2) marks. Attempt All questions.
2. Let A be a linear transformation
1. Which of the following sequences
from R n to Rn. If x ∈ R n, then
has a convergent subsequence ? the derivative of A at x is given
by :
(A) an = (–1)n n ()
(A) A′ x = 0, the zero
transformation
(B) an = n + cos n
()
(B) A′ x = I, the identity
(C) an = cos n2 + i sin n2 transformation
()
(C) A′ x = A
e n
(D) an = 5
n ()
(D) A′ x = A.A
Or Or
The value of :
1
Gauss-Seidel method converges as 1
∫ 1 + x dx ,
0
fast as the Jacobi method :
correct to three decimal places, by
the Simpson’s rule with h = 0.5 and
(A) Fourth
x : 0.0 0.5 1.0
1
(B) Thrice = y : 1.000 0.6667 0.5000
1+ x
(A) 0.894
(C) Twice
(B) 0.794
(C) 0.694
(D) Fifth (D) 0.594
1 [P.T.O.
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3. The Taylor series for sin x about 4. A function f : [a, b] → R is Riemann
O is : integrable if and only if it is bounded
x2 x4 and :
(A) 1 − + − .....
2! 4! (A) it is discontinuous only at a
x3 x5 finite number of points
(B) x − + − .....
3! 5!
(B) it is either monotone or it has
x3 x5 only a finite number of
(C) x − + − .....
3 5 discontinuities
x2 x3 x4 (C) the set of its discontinuities is
(D) x − + − + .....
2! 3! 4!
of Lebesgue measure zero
Or
The Lagrange interpolating (D) the set of its discontinuities is
polynomial of degree two countable
approximating the function y = lnx, Or
defined by the following table
If f(x) is continuous function in
values :
x0 ≤ x ≤ xn then given any ∈ > 0,
x y = lnx there exists a polynomial p(x) such
2 0.69315 that :
2.5 0.91629
(A) |f(x) – p(x)|< ∈; for some x in
3.0 1.09861
(x0, xn)
(A) l2(x) = – 0.81366x2 + 0.8164x
– 0.60761 (B) |f(x) – p(x)| ≤ ∈; for some x in
(B) l2(x) = – 0.8164x2 + 0.81366x (x0, xn)
– 0.60761
(C) |f(x) – p(x)|< ∈; for all x in
(C) l2(x) = 0.8164x2 + 0.81366x
(x0, xn)
– 0.60761
(D) l2(x) = 0.81366x2 + 0.8164x (D) |f(x) – p(x)|≤ ∈; for all x in
– 0.60761 (x0, xn)
2
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5. The solution of the following linear 7. Inclusion of an additional constraint
in the existing set of constraints
programming problem is :
will cause a(n) :
Max. z = 3x1 + 2x2
(A) change in objective function
Subject to constraints coefficients
(B) change in constraint coefficients
x1 + x2 ≥ 3
(C) increase in the optimal objective
x1 – x2 ≤ 1
function value
and x1, x2 ≥ 0 (D) decrease in the optimal objective
function value
(A) unbounded
8. The optimal solution of the quadratic
(B) x1 = 2, x2 = 1
programming problem :
(C) x1 = 1, x2 = 2 Max. z = 10x1 + 25x2 – 10x12 − x22
(D) x1 = 0, x2 = 3 – 4x1x2
Subject to constraints
6. If the pth variable of the primal is
x1 + 2x2 + x3 = 10
unrestricted in sign, then the pth
x1 + x2 + x4 = 9
constraint of the dual is :
and x1, x2, x3 x4 ≥ 0
(A) Less than type
(A) x1 = 0, x2 = 5, x3 = 0, x4 = 4
(B) More than type
(B) x1 = 4, x2 = 0, x3 = 6, x4 = 5
(C) Equality type (C) x1 = 2, x2 = 4, x3 = 0, x4 = 4
(D) Not necessary equality (D) x1 = 9, x2 = 0, x3 = 1, x4 = 0
3 [P.T.O.
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10. The value of the integral :
9. Let f(z) be an analytic function
within and on |z – 2| = 3. Suppose
∫ (x + y − 2ixy) dz
c
2 2
|f(z)| has maximum value 2 on
along c, where c is a line from
|z – 2| = 3. Then the value of
z = 0 to z = 1 + i is :
|f ′′(2)| is :
4
(A)
5 3
(A) >
9 2π
(B)
3
4
(B) ≤
9 (1 + i )(2 − i )
(C)
3
1
(C)
2 (1 − i )(2 + i )
(D)
3
2
(D) Or
3
A researcher studies achievement
Or by children in poorly funded
elementary schools. She believes
that parent involvement has an
In an experimental research study,
impact on children by increasing
the primary goal is to isolate their motivation to do school work.
Thus, in her model, greater parent
and identify the effect produced involvement leads to higher student
motivation, which in turn creates
by the ....................... .
higher student achievement.
Student motivation is what kind of
(A) dependent variable variable in this study ?
(A) Mediating or intervening
(B) independent variable
variable
(C) interaction effect (B) Confounding variable
(C) Control variable
(D) confounding variable (D) Independent variable
4
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11. The degree of the extension field 12. Let M 2(z) be the ring of 2 × 2
matrices over z.
Q ( 5, 3 ) over Q is : Consider the following statements :
(1) M2(z) is a field
(A) 4 (2) M2(z) is an integral domain
(3) M2(z) is not an integral domain
(B) 3 (4) M2(z) is a division ring
Then :
(C) 2
(A) Only statement (3) is true
(B) Only statements (1) and (2)
(D) 1
are true
Or (C) Only statement (4) is true
(D) All the statements are true
Select the most suitable words to Or
If most of the measurements in a
complete the statement : “The entire large data set are of approximately
group of objects or people about the same magnitude except for a few
measurements that are quite a bit
which information is wanted is larger, how would the mean and
called the ....................... . Individual median of the data set compare and
what shape the histogram of the
members are called..................... . The data set have ?
..................... is the part that is (A) The mean would be larger than
the median and the histogram
actually examined in order to gather would be skewed with a long
information.” right tail
(B) The mean would be larger than
(A) population, units, sample the median and the histogram
would be skewed with a long
(B) sample, units, target population left tail
(C) The mean would be equal to the
(C) whole, items of interest, median and the histogram
would be symmetrical
response group
(D) The mean would be smaller
than the median and the
(D) population, explanatory
histogram would be skewed
variable, subgroup with a long right tail
5 [P.T.O.
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13. Consider the ring z36. This ring 14. Which of the following statements
has : is true ?
(A) only one prime ideal (A) There exists a field of order 24
(B) exactly two prime ideals (B) There exists a field of order
(C) only one maximal ideal pq with pq distinct primes.
(D) no proper ideals (C) There exists a field of order
2k, (k > 2)
Or
(D) There exists a field of order 25
Let Xn = X and Yn = – X, where
X ~ N(0, 1). Which of the following Or
statements are correct ? Let Ω = R, F = B(R) = Borel σ-field :
~ ~
d
(i) Xn → X , where X is an 1
0, n n odd
iid copy of X
An =
(ii) Y d 1
n
→ X 1 − , 1 n even
~ n
d
(iii) X + Y
n n
→ X + X
Then which of the following is
p
(iv) X + Y
n n
→ 0 true ?
(A) (i), (ii) and (iii) (A) lim A n = lim A n = φ
(B) (i), (ii) and (iv) (B) lim An = {1}
(C) (i) and (ii) (C) lim A n = lim A n = R
(D) (ii), (iii) and (iv) (D) lim An = {0, 1}
6
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15. The matrix of the linear mapping Or
f : R2 → R3, given by : Let X 1 , X 2 , ................. X n be
iid realization from F. Consider
f(x, y) = (2x, y, x – y) the following sequence :
n
∑I
1
with respect to the standard bases Tn(x) = n Xi ≤ x ,
i=1
is :
where I A denote the indicator
function of the set A and x ∈ R fixed.
2 0
Which of the following statements
0 1 are correct ?
(A)
1 −1 (i) p
Using WLLN, Tn(x) → F(x)
d
(ii) Using WLLN, Tn(x) → F(x)
2 0 1
a.s.
(iii) Using SLLN, Tn(x) →
(B)
0 1 −1
F(x)
(iv) E(Tn(x)) = F(x), Var(Tn(x)) =
2 2 1
(C) F ( x ) (1 − F ( x ))
1 1 2
n
(A) (i), (iii) and (iv)
2 1
(B) (i), (ii) and (iii)
2 1
(D)
(C) (ii), (iii) and (iv)
1 2
(D) (i), (ii) and (iv)
7 [P.T.O.
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16. Which of the following subspaces 17. Which of the following is false ?
of R are not homeomorphic ? (A) If f : R → R is a Lipschitz
map, then f is uniformly
(A) [0, 1) and [0, ∞)
continuous
(B) (0, 1) and [0, 1] (B) Any map f : N → R is uniformly
continuous
(C) The set of all integers and the
(C) If f : R → R is continuous and
set of all even integers
f(x) → 0 as x → ± ∞, then f is
(D) [1, 2] and [2, 4] uniformly continuous.
Or (D) If f : (0, 1) → R is continuous,
then f is uniformly continuous
Let X be a random variable with
Or
probability distribution :
A card is drawn randomly from a
P[X = k] = pk, k = 1, 2, .......... . deck of ordinary playing cards. You
win Rs. 10 if the card is a spade or
Then :
an ace. What is the probability that
∞ you will win the game ?
(A) E (X ) ≤ ∑
k =1
P [X ≥ k]
1
(A)
13
∞
(B) E (X ) = ∑ P [X ≥ k]
k =1
(B)
13
52
∞ 17
(C) E (X ) ≥ ∑ P [X ≥ k]
k =1
(C)
52
4
(D) E(X) does not exist (D)
13
8
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18. Let {En} be a sequence of measurable 19. Which of the following is false ?
sets. Which of the following is
(A) If f is measurable then so
false ?
is |f|
(A) If E n ⊃ E n+1 for each n,
(B) If |f| is measurable so is f
∞
(C) If f is integrable so is |f|
then m I E ( )
= lim m En
n = 1 n n →∞
(D) If f is measurable and |f| is
integrable, so is f
(B) If E n ⊂ E n+1 for each n,
Or
∞
then m U ( )
En = lim m En A restaurant manager is considering
n = 1
n →∞
a new location for his restaurant.
The projected annual cash flow for
(C) m(lim infn → ∞ En) ≤ lim
the new location is (in Rs.)
infn → ∞ m(En)
Cash flow Probability
(D) m(lim sup n → ∞ E n ) ≥ lim
10,000 0.10
supn → ∞ m(En), provided that
30,000 0.15
∞
m U E < ∞ for some k ≥ 1.
n = k n
70,000 0.50
90,000 0.15
Or
1,00,000 ?
Suppose a die is tossed 5 times. What
The expected cash flow for the new
is the probability of getting exactly
2 fours ? location is :
(A) 0.161 (A) Rs. 12,800
(B) 0.171 (B) Rs. 64,000
(C) 0.250 (C) Rs. 70,000
(D) 0.333 (D) Rs. 60,000
9 [P.T.O.
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20. A group of order pe, where p is 21. The field with 16 elements contains
prime and e > 1, is never : a subfield with k elements if :
(A) Simple (A) k = 8
(B) k = 4
(B) Abelian
(C) k = 9
(C) Cyclic
(D) k = 11
(D) Solvable
Or
Or
Suppose a die is tossed 5 times. What
The chances that you will ticketed is the probability of getting exactly
for illegal parking on campus are 2 fours ?
1
about . During the last nine days, (A) 0.028
3
you have illegally parked every day
(B) 0.161
and have not been ticketed. Today,
(C) 0.177
on the 10th day, you again decide
(D) 0.333
to park illegally. The chances that
22. The multiplicative group of non-zero
you will be caught are :
elements of a field F is cyclic if :
1
(A) still equal to
3 (A) F = R
1
(B) greater than (B) F = C
3
1
(C) less than (C) F = F625
3
1
(D) equal to (D) F = Q
10
10
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Or Or
A public opinion poll surveyed a
It has been estimated that about
simple random sample of voters.
30% of frozen chicken contain
Respondents were classified by
gender (male and female) and by enough salmonella bacteria to cause
voting preference Republican, illness if improperly cooked. A
Democrat or Independent. Results consumer purchases 5 frozen
are shown below : chickens. What is the probability
Voting preferences that the consumer will have less
Republican Democrat Independent Total than 2 contaminated chickens ?
Male 200 150 50 400
(A) 0.4206
Female 250 300 50 600
(B) 0.3233
Total 450 450 100 1000
If you conduct a chi-square test of (C) 0.1181
independence, what is the expected (D) 0.5282
frequency count of male
24. Let F be a linear operator on a
independents ?
normed space X. Then which of the
(A) 10
following statements is not true ?
(B) 25
(C) 40 (A) F is bounded whenever X is
(D) 60 finite dimensions
23. The roots of which polynomial (B) F is bounded whenever it is
cannot be expressed by radicals
continuous at a point of X
over Q ?
(C) F is bounded whenever X is a
(A) x(x4 + x3 + 3) + 3
Banach space
(B) x3(x3 + 2) + 1
(C) x(x4 + x2 + x) + 1 (D) F is bounded whenever it is
(D) x(x2 – 4) (x2 + 4) + 2 uniformly continuous
11 [P.T.O.
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Or
Or
A study was conducted to estimate
Which of the following statements the effectiveness of assignments in
an introductory statistics course.
is not true about Poisson probability
Students of teacher A received no
distribution with parameter λ. assignments, while students of
teacher B received assignments. The
(A) The mean of the distribution is λ final grade of each student was
recorded and it was found that the
(B) The variance of the distribution 95% confidence interval for the
difference in the mean grades
is λ (Group A—Group B) was computed
to be –3.5 ± 1.8. This means :
(C) The coefficient of variation
(A) there is evidence that
is 1 assignments caused better
grades because the 95%
(D) The parameter λ must be confidence interval does not
cover 0
greater than zero
(B) there is evidence that
25. Let X be a normed linear space and assignments caused better
grades because the difference in
x, y ∈ X. Then there may not be a the population means is less
bounded linear functional f on than zero
(C) there is evidence that
X such that :
assignments do not cause better
grades because the 95%
(A) f(x – y) = ||x – y|| + 1
confidence interval does not
cover zero
(B) f(x) = f(y)
(D) there is little evidence that
(C) f(x – y) = ||x – y|| assignments caused better
grades because the 95%
x− y confidence interval does not
(D) f ( x − y ) =
2 cover zero
12
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26. Let A be a bounded linear 27. Let <xn> be a bounded sequence of
transformation from a Banach space elements in a separable Hilbert
X to a Banach space Y. Suppose space. Then :
X
is isomorphic to image of A. (A) there is a subsequence of <xn>
ker A
which converges weakly
Then :
(B) every subsequence of <x n >
(A) ker A is closed and Im A is open
converges weakly
(B) ker A is closed and Im A is also
(C) there is a subsequence of <xn>
closed
which converges strongly
(C) ker A is open
(D) every subsequence of <x n >
(D) Im A is open
converges strongly
Or
Or
With regard to the chi-squared
The correlation coefficient :
test :
(A) describes the association
(A) it is used as an alternative to
the t-test to determine the between two variables
difference between two means (B) is measured on a scale of 0
(B) the number of degrees of to 1
freedom is the number of (C) describes the degree of
independent comparisons agreement between two
(C) the larger the value of the chi- variables
squared test, the less likely it (D) is positive when a positive value
is to be significant of one variable implies that the
(D) the null hypothesis is not other variable also takes a
required positive value
13 [P.T.O.
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28. The space [0, 1] × [0, 1] in the Or
dictionary order topology is not :
Let x 1, x 2 , .... x n be a random
(A) connected
sample from uniform distribution
(B) locally connected
v(0, θ). Let δ(x) = 2 x . Which of the
(C) locally path connected
following statements is true :
(D) Hausdorff
θ2
Or (A) δ(x) is biased and v(δ(x)) >
n
θ2
In a normal distribution : (B) δ(x) is biased and v(δ(x)) <
n
(A) the coefficient of variation is the
(C) δ(x) is unbiased and v(δ(x))
same as the standard deviation
θ2
>
(B) the mean is higher than the n
median (D) δ(x) is unbiased and v(δ(x))
(C) 95% of observations lie within θ2
<
one standard deviation of the n
mean 30. Which of the following spaces is
(D) Mann-Whitney test is suitable normal ?
for analysis
(A) Product of two normal spaces
29. Suppose X is Lindelöf and Y is
(B) Closed subspace of a normal
compact. Then X × Y is :
space
(A) metrizable
(C) The product space RJ, where J
(B) second countable
is uncountable
(C) compact
(D) Lindelöf (D) Subspace of a normal space
14
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Or
32. Which of the following numbers can
Let x1, x2, ...... xn be a random
sample from exponential distribution not be expressed as a sum of two
with mean λ. Let x be an estimator
squares ?
of λ. Then x is :
(A) biased and efficient (A) 23
(B) unbiased, MLE and efficient
(B) 24
(C) biased, MLE but not efficient
(D) unbiased but not efficient (C) 25
31. The fundamental group of a
(D) 26
topological space X is trivial if X
is :
Or
(A) {(x1, x2) ∈ R2/ 0 < x12 + x22 ≤ 1 }
Let x1, x2, .... xn be a random sample
(B) convex subset of R2
(C) R2 – {0} from N(θ, 1). Then the critical region
(D) Circle in R2 for testing H 0 : θ = θ0 against
Or
H1 : θ > θ0 is of the form :
Let X ~ N(0, σ2) then :
(A) X is complete but X2 is not (A) x > k
complete
(B) x < k
(B) X is not complete but X2 is
complete
(C) k1 < x < k2
(C) X and X2 are both complete
(D) X and X2 are both not complete (D) Σ(xi – x )2 > k
15 [P.T.O.
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33. For which value of n, the statement 34. Which of the following statements
“In a party of n persons, there are is not true ?
three persons, who know each other (A) Every connected graph has a
or there are three persons who are spanning tree
unknown to each other” is always (B) There is no odd degree vertex
true ? in a Eulerian graph
(C) Complement of a disconnected
(A) 3
graph is connected
(B) 4
(D) In a Hamiltonian graph, there
(C) 5
is a cycle passing through every
(D) 7 edge
Or Or
X1, X2, .... Xn are i.i.d. r.v.s. with
The degrees of freedom of chi-square
E(Xi) = θ and V(Xi) = θ. Assymptotic
for testing goodness of fit with n
n
observations and k classes are : distribution of ∑ X is :
i=1
i
(A) n – 1
(A) N(nθ, θ)
(B) n – k θ
(B) N(θ, )
n
(C) k
(C) N(θ, nθ)
(D) k – 1 (D) N(nθ, nθ)
16
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35. Let P be a poset and f : P → P 36. The solution of the initial value
problem y′ = f(x, y), y(x 0) = y 0
be order preserving map. There
in a rectangle R : |x – x0| ≤ a,
exists an element x ∈ P such that |y – y0| ≤ b exists and unique
if :
f(x) = x if :
(A) f(x, y) is continuous in R and
(A) P is a lattice bounded
(B) P is a finite poset (B) f(x, y) is continuous, bounded
and satisfies Lipschitz condition
(C) P is a chain on R
(D) P is a bounded poset (C) f(x, y) is continuous and satisfies
Lipschitz condition in R
Or
(D) f(x, y) is bounded and satisfies
x1, x2, ..... xn is a random sample Lipschitz condition in R
of size n from exponential Or
1
distribution with mean . Let x(1) X1, X2, ..... Xn are i.i.d. r.v.s. from
θ
U(0, 1) approximate distribution of
be the first ordered statistic.
12
E(x(1)) is : ∑ X − 6 is :
i =1
i
(A) nθ
n
(A) N 0,
(B)
1 12
nθ
(B) N(0, 1)
(C) θ
(C) N(6, 1)
1
(D)
θ (D) N(0, n)
17 [P.T.O.
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37. The eigen values of the one- 38. The integral surface of the equation
dimensional wave equation
yp – xq = 0
ytt = c2yxx, 0 < x < 1 are :
passing through the equations x = 0,
(A) π, –π
π z = y2 is :
(B) π,
2
(A) x2 – y2 = z2
(C) π, 2π, 3π, .....
(D) 2π, 3π (B) x2 + y2 + z2 = 5
Or (C) x2 + y2 = z
If (D) x2 + y2 = 4z
Or
1 1
1 2 2
To test the significance of
1 1
ρ = 1
2 2 , correlation coefficient, the
1 1
1
2 2 appropriate test statistic is :
then ρ1.23
2 = r n−3
(A)
1 − r2
1
(A)
4 r n −1
(B)
2 1 − r2
(B)
3
r n−2
1 (C)
(C) 1 − r2
3
r n
3 (D)
(D) 1 − r2
4
18
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39. Let J n (x) be a Bessel function 40. The number of non-negative
of order n. Consider :
integer solutions to the equation
(1) J0′ ( x ) = − J1 ( x ) , 5x + 3y = 42 is :
n (A) 1
(2) Jn′ ( x ) = J ( x ) − J n +1 ( x ) ,
x n
(B) 3
n
(3) Jn′ ( x ) = Jn −1 ( x ) − J (x) . (C) 4
x n
Then : (D) 2
(A) only (1) is true Or
(B) only (2) and (3) are true
Let X = (X1, X2, X3)′ with V(X) = Σ.
(C) all (1), (2) and (3) are true
Eigen values of Σ are .6, .4 and 2.
(D) none of (1), (2) and (3) is true
Then the proportion of variance
Or
explained by the first principal
If (X 1 , X 2 , X 3 )′ follow N(µ, Σ)
component is :
where µ′ = [1, 2, 3],
(A) .2
1
1 + 0 (B) .6666
2
1
Σ = + 1 0 (C) .5
2
0 0 1 (D) .3333
41. The unit digit in the decimal
then X1 – X2 + X3 follows : expansion of 7100 is :
5 (A) 5
(A) N 2,
2
(B) 3
(B) N(0, 2)
(C) N(2, 2) (C) 2
(D) N(2, 3) (D) 1
19 [P.T.O.
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Or Or
The least square regression line is There is an approximate linear
the line : relationship between the height of
females and their age (from 5 to 18
(A) which is determined by use of
years) described by :
a function of the distance
between the observed Y’s and height = 50.3 + 6.01 (age)
the predicted Y’s where height is measured in cm and
(B) for which the sum of residuals age is in years
about the line is zero Which of the following is not
(C) for which the sum of squares of correct ?
the residuals about the line is (A) The estimated slope is 6.01
zero which implies that children
(D) which has the smallest sum of increase by about 6 cm for each
the squared residuals of any line year they grow older
through the data values (B) The estimated height of a child
42. Consider the following congruences : who is 10 years old is about
110 cm
(1) x2 ≡ 31(mod 7)
(C) The average height of children
(2) x2 ≡ 7(mod 31)
when they are 5 years old is
Then :
about 50% of the average
(A) (1) has a solution but not (2) height when they are 18 years
(B) Both the congruences have a old
solution (D) The estimated intercept is 50.3
(C) None of (1) and (2) has a cm which implies that children
solution reach this height when they are
50.3
(D) (2) has a solution but not (1) = 8.4 years old
6.01
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43. The number of solutions to the 44. Let a one-dimensional harmonic
congruence x3 ≡ 3(mod 7) is : oscillator has Lagrangian
(A) 3 1 1
L= mx& 2 − kx 2 ,
2 2
(B) 2
then the path of the oscillator in the
(C) 1
phase space is :
(D) No solution
(A) a circle
Or
(B) a hyperbola
Which of the following statements
is not true ? (C) an ellipse
(A) The ANOVA problem is referred (D) a straight line
to non-parametric hypothesis
Or
testing
(B) The ANOVA refers to a In a single factor ANOVA problem
collection of experimental involving five populations, with a
situations and statistical random sample of four observations
procedures for the analysis of from each one, it is found that
quantitative responses from
SSTr = 16.1408 and SSE = 37.3801.
experimental units
Then the value of the test statistic
(C) The simplest ANOVA problem
is referred to as one-way is :
ANOVA (A) 0.432
(D) Single-factor ANOVA focusses (B) 0.812
on a comparison of more than
(C) 1.619
two populations or treatment
means (D) 2.316
21 [P.T.O.
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45. A bead sliding on a uniformly 46. A Lagrangian of a particle of mass
rotating wire in a force-free
m constrained to move on the plane
space. Then the Hamiltonian of the
curve xy = c, where c is constant,
bead :
is :
(A) represents total energy and
constant of motion
(A) L =
1
2
( )
m x& 2 + &y2 − mgy
(B) represents total energy but not
a constant of motion 1 c2 c
(B) L = m 1 − 4 x& 2 + mg
(C) does not represent total energy 2 x x
but represents constant of
1 c2 c
motion (C) L = m 1 + 4 x& 2 − mg
2 x x
(D) neither represents total energy
1 mgc
nor a constant of motion (D) L = mx& 2 −
2 x
Or
Or
Stratified sampling will achieve the
maximum gain in precision if : Cluster sampling provides an
(A) Sample is allocated to strata unbiased estimator of the population
according to proportional mean only if :
allocation
(A) all clusters are of equal size
(B) Sample is allocated to strata
according to Neyman allocation (B) all clusters are homogeneous
(C) Stratum mean squares are
(C) cluster sizes are known
minimized
(D) Stratum sizes are proportional (D) all clusters are included in the
to stratum means sample
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47. Which one of the following 48. The tangent vector field(s) to the
sphere x2 + y2 + z2 = r2
Lagrangians does not produce the
is (are) :
equation of motion &q& – q = 0.
(1) (x2, x1, 0)
(A) L = (q + q& )2
(2) (x2, –x1, 0)
(B) L = q2 + q& 2 (3) (x3, 0, – x1)
(4) (x1, x2, x3)
(C) L = q2 – q& 2
Then :
(D) L = (q – q& )2
(A) only (1) is true
Or (B) both (1) and (2) are true
The ratio estimator can be used in (C) only (2) and (3) are true
stratified sampling only if : (D) only (4) is true
Or
(A) Sample allocation is
Horvitz-Thompson estimator is
proportional allocation
unbiased for population mean
(B) The concomitant variable is because :
observed on all selected (A) it assigns unequal weights to
different sampling units
sampling units
(B) it assumes sample selection by
(C) Stratum means for the simple random sampling
concomitant variable are known (C) it compensates for the unequal
probabilities of selection in
(D) Strata are homogeneous with
construction of the estimator
respect to the concomitant
(D) it takes the sum over all
variable population units
23 [P.T.O.
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49. Consider a curve α : R → E2, defined
50. The values of c for which
by
α(t) = (x(t), y(t)) M : z(z – 2) + xy = c
Then the curvature of the curve is
represents a surface are :
given by :
xy& − yx& (A) different from zero
3
(A)
( x& 2 + y& 2 2) (B) different from 1
xy& 2 − yx& 2
(C) different from 2
3
(B)
(x& 2 + ) 2
y& 2
(D) different from –1
&& − yx&&
xy
3 Or
(C)
( x& 2 + )
2
y& 2
If a 3 4-factorial experiment is
&&& − yx
xy &&&
3
conducted in a block of size 9, then
(D)
( x& 2 + y& 2 2)
the number of generalized
Or
interactions confounded will be :
In a connected design, the diagonal
elements of C-matrix are all may
(A) 3
be :
(A) Negative (B) 4
(B) Non-negative
(C) 1
(C) Zero
(D) 2
(D) Positive
24
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51. The Gaussian curvature at any 52. The extremal of the functional
point of a surface r = a(sin u cos v,
2
sin u sin v, cos u) is : x3
I ( y ( x )) =
(A) a
∫
1
y ′ 2 dx
1
(B) satisfying y(1) = 0, y(2) = 3 is :
a
1 (A) y = x – 1
(C)
a2
a2 x2
(B) y = −1
(D) 2
2
Or (C) y = x2 – 1
Consider the following statements :
x2
(1) If at least one element of (D) y = + x −1
2
incidence matrix of a block
design is zero, then the design Or
is incomplete block design.
With n-variables the number of
(2) If at least one element of
linearly independent contrast are :
incidence matrix of a block
design is zero, then the design
n
is non-orthogonal design.
(A)
2
Of these statements :
(A) Only (1) is true
(B) n + 1
(B) Only (2) is true
(C) n – 1
(C) Both (1) and (2) are true
(D) Neither (1) nor (2) is true (D) n
25 [P.T.O.
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53. The functional 54. A curve of fixed perimeter l that
π encloses maximum area is :
4
I ( y ( x )) =
∫ ( y − y′ )dx
0
2 2 (A) a circle centred at (0, 0) and of
radius l
satisfying y(0) = 0 attained its l
(B) a square with diagonal
extremal on : 2π
(A) y = x
(C) a circle centred at arbitrary
(B) y = 0
l
(C) y = sin x point (a, b) and of radius
2π
(D) y = cos x
(D) a rectangle with diagonal 2πl
Or
Or
Consider a stationary time series
Xt = µ + φ Xt–1 + Zt, t = 0, ±1, Let {Xt} and {Yt} be a stationary
±2, .........., Zt ~ iid Normal (0, σ2). AR(1) and AR(2) process
Which of the following statements respectively. Suppose Xt and Ys are
are correct ? uncorrelated at all leads and lags.
(i) Best linear forecast Pt Xt+k is Then, which of the following
a weighted average of E(X t)
statements are correct ?
and latest observation Xt
(i) Xt + Yt is a stationary process
(ii) Pt Xt+k → E(Xt) as k → ∞
(iii) Forecast mean square error (ii) Xt + Yt is AR(3)
of P t X t+k approaches to
(iii) Xt – Yt is MA(1)
σ2/(1 – φ2)
(iv) Best linear forecast P t X t+k (iv) Xt + Yt is ARMA(3, 2)
will be Xt itself for all k (A) (i) and (ii)
(A) (i) and (ii)
(B) (i) and (iii)
(B) (i), (ii) and (iii)
(C) (i) and (iv)
(C) (ii), (iii) and (iv)
(D) (i) and (iv) (D) (iii) and (iv)
26
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55. The extremal of the functional 56. Consider the following statements :
x2 (i) If a kernel is symmetric then all
1 + y′2 its iterated kernels are
I ( y ( x )) =
∫
x1
x
dx symmetric
(ii) The eigen functions of a
is a : symmetric kernel corresponding
(A) cycloid to different eigen values need
not be orthogonal.
(B) circle
(C) catenary Then :
(D) parabola (A) both (i) and (ii) are true
Or (B) both (i) and (ii) are false
Given the following functions, (C) only (i) is true
which of them can be auto-
(D) only (ii) is true
correlation functions of a weak
stationary process ? Or
(i) ρ(h) = 1 + |h|, h = 0, ±1, Consider two MA(1) processes Xt =
± 2, .... 1
(ii) ρ(h) = φ|h| , |φ| < 1, h = 0, Zt – Z and Yt = Zt – θ Zt–1,
θ t–1
± 1, ± 2, .... where Z t ~ white noise (0, σ 2).
Which of the following statements
1 if h=0
are true ?
(iii) ρ (h ) = (i) Both {Xt} and {Yt} processes
0 otherwise
have the same covariance
function
θ (ii) When |θ| < 1, {Yt} process is
1 + θ2 if |h| = 1
preferred
(iii) Both {Xt} and {Yt} processes are
(iv) ρ (h ) = 0 |h| > 1 causal
1 (iv) Both {Xt} and {Yt} processes
h=0
have exponentially decaying
ACF function
(A) (i), (iii) and (iv)
(A) (ii), (iii) and (iv)
(B) (i), (ii) and (iii) (B) (ii), (iii) and (iv)
(C) (i), (iii) and (iv) (C) (i), (ii) and (iii)
(D) (i), (ii) and (iv) (D) (i), (ii) and (iv)
27 [P.T.O.
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57. The initial value problem 58. The resolvent kernel of the Volterra
corresponding to the integral integral equation with kernel
equation
k(t, s) = 1 is :
t
(A) λ(t – s)2
x (t )
∫ (t − s) x (s) ds
2
= t3 +
2
0 (B) eλ(t – s)
is :
(A) x′′′(t) + 2x(t) = 0, x(0) = x′(0) = (C) eλ(t – s)
x′′(0) = 0 (D) λ(t – s)
(B) x′′′(t) – 2x(t) = 6, x(0) = x′(0) =
x′′(0) = 0 Or
(C) x′′′(t) – 2x(t) = 0, x(0) = x′(0) =
Let {Xn , n ≥ 0} be a branching
x′′(0) = 0
(D) x′′′(t) + 2x(t) = 6, x(0) = x′(0) = process with E(X 1 ) = m. If
x′′(0) = 0 Xn
Wn = , then E(Wn) is :
Or mn
Patients arrive at doctor’s clinic (A) m
(B) m n
according to Poisson process with
(C) 1
1
rate λ = minute. The doctor will (D) m n–1
10
not examine the patient until at 59. Consider the integral equation
1
least 3 patients are in the waiting
room. Then the expected waiting
x (t ) = 1 + λ
∫ (1 − 3ts) x (s) ds ,
0
time until the first patient is where λ is a real parameter. Then
the Neumann series for the integral
admitted to see the doctor is :
equation converges for :
(A) 10 minutes
(A) – 2 < λ < 2
(B) 30 minutes (B) 0 < λ < 3
(C) 20 minutes (C) – 3 < λ < 0
10
(D) minutes (D) – 3 < λ < 3
3
28
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Or Or
Consider a state Markov chain with
Let a random process be defined as
states 0 and 1, and transition
x(t) = y cos ωt, t ≥ 0
probability matrix
where ω is a constant and y is a
1 0
uniform random variable over
P = 1
1
(0, 1), then the covariance function 2 2
kx(t, s) of x(t) is :
Then the state is :
1 (A) Ergodic
(A) cos ωt. cos ωs
12 (B) Absorbing
1 (C) Transient
(B) cos ωt. cos ωs
4 (D) Recurrent
1 61. The nth order divided difference
(C) cos ωt. cos ωs
3 1
[x1, x2, ......, xn+1] of the function
x
1
(D) cos ωt is .......................... .
2
−1
60. The order of convergence in Newton- (A) x x ..... x
1 2 n +1
Raphson method is :
(−1)n
(A) 0 (B)
x1x2 ..... xn +1
(B) 1 −1
(C) x x ..... x
1 2 n
(C) 2
( −1)n
(D) 3 (D)
x1 x2 ..... xn
29 [P.T.O.
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Or
62. The area of a circle of diameter d
Pearle’s vital Index in the context
is given for the following values :
of vital statistics is :
d area
(A) Ratio of the number of births
in the given period and the 80 5025
number of deaths in the same
85 5674
given period, multiplied by 100
90 6362
(B) Ratio of the number of deaths
in the given period and the
95 7088
number of births in the same
100 7854
given period, multiplied by 100
(C) Ratio of the number of births The area of a circle of diameter 105
in the given period and the by interpolation formula is :
number of deaths in the same
(A) 8666
given period, multiplied by 1000
(D) Ratio of the number of deaths (B) 8546
in the given period and the
(C) 8425
number of births in the same
given period, multiplied by 1000 (D) 8353
30
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Or Or
The number of persons dying at
The crude and standardised death age 75 is 476 and the complete
expectation of life at 75 and 76 years
rate from the following data is :
are 3.92 and 3.66 years. Then
Age-group Population Deaths what are the numbers living at ages
75 and 76 ?
(years) (A) 2660 and 2181
(B) 2675 and 2199
under 10 20,000 600
(C) 2575 and 2099
10—20 12,000 240 (D) 2565 and 2081
20—40 50,000 1250 64. The Laplace transform of the
periodic function F(t) with period
40—60 30,000 1050 T > 0 is :
Above 60 10,000 500 T
(A) 29.80 and 29.83 ∫ e F (t) dt
0
− pt
(A)
(B) 27.40 and 31.40 T
(C) 31.40 and 27.40 T
(D) 30.50 and 34.50 ∫
e F (t ) dt
− pt
(B) 0
63. An approximate value of y when (
1 − e − pT )
x = 0.2 by Runge-Kutta fourth order
method is .......................; T
given that y′ = x + y and y(0) = 1.
∫
e F (t ) dt
(C) 0
pt
T
(A) 1.5418
(B) 1.2428 T
(C) 1.2534 ∫
e F (t ) dt
pt
(D) 0
(D) 1.1327 (
1 − e − pT )
31 [P.T.O.
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Or
Or
Let Pn is the probability of the mean
of a sample of size n falling outside If Rs(t), Rp(t) and Rnk (t ) are the
the control limits. Then what is the
probability that at most X-sample reliabilities of a series, parellel and
are to be taken for at least s-points
to go out of control : k-out-of-n systems respectively then
for a system having identical
r −1 X
(A)
∑ (
Ps 1 − P X − s
n
s=0 s
n
) components, which of the following
is true ?
r −1 X (A) Rs(t)/Rp(t) ≥ 1, Rp(t) – Rs(t) ≥ 0
∑ ( )
X−s
(B) 1 − P 1 − Pn
s
n
s =0 s (B) Rnk (t ) /Rp(t) ≥ 1, Rp(t) – Rnk (t ) ≥ 0
1 (C) Rs(t)/ Rnk (t ) ≥ 1, Rs(t) – Rp(t) ≥ 0
(C) r −1 X
(D) Rp(t)/ Rnk (t ) ≥ 1, Rnk (t ) – Rs(t) ≥ 0
∑ (
Ps 1 − P X − s
n
s= 0 s
n
)
66. The Fourier cosine transform of
1 1
is :
(D) r −1 X 1 + t2
X − s
1 −
∑ Pn 1 − Pn
s
( )
π − p3
s =0 s (A) e
2
65. Which of the following is the
inverse Laplace transform of π − p2
(B) e
3p + 6 2
?
p2 + 8 p + 25
π p2
(A) e–4t(3cos(3t) + 9sin(3t)) (C) .e
2
(B) e–4t(3cos(3t) + 6sin(3t))
(C) e–4t(3cos(3t) – 3sin(3t)) π −p
(D) .e
(D) e–4t(3cos(3t) – 2sin(3t)) 2
32
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Or Or
Three components with failure
Three components with failure
rates as 2 × 10–3, 3 × 10–3 and
rates 5 × 10 –4 , 3 × 10 –3 and 5 × 10–3 are arranged in series, then
4 × 10–3 respectively are arranged the system failure rate will be :
(A) 3 × 10–8
in series, the MTSF will be :
(B) 5 × 10–3
31
(A) × 103 (C) 10–2
12
(D) 2 × 10–2
31
(B) × 104
12 68. Which one of the following is not
an entire function ?
12
(C) × 104 (A) f(z) = ez
31
(B) f(z) = e–z
2
(D) × 103 (C) f(z) = eiz
15
(D) f(z) = e z
67. The inverse Fourier transform of
Or
e–|p| is :
The ASN of a double sampling
2 plan reduces to that of single
(A)
π (1 − t) sampling plan if the probability of
making a decision on the basis of
2 first sample is :
(B)
( π (1 − t )) 2
3
(A)
4
2
(C)
(
π 1 + t2 ) (B)
1
2
2 (C) 0
(D)
π (1 + t) (D) 1
33 [P.T.O.
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69. Consider the following two 70. Which one of the following is
statements :
(i) Every function analytic in the true ?
()
extended plane is constant
(A) f(z) and f z are
(ii) If f(z) is entire function
such that f(z) = u + iv and simultaneously analytic
u2 ≤ v2 + 2012, then f(z) is
constant (B) f(z) and f (z ) are
Then :
simultaneously analytic
(A) both (i) and (ii) are false
(B) both (i) and (ii) are true (C) f(z) and ()
f z are
(C) only (i) is true simultaneously analytic
(D) only (ii) is true
Or ()
(D) f z and ()
f z are
In the context of queueing theory simultaneously analytic
which of the following statements
are correct ? Or
(i) queueing theory deals with
situations where customers Average number of customers in
arrive, wait for the service, get
the service and leave the system the system for (M|G|1) queueing
λ
(ii) customers in queueing theory model, where ρ = is :
µ
might include humans
machines, ships, letters etc. λ 2σ 2 + ρ 2 1
(A) 2λ 1 − ρ + µ
(iii) A queue refers to physical ( )
presence of the customers
waiting to be served λ2σ 2 + ρ2
(B)
(iv) A study of queueing theory 2 (1 − ρ)
helps the manager to establish
an optimum level of service
λ2σ 2 + ρ2
(A) (i), (ii) and (iii) (C) +ρ
2 (1 − ρ)
(B) (i), (ii) and (iv)
(C) (i) and (iv) λ2σ 2 + ρ2
(D) 2λ 1 − ρ
(D) (i) and (iii) ( )
34
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Or
71. Consider the following two
In the context of Inventory models,
statements : which of the following statements
is correct ?
(i) If a function f : C → C is entire (A) Re-order quantity in a “fixed-
order interval system” equals
and bounded then f(z) is economic order quantity
(B) Review period of the items is
constant.
always kept higher than its lead
time
(ii) If a function f : C → C is entire
(C) In periodic review system of
and real part of f(z) is bounded, inventory control, the stock is
usually replenished at unequal
then f(z) is constant. time interval
(D) The (S, s) system requires high
Then :
safety stock volumes.
72. Let S be the set of real numbers
(A) both the statements (i) and (ii)
which can be represented in the
form
are false
a1 a2 an
+ 2
+ ..... +
+ ......,
(B) only (i) is true 5 5 5n
where an = 0 or 2 for each n.
Then m(S) is :
(C) only (ii) is true (A) 0
(B) 1
1
(D) both the statements (i) and (ii) (C)
2
are true 2
(D)
5
35 [P.T.O.
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Or 73. For f(x) = |x|, the four Dini
derivatives are given by :
The probability distribution of
(A) D + = 1, D – = 1, D + = –1,
monthly sales of a certain item is D– = –1
as follows :
(B) D+ = D+ = 1, D– = D– = –1
Monthly Sale Probability (C) D + = 1, D – = 1, D + = 1,
D– = –1
0 0.02
(D) D + = 1, D – = –1, D + = 1,
1 0.05 D– = 1
2 0.30 Or
4 0.20 For any multi-stage problem, the
solution by dynamic programming
5 0.10 involves :
6 0.06 (i) The recurrence relation
connecting optimum decision
The cost of carrying inventory is function for the (n – 1) stage
Rs. 10 per unit per month. The process
current policy is to maintain a stock (ii) The relation giving the
of 4 items at the beginning of each optimum decision function for a
one-stage process
month. Assuming that the cost of
shortage is proportional to both time (iii) The optimum decision function
for n-stage process
and quantity short, then what is the
(iv) If the number of stage is large,
imputed cost of shortage of one item
take a limiting process and
for one unit of time. solve the resulting functional
equation
(A) Rs. 115 < C2 < Rs. 390
(A) (i) and (ii)
(B) Rs. 100 < C2 < Rs. 290
(B) (i), (ii), (iii) and (iv)
(C) Rs. 120 < C2 < Rs. 410
(C) (i) and (iv)
(D) Rs. 110 < C2 < Rs. 290 (D) (i) and (iii)
36
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75. Let υ be a signed measure on [[X, S]].
74. Let {f n } be a sequence of non-
Consider the following statements :
negative measurable functions and (i) Hahn decomposition of the
space X is unique
lim fn = f a.e., then :
(ii) Jordan decomposition of the
signed measure υ is unique.
(A) ∫ fdx = lim inf ∫ f dx n
Then :
(B) ∫ fdx ≤ lim inf ∫ f dx n (A) Only (i) is true
(B) Only (ii) is true
(C) ∫ fdx ≥ lim inf
∫ fn dx
(C) Both (i) and (ii) are true
(D) ∫ fdx = lim sup ∫ f dx n
(D) Both (i) and (ii) are false
Or
Or
In a decision-making situation
the :
Using dynamic programming, the
(A) Total number of courses of
maximum value of x12 + 2x22 + 4 x3 action cannot be more than the
number of events
subject to x1 + 2x2 + x3 ≤ 8 and
(B) Laplace principle is based on
x1, x2, x3 ≥ 0 is obtained as : the premise of equally-likely
occurrence of possible events
(A) 46 (C) Minimax is an optimist’s choice
while minimum is a pessimist’s
(B) 64 criterion
(D) For any utility function, the
(C) 32 units of measuring utility
should always range between
(D) 81 zero and 100
37 [P.T.O.
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ROUGH WORK
38
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Source: Savitribai Phule Pune University. Question paper published by Savitribai Phule Pune University.