MH SET 2013: 17 February, Paper II, Mathematical Sciences
Text of page 1
FEB - 30213/II
Mathematical Science
Paper II
Time Allowed : 75 Minutes] [Maximum Marks : 100
Note : This Paper contains Fifty (50) multiple choice questions. Each question
carries Two (2) marks. Attempt All questions.
1 4. Let f be analytic in a bounded
1. Let Gn = 0, 1 + for n ∈ N. Then
n
∩ Gn is : domain D with f(z) = f(2z) for every
(A) closed z ∈ D. Then
(B) open
(C) both open and closed (A) f(z) is a non-zero constant
(D) neither open nor closed
∞ (B) f(z) is the identity function of D
1
2. Let A n = , 1 . Then
n U A is :
n=1
n
(C) f(z) 0 in D
(A) (0, 1)
(D) f(z) ≡ 0 in D
(B) (0, 1]
(C) [0, 1] 5. A monotone function :
(D) [0, 1)
(A) has discontinuities everywhere
3. The series Σ an zn represents an
entire function, if : (B) is continuous everywhere
1/ n
(A) lim| an | = 0
n→∞ (C) has countably many
1/ n
(B) nlim| an | is a positive real
→∞ discontinuities
number
(D) has countably many points of
(C) nlim| an |1/ n = ∞
→∞
continuity
(D) nlim| an |1/ n = 0
→∞
1 [P.T.O.
Text of page 2
FEB - 30213/II
6. Let S = {u1, u2, ......., up} be a linearly 9. The dimension of the space of
independent subset of a vector space
diagonal n × n matrices is :
V = < v1, ......., vq>. Then :
(A) p < q (A) n
(B) p = q
(B) n 2
(C) p < q
(C) n(n – 1)/2
(D) p > q
7. If T : R3 → R3 is the identity map, (D) n(n + 1)/2
then nullity T = ?
10. Let A1, ...., An be column vectors of
(A) 0
size m. Assume that they have
(B) 1
coefficients in R, and they are
(C) 2
(D) 3 linearly independent over R.
8. In which of the following Then :
alternatives a subset T of the set :
(A) They are linearly independent
S = {(2, 0, 0), (2, 2, 2), (2, 2, 0),
over C
|
(0, 2, 0)}
is not a basis of R3(R) ? (B) They are linearly dependent
(A) T = {(2, 0, 0), (2, 2, 0),(2, 2, 2)} over
(B) T = {(2, 0, 0), (2, 2, 2),(0, 2, 0)}
(C) They form a basis for R
(C) T = {(2, 0, 0), (2, 2, 0),(0, 2, 0)}
(D) They form a subspace for R
(D) T = {(2, 2, 0), (2, 2, 2),(0, 2, 0)}
2
Text of page 3
FEB - 30213/II
11. Let E, F and G be mutually 13. The probability that a certain
exhaustive events such that E and machine will produce a defective
F are mutually exclusive and F and 1
item is . If a random sample of 8
4
G are independent events. Then a
items is taken from the output of the
feasible assignment of probabilities
machine, what is the probability that
is :
there will be 7 or more defectives in
(A) P(E) = 0.5, P(F ∪ G) = 0.5, the sample ?
P(F) = 0.2 25
(A)
(256)2
(B) P(E) = 0.2, P(F ∪ G) = 0.65,
P(F) = 0.5 4
(B)
(256)2
(C) P(E) = 0.2, P(F ∪ G) = 0.8,
24
P(F) = 0.4 (C)
(256)2
1
(D) P(E) = 0.5, P(F ∪ G) = 0.36, 5
n(n + 1) (D)
(256)2
P(F) = 0.2
14. Let X follow a Poisson (2)
12. Let N be a random variable with
distribution. Then :
(A) The r.v. 2X follows Poisson (4)
P(N = n) = , n = 1, 2, ..... .
X
and follows Poisson (1)
Then E[N] is : 2
X
(B) Both 2X and are not Poisson
1 2
(A) r.v.s.
10
(C) The r.v. 2X follows Poisson (4)
3
(B) X
10 but is not a Poisson r.v.
2
(C) 1 (D) The r.v. 2X is not a Poisson r.v.
X
but is Poisson (1)
(D) infinity 2
3 [P.T.O.
Text of page 4
FEB - 30213/II
15. Consider the Linear Programming 1
17. Consider the function f(x) = on
Problem x
Maximize Z = x1 + x2 1
[1, ∞] and g(x) = on x > 0.
x
Subject to x1 + 3x2 < 4
Then :
3x1 + x2 < 4
x1, x2 > 0 (A) both f and g are uniformly
For this problem the value of the continuous
objective function at the optimal
(B) g is uniformly continuous but
solution is :
f is not
(A) 2
(B) 4 (C) neither f nor g is uniformly
(C) 1 continuous
(D) 3
(D) f is uniformly contrinuous but
16. Maximization assignment problem
g is not
is transformed into a minimization
problem by : Or
(A) Substracting all the elements of
Let X and Y be two random variables
a column from the highest
element of that column with E[Y|X] = X with probability 1.
(B) Substracting each element of Then :
the profit matrix from the
(A) Cov(X, Y) = 0
highest element of the matrix
(C) Substracting all the elements in (B) Cov(X, Y) = E[Y|X]
a row from the highest element
(C) Cov(X, Y) = Var(Y)
of that row
(D) Any of the above (D) Cov(X, Y) = Var(X)
4
Text of page 5
FEB - 30213/II
18. If f(x) is monotonic increasing on 1 1
19. In R let Fn = − , , ∀ n ∈ N.
n n
(a, b) and a < c < b, then lim− f(x) =
x→ c Then, ∩ Fn is :
(A) inf{f(x)| x < c} (A) {0}
(B) sup{f(x)| x > c} (B) φ
(C) sup{f(x)| x < c} (C) both open and closed
(D) inf{f(x)| x > c} (D) neither open nor closed
Or
Or
Let X1, X2, ....., Xn be a random
Let X and Y be two independent r.v.s
sample from N(µ, µ), where mean =
such that X follows the exponential
variance = µ is unknown. Then
distribution with mean 2 and Y
which of the following statements is
1
follows Binomial 8, . Then the not true ?
2
variance of X + 2Y :
(A) (ΣX 2i ) is sufficient for µ
(A) is 10.
(B) (ΣX i ) is sufficient for µ
(B) can not be computed from the
(C) (ΣX i , ΣX i2 ) is jointly sufficient
given information.
for µ
(C) is 12.
(D) Sufficient statistics do not exist
(D) is 8.
5 [P.T.O.
Text of page 6
FEB - 30213/II
20. Consider the sequence : Or
n
(−1)n 107 5 Let the random variable X follow
un = and vn =
n n
U(θ, θ + 1). Then which of the
Then,
following statements is not correct ?
(A) only {un} is convergent
(A) Min Xi = X(1) is an mle which
(B) only {vn} is convergent i
(C) both of these have the same is sufficient for θ
limit
1
(B) X – is an unbiased
(D) {un} is oscillating and {vn} is 2
convergent estimate of θ
Or
(C) UMVUE will not exist for θ
Let the random variables X1, X2 be
(D) Any value of θ in the interval
distributed as Poisson variates with
[X(n)–1, X(1)] is an m/e
mean λ. Then number of unbiased
estimators of λ is : 22. Let R be the set of real numbers.
(A) 3 Let f : R → R be such that
(B) 2 |f(x) – f(y)| < |x – y|3, for all
(C) infinity x, y ∈ R.
(D) 4
Then the value of the function f(x)
1 2
2x
x x
21. ∫
0
1 − +
1! 2!
− ....
e dx = is :
(A) x
(A) e – 1
(B) e (B) x2
(C) e 2 (C) Zero
(D) e + 1 (D) a constant
6
Text of page 7
FEB - 30213/II
Or Or
Let z 1 and z 2 be independent Let X1, X2, ...... Xn be iid r.v.s with
standard normal variables and let pdf f(x), where
y1 = z1 z22 then the correlation
between z1 and y1 equals :
Then :
3 n
(A)
(A) EX (1) =
2 n−1, where
(B) 0 X (1) = Min X i
i
(C) 5–1/2 (B) EX1 is finite
n
(D)
1 (C) EX(n) = n + 1 , X ( n) = Max
i
Xi
3
fC
|
( x) = x −2I(1, ∞ ) ( x)
23. Consider the following two (D) EX(1) and EX(n) do not exist
statements : 24. The non-zero roots of the equation
(a) ez, z ∈ C
| is a one-one function (1 + z)5 = (1 – z)5 :
(b) sin z, z ∈ is a bounded entire (A) are real
function
(B) some are real and some are
(A) both (a) and (b) are false purely imaginary
(B) both (a) and (b) are true
(C) are purely imaginary
(C) only (b) is true
(D) are complex numbers not lying
(D) only (a) is true on x-axis or y-axis
7 [P.T.O.
Text of page 8
FEB - 30213/II
Or
Or
In the usual two way classification
To obtain a critical (region|value (or model with one observation per cell
E(yij) = αi = βj, V(yij) = σ2,
cut-off point) in testing a statistical
i = 1......a;
hypothesis, we need the distribution
j = 1.....b
of a test statistic : Which parametric function is not
estimable ?
(A) without any assumption
(A) α1 – β1
(B) under H1 (B) α1 + β1
(C) α1 – α2
(C) under H0
(D) β1 – β2
(D) all of the above
26. f(z) = cosec has :
25. Let z ∈ C
| . The inequality |z + 1|
(A) infinitely many simple poles
> |z – 1| is : and z = ∞ a double pole
(B) z = 0 an essential singularity,
(A) true iff Re z > 0
z = ∞ a double pole
(B) Always true (C) infinitely many simple poles
and double pole at z = ∞ and
(C) Never true essential singularity at z = 0
(D) infinitely many simple poles
(D) True iff Im z > 0
and z = 0 essential singularity
8
Text of page 9
FEB - 30213/II
Or
Or
In a linear model E(yi) = α + βxi;
i = 1.....n and V(yi) = σ2. and β̂
Which of the following pairs
are least squares estimators of α and
represents linear regression line ? β respectively.
n
(A) ˆy = 3 x + 5; xˆ = 2 y − 3
If s2 = ∑ ( y − αˆ − βˆ x )
i=1
i i
2
then
unbiased estimator of σ2 is :
(B) ˆy = 2 x + 4; xˆ = 0.5 y − 2
s2
(A)
(n − 2)
(C) yˆ = 1.5 x + 1; xˆ = − 0.5 y + 0.8
s2
(B)
(n − 1)
(D) ˆy = 3 x2 + 4; ˆx = 0.2 y2 + 1
s2
α̂iπi
π
3 (C)
n
27. If C is the circle |z – 2| = 2, then
ns2
(D)
(n − 1)
dz
∫
C
z−5
= 28. Let f(z) be analytic in D = {z||z|
1
< 1} and f = 0 n = 2, 3, ..... .
n
(A) 0 Then :
(A) f may not have any zero other
(B) 2πi than ones given
(B) f ≡ 0 in D
1
(C) in addition to { |n = 2, 3, ....},
(C) n
the only zero of f in D is 0
(D) f(z) = 0 for z ∈ (–1, 1) but f
(D)
may not be zero at any other
point
9 [P.T.O.
Text of page 10
FEB - 30213/II
Or
Or
Given below is a 23 design with 3
In a RBD with b blocks and v factors A,B,C in two blocks B1 and
B2. Identify x and y so that the block
treatments a single yield is missing.
contrast B1 – B2 represents main
If B and T respectively denote the effect A :
B1 : (1) x c bc
total of block and treatment which
B2 : a ab y abc
contain the missing value and G is (A) x = ac, y = b
the grand total, then estimate of the (B) x = a, y = ac
(C) x = b, y = ac
missing value is given by :
(D) x = ac, y = a
(A) (vb + bT – G)/(b − 1)(v − 1) 30. The value of the integral
1+ i
(B)
∫ 0
( x − y + ix2 ) dz
(C)
along the straight line from z = 0
(D)
to z = 1 + i is :
29. The value of is :
i
(A)
3
(A) 2 99
(B) i(1 + i)
(B) 3 99
1
(C) (i – 1)
3
(C) –299
1
(D) (i – 3)
6
(D) –399
10
Text of page 11
FEB - 30213/II
Or Or
The degrees of freedom for the error When data are collected in a
in a Latin Square design with 5 statistical study for only a portion
rows, 5 columns and 5 treatments or subset of all elements. If interest
with two missing observations is : we are using a :
(A) 22 (A) Census
(B) Sampling frame
(B) 10
(C) Population
(C) 12
(D) Sample
(D) 14
32. In Sn, the number of distinct cycles
31. Let G = {1, –1}. Then the group
of length r < n is :
(G, •) is :
(A) r !
(A) isomorphic to (Z, +)
(B) (n – r) !
(B) homomorphic image of (Z5, +)
(C) n !/(n – r) !
(C) isomorphic to (Z5, +)
1 n!
(D) r . (n − r)!
(D) a homomorphic image of (Z, +)
11 [P.T.O.
Text of page 12
FEB - 30213/II
Or Or
Which of the following statements
Which of the following is not the goal
is correct ?
of descriptive statistics ?
(A) In a statistics problem,
(A) Summarizing data characteristics of a sample are
assumed to be unknown
(B) Displaying aspects of the
(B) Probability reasons from the
collected data
population to the sample
(C) Reporting numerical findings (deductive reasoning) whereas
inferential statistics reasons
(D) Estimating characteristics of
from the sample to the
the population population (inductive
reasoning)
33. If p is a group of order pn, (n > 1),
(C) Hypothesis testing and
(A) 0(Z(G)) need not be a power
estimation by confidence
of p intervals are the least impor-
tant types of inferential
(B) Z(G) is not singleton
statistical procedures
(C) Z(G) is not commutative
(D) In a probability problem,
(D) Z(G) need not be a normal characteristics of sample are
assumed to be unknown
subgroup of G
12
Text of page 13
FEB - 30213/II
34. If G is a group of order 108
35. G is a group of order pq where p and
then :
q are primes.
(A) G has a unique normal
subgroup
Then :
(B) G is cyclic
(C) G is non-communicative (A) G is solvable only if G is
(D) G is not simple abelian
Or
(B) G is solvable only if p > q and
Which of the following statements
is correct ? q | p – 1
(A) Color of ten automobiles
(C) G is solvable only if p = q
recently purchased at a certain
dealership is an example of a
(D) G is always solvable
univariate data set
(B) Height and weight for each Or
basketball player on Pune
University team is an example The expected number of heads in
of bivariate data set
300 tosses of a fair coin is :
(C) The systolic blood pressure,
diastolic blood pressure, and (A) 300
serum cholesterol level for each
patient participating in a (B) 250
research study is an example of
multivariate data set (C) 200
(D) All of the above statements are
(D) 150
correct
13 [P.T.O.
Text of page 14
FEB - 30213/II
36. Let R be a commutative ring.
Or
Then :
Economic Order Quantity (EOQ) in
(A) Every ideal of R is maximal
(B) R is a field if and only if R Inventory problem, results in :
does not have a proper non-zero
(A) reduced chances of stock outs
ideal
(C) Every proper ideal of R is (B) miximization of set-up cost
prime
(C) equalization of carrying cost
(D) A proper ideal of R is maximal
if and only if it is prime and procurement cost
Or
(D) favourable procurement price
Which of the following is not a
measure of center ? 38. Let D denote the derivative which
(A) The mean
we view as a linear map on the space
(B) The variance
of differential functions and k be a
(C) The median
non-zero integer. Then the eigen-
(D) The trimmed mean
37. In which of the alternatives a subset vectors of D2 are :
W of a vector space R3(R) is not a
subspace ? (A) sin x and sin kx
(A) W = {(a, b, 0)|a, b ∈ R} (B) cos x and cos kx
(B) W = {(a, b, c)|a + b + c = 0}
(C) k sin x and k cos x
(C) W = {(a, b, c)|a2 + b2 + c2 < 1}
(D) W = {(a, a, 0)|a ∈ R} (D) sin kx and cos kx
14
Text of page 15
FEB - 30213/II
Or Or
For a two person game, in game Sequencing problem involving
theory with A and B, the minimizing
processing of two jobs on ‘n’
and the maximizing players, the
machines :
optimal strategies are :
(A) cannot be solved graphically
(A) maximax for A and minimax
for B (B) can be solved graphically
(B) minimin for A and maximin (C) has a condition that the
for B processing of two jobs must be
(C) maximin for A and minimax in the same order
for B
(D) none of the above
(D) minimax for A and maximin
40. Let V be a finite dimensional space
for B
C and let T :V → V be a linear
over |
39. If T : R3(R) → R2(R) is defined by
map. Assume that all the eigen-
T(x1, x2, x3) = (x1, x2), then which
values of T are equal to 0. Then :
of the following alternatives is not
true for T. (A) T is not nilpotent
(A) T is a linear transformation (B) T is diagonalizable
(B) T is an isomorphism (C) There is an integer r > 1 such
(C) Range of T = R2 that Tr = 0 (zero map)
(D) Ker T = {(0, 0, x3)|x3 ∈ R} (D) T is a zero map
15 [P.T.O.
Text of page 16
FEB - 30213/II
Or 42. The dimension of the subspace
For a “Poisson exponential single
of Kn consisting of those vectors
server and infinite population”
queuing model, which of the A = (a1, ....., an) such that a1 + .... +
following is not correct :
an = 0 is :
(A) E(m) = λE(w)
(B) E(n) = λE(ν) (A) n
(C) E(n) = E(m) –
1 (B) n – 1
(D) E(ν) = E(w) +
µ
41. Let dim V > dim W and let (C) n/2
T : V → W be a linear map. Then :
n−1
(A) Ker T = {0} (D)
2
(B) dim Ker T = dim Im T
Or
(C) Ker T is not {0}
(D) T is invertible A cluster sampling is when :
Or
(A) units are clustered together
The types of probability sampling
after the study to enhance data
are :
(A) random sampling, snowball analysis
sampling and lottery method
(B) in the first instance groups of
(B) computer methods, lottery
people are chosen for the study
methods and snowball
sampling (C) a quota of people is chosen for
(C) simple random and systematic the study
sampling
(D) random numbers, random (D) units are clustered together
sampling and computer after sample selection for data
methods analysis
16
Text of page 17
FEB - 30213/II
43. Let a ∈ K and a ≠ 0. For the matrix
44. Consider a homogeneous equation
1 a y" + ay' + by = 0. Its characteristic
:
0 1 equation has a root r of multiplicity
two. Then the Wronskian W of the
(A) the eigen-vectors of the matrix
solutions of the equation is :
generate 2-dimensional space
(A) W = e2rx
(B) the eigen-vectors of the matrix
generate 1-dimensional space (B) W = xe2rx
2
(C) if K = R, then the characteristic (C) W = xer x
polynomial and minimal 2
(D) W = er x
polynomial are same
Or
(D) the eigen-vectors are orthogonal
A researcher decides to increase the
Or
size of his random sample from 1500
In a study of attitudes to university
to 4000. The effect of this increase
policies, a researcher interviewed
is to :
150 first-year students, 130 second-
(A) reduce the variability of the
year students and 100 third-year
estimate
students. The sampling procedure
used in this study was : (B) reduce the bias of the estimate
(A) probability sampling (C) increase the standard error of
(B) stratified sampling the estimate
(C) quota sampling (D) have no effect because the
(D) temporal sampling population size is the same
17 [P.T.O.
Text of page 18
FEB - 30213/II
45. The partial differential equation 46. The order of the differential equation
which represents all surfaces of
of the family of circles of variable
revolution about z-axis is
radius r with centres on the x-axis
represented by :
is :
∂z ∂z
(A) x + y =0
∂x ∂y (A) 2
∂z ∂z
(B) x − y =0 (B) 3
∂x ∂y
∂z ∂z (C) 4
(C) y +x =0
∂x ∂y
(D) 5
∂z ∂z
(D) y −x =0 Or
∂x ∂y
Or Multiple correlation coefficient
Which of the following statements cannot be negative because :
is incorrect about the sampling
(A) it is the maximum among all
distribution of the sample mean ?
possible correlation coefficients
(A) The standard error of the
sample mean will decrease as between the dependent variable
the sample size increases and a linear combination of the
(B) The sample mean is unbiased independent variables
for the true (unknown)
(B) There are enough independent
population mean
variables having positive
(C) The sampling distribution
correlation with the dependent
shows how the sample mean will
variable
vary among repeated samples
(D) The sampling distribution (C) We take the positive square
shows how the sample is root
distributed around the sample
(D) We reject the negative value
mean
18
Text of page 19
FEB - 30213/II
47. The solution of homogeneous initial
Or
value problem is
y = 2e10x + sin 3x, Which of the following statements
then the least possible order of the
is false ?
differential equation is :
(A) 1 (A) A physical interpretation of the
(B) 2 sample mean x demonstrates
(C) 3
how it measures the centre of
(D) 4
a sample
Or
Hotelling T 2 statistic is a (B) The sample median is very
multivariate generalization of :
sensitive to extremely small or
(A) Chi-square statistic
(B) Student t-test extremely large data values
(C) Snedecor’s F-statistic (outliers)
(D) Mahalanobis' D2-statistic
(C) The sample median is the
48. The set of all spheres with centres
on the z-axis and of radius a is middle value when the
represented by the :
observations are ordered from
(A) first order ordinary differential
equation smallest to largest
(B) first order partial differential
equation (D) The sample mean is very
(C) second order ordinary sensitive to extremely small or
differential equation
extremely large data values
(D) second order partial differential
equation (outliers)
19 [P.T.O.
Text of page 20
FEB - 30213/II
49. Consider the equation L(y) = y" + 50. Let φ 1 and φ 2 be differentiable
a1y' + a2y = 0, where a1 and a2 are functions on an interval I and W(φ1,
real constants. Then every solution φ2) be the Wronskian of φ1, φ2.
of L(y) = 0 tends to zero as x → ∞ Consider the following two
if : statements
(A) a1 > 0 (I) W(φ 1, φ 2) (x 0) ≠ 0 for some
x0 ∈ I ⇒ φ1, φ2 are linearly
(B) a1 < 0
independent
(C) a1 = 0
(II) φ1, φ2 are linearly independent
(D) a1 ≠ 0, a2 > 0 functions on I ⇒ W(φ 1, φ 2)
Or (x) ≠ 0
Let {yn, n > 1} be a sequence of then :
independent standard normal (A) both (I) and (II) are false
variables.
(B) both (I) and (II) are true
Let Xn = Yn3 – 1, n > 1. Then (C) only (I) is true
(D) only (II) is true
is : Or
In almost all non-parametric tests,
(A) 1 which of the following assumptions
is always true ?
1
0
1 − x2
(B) ∫−∞ 2π
e 2 dx (A) The form of the distribution
function (df) is known
−1 1
1 − x2
(C) ∫−∞ 2π
e 2 dx (B) The df is discrete
(C) The distribution is normal
1
(D) (D) The df is continuous
2
20
Text of page 21
FEB - 30213/II
ROUGH WORK
21 [P.T.O.
Text of page 22
FEB - 30213/II
ROUGH WORK
22
Official key not published for this paper.
Page 1 of 23
Source: Savitribai Phule Pune University. Question paper published by Savitribai Phule Pune University.