MH SET 2011: 27 November, Paper II, Mathematical Sciences
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Mathematical Sciences
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. Let f(z) = sin z, z C
| . Then f(z) : 3. Let the sequence {a n } be given
(A) is bounded in the complex plane by
1 1 1
(B) assumes all complex numbers 1, 2, 3, 1 + , 2 + , 3 + ,
2 2 2
(C) assumes all complex numbers 1 1 1
1 + , 2 + , 3 + , ....... .
3 3 3
except i
Then
(D) assumes all complex numbers lim sup an
nof
except i and –i
is :
2. The radius of convergence of the
(A) 3
series
(B) f
n ! zn
¦ nn (C) 1
is : (D) –1
4. If I z E F [R , then :
(A) 1
(A) inf E d inf F
(B) f
(B) inf E > inf F
(C) 1/4
(C) inf E t inf F
(D) e
(D) inf E < inf F
1 [P.T.O.
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5. If a and b are real numbers, then 7. The matrix R(T) associated with the
inf {a, b} = rotation by T = S/4 is :
ª 2 2 2 2º
ab ab
(A) (A) « »
2 «¬ 2 2 2 2»¼
ab ab ª0 1º
(B) (B) « »
2
«¬1 0 »¼
ab ab ª 1 0º
(C)
2 (C) « »
«¬ 0 1»¼
ab ab
(D) ª 2 2 2 2º
(D) « »
2
«¬ 2 2 2 2 »¼
6. The dimension of the space of
8. Let W be a subspace of a vector
n × n matrices all of whose
space V and let T : W o Vc be a
components are 0 expect possibly
linear map, then :
the diagonal components is :
(A) T can be extended to a linear
(A) n 2 transformation from V to Vc
(B) T is necessarily linear map from
(B) n – 1
V to Vc
(C) n2 – 1
(C) Ker T is not a subspace of V
(D) n (D) Im T is a subspace of V
2
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9. Let S, T L(V1, V2), V1, V2 are 11. A sample space consists of five
finite dimensional vector spaces. simple events E1, E2, E3, E4 and E5.
If P(E1) = P(E2) = 0.1, P(E3) = 0.4
Then :
and P(E4) = 3P(E5). Then P(E4)
(A) rank S + rank T d rank (S + T)
and P(E5) :
(B) Im (S + T) = Im S + Im T
(A) are 0.06 and 0.02, respectively
(C) Im (S + T) Im S + Im T (B) cannot be determined from the
(D) min{rank S, rank T} d rank ST, given information
where V1 = V2 (C) are 0.6 and 0.2, respectively
10. Let A be a matrix similar to a square (D) are 0.3 and 0.1, respectively
matrix B. Then which one of the 12. If independent binomial
following is false ? experiments are conducted with
n = 10 trials. If the probability
(A) If A is self-adjoint then so is B
of success in each trial is P = 0.6,
(B) If A is non-singular then so then the average number of
is B successes per experiment is :
(C) Determinant of A is the same (A) 4
as the determinant of B (B) 6
(D) Trace of A is the same as the (C) 8
trace of B (D) 10
3 [P.T.O.
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13. The joint distribution of r.v.s X and 15. The objective ‘Linear’ in Linear
Y is given by :
Programming Problem implies :
X\Y 2 0 2 (A) The objective function and the
1 1 1 constraints are both linear in
0
4 4 4 the variables
1 1 (B) The constraints are linear but
2 0
8 8
not the objective function
Then P[X + Y = 0] is : (C) The objective function alone is
(A) 3/4 linear in the variables
(B) 1/4
(D) None of the above
(C) 3/8
16. The optimal solution to the Linear
(D) 5/8
Programming Problem :
14. Let X and Y be two r.v.s such
that E[X] = 2, E[X2] = 5, E[Y] = 4, Maximize : Z = 2x1 + x2
E[Y2] = 16 and E[XY] = 12.
Consider the statements : Subject to : x1 + x2 d 1 o (1)
(I) The r.v.s X and Y are 3x1 + x2 d 2 o (2)
independent
(II) The r.v. Y is degenerate x1 + 2x2 d 3 o (3)
(III) Given expected values are not x1, x2 t 0 o (4)
compatible
Then : (A) lies at the intersection of (1)
(A) Only statements I and II are and (3)
correct
(B) lies at the intersection of (2)
(B) Only statements II and III are
and (3)
correct
(C) Only statements I and III are (C) lies at the intersection of (1)
correct and (2)
(D) All three statements are
correct (D) cannot be determined
4
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17. If an = (–1)nn, for n = 1, 2, 3, 4, ......., 18. If I z E F [R , then :
then : (A) sup E d sup F
lim sup an (B) sup E t sup F
nof
(C) sup E < sup F
is equal to :
(D) sup E > sup F
(A) 1
Or
(B) f The characteristic function of a
(C) –1 standard Cauchy distribution :
(A) is e–|t|
(D) –f
(B) is eit
Or
(C) is eSt
Let F and G be probability
(D) does not exist
distribution functions. Which of
19. Radius of convergence of the
the following may not be a
series :
probability distribution function ?
n ! zn
¦ nn
(A) H(x) = F2(x)G3(x)
is :
(B) H(x) = F(G(x))
(A) 1
x
1
(C) H(x) = ³ G u dF u (B)
4
f
(C) e
F x G x
(D) H(x) =
2 (D) f
5 [P.T.O.
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Or Or
The p.d.f. of the random variable
Let the random variables X1, X2
X follows f(x), where :
be distributed as N(T, 1). Then
1 ª xTº
f(x) = exp « »,
2T ¬ T ¼ number of unbiased estimators of
–f < x < f T is :
Mean of the distribution is :
(A) infinity
1
(A)
T
(B) 3
(B) 2T
(C) 2
(C) T
(D) 0 (D) 4
20. The interval of convergence of the 21. In order that f defined by
series
f(x) = (1 + x)cot x x z 0 be continuous
x 2 x3 x 4
x ......... at x = 0, how f(0) be defined ?
2 3 4
is : (A) f(0) = 0
(A) –1 < x d 1 1
(B) f(0) =
e
(B) –1 < x < 1
(C) f(0) = e
(C) –1 d x < 1
(D) 0 d x d 2 (D) f(0) = 1
6
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Or Or
Let the random variables X1, X2 are For testing H0 : FX(x) = HY(y), when
two independent random observations
Poisson variates with parameter O.
on X and Y are available, which of
Then which of the statements is
the following non-parametric test
true ?
cannot be used ?
(A) X1 is sufficient for O
(A) Mann-Whitney test
(B) X1 – X2 is sufficient for O
(B) Sign test
(C) X1 + 2X2 is not sufficient for O (C) Kolmogorov-Smirnov test
(D) X2 is sufficient for O (D) Wald-Wolfowitz run test
22. If g(x) = sin x and f(x) = cos x, then 23. If f(x) is continuous on [a, b], then
the incorrect statement among the
the point at which the conclusion of
following is :
Cauchy’s Mean Value Theorem
(A) f(x) is bounded on [a, b]
holds in [–S/4, S/4] is :
(B) f(x) assumes all values between
(A) S/6
f(a) and f(b)
(B) 0
(C) f(x) is increasing on [a, b]
(C) S/4
(D) f(x) is uniformly continuous on
(D) –S/6 [a, b]
7 [P.T.O.
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Or Or
Let X1, X2, ........, Xn be a sample If, for a given D, 0 d D d 1, non-
from N(P, 1). Then the shortest randomized Neyman-Pearson and
confidence interval based on likelihood ratio tests of a simple
TP(X) = n XP is : hypothesis against a simple
§ Z1 D / 2 Z1 D / 2 · alternative exist, then :
(A) ¨ X , X ¸
© n n ¹
(A) They are equivalent
§ Z1 D / 2 ZD / 2 ·
(B) ¨ X , X ¸ (B) They are one and the same
© n n ¹
(C) They are exactly opposite
§ ZD / 2 Z ·
(C) ¨ X , X D/2 ¸
© n n ¹
(D) One can’t say anything about
§ ZD / 2 Z1 D / 2 · it
(D) ¨ X , X ¸
© n n ¹
25. Let
C o |
24. Let f : | C be a complex valued
1 1
function. If f(z) and f z are both x 1 i 3 , y 1i 3 .
2 2
analytic, then :
Then :
(A) f(z) is a constant function
(A) x2 + y2 = 1
(B) f(z) is the identity function
(B) x2 – y2 = x – y
(C) f(z) is unbounded
(C) (xy)2 = xy
(D) f(z) is a non-constant entire
function (D) x3 = x, y3 = y
8
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26. If |z1 + z2| = |z1| + |z2| for
Or
z1, z2 C
| , then :
Let y1, y2, y3 be three independent
(A) One of z 1 or z 2 is positive
observations having expectations,
multiple of the other
E(y1) = E0 – E1 + E2, E(y2) = E0 –
(B) z1 and z2 have the same length
2E 2 , E(y 3 ) = E 0 + E 1 + E 2 and
V(yi) = V2 for i = 1, 2, 3. The least (C) z1 is real and z2 is imaginary
squares estimates of E0, E1 and (D) z1, z2 are real
E 2 are Ê0 = (y 1 + y 2 + y 3 )/3,
Or
Ê1 = (–y1 + y3)/2, Ê2 = (y1 – 2y2
For the two way ANOVA model :
+ y 3)/6. Which of the following
y ijk = P + D i + E j + J ij + e ijk
statements is false ?
i = 1, ....., 4; j = 1, ....., 5; k = 1, 2,
(A) Variances of Eˆ i are unequal
the error degrees of freedom are :
(B) Eˆ i ’s are unbiased estimators
(A) 20
of Ei
(B) 24
(C) It is possible to obtain unbiased
estimator of V2 (C) 12
(D) cov Eˆ i , Eˆ j = 0 (D) 27
9 [P.T.O.
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27. For z C
| the inequality : 28. Consider the statements :
|z + 1| > |z – 1| (a) If a function is analytic in a
is : bounded domain, then it is
(A) always true bounded.
(B) never true (b) If u(x, y) is harmonic in a
domain D, then there exists a
(C) true iff Re z > 0
harmonic function v(x, y) such
(D) true iff Im z > 0
that u(x, y) + iv(x, y) is analytic.
Or
Then :
Let y 1 , y 2 be two independent
(A) both (a) and (b) are true
observations having expectations
(B) both (a) and (b) are false
E(y1) = T1 + T2, E(y2) = T1 – T2 and
V(y1) = V(y2) = V2. Let ˆTi be least (C) only (a) is true
squares estimate of Ti. Which of the (D) only (b) is true
following statements is correct ? Or
y1 y2 V2
(A) Tˆ 1 and V Tˆ 1 A block design is said to be
2 2
connected if :
y1 y2 V2
(B) Tˆ 2 and V Tˆ 2 (A) R(C) < v – 1
2 4
y1 y2 V2 (B) R(C) = v
(C) Tˆ 1 and V Tˆ 1
2 2
(C) R(C) < v – 2
y1 y2 V2
(D) Tˆ 2 and V Tˆ 2 (D) R(C) = v – 1
2 4
10
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29. The value of (i)i – (–i)–i is : Or
(A) zero Layout of a block design with 4
blocks and four treatments A, B, C,
(B) non-zero real
D is given below :
(C) purely imaginary
Block 1 : A, B, C, x
(D) simply a chaos
Block 2 : A, B, D, y
Or
Block 3 : A, C, D, w
A RBD is :
Block 4 : B, C, D, z
(A) Connected and balanced
Identify x, y, w and z so that given
(B) Connected but not balanced design is RBD.
(C) Not connected but balanced (A) x = D, y = C, w = B, z = A
(D) Not connected not balanced (B) x = B, y = A, w = B, z = A
| o C
30. Define a function f : C | by (C) x = B, y = C, w = B, z = A
f(z) = |z|. Then f(z) is : (D) x = D, y = B, w = B, z = A
(A) continuous everywhere but not 31. Which of the following ring is
differentiable at the origin isomorphic to the field of complex
(B) continuous everywhere but numbers ?
differentiable only at the origin (A) R[x]
(C) continuous and differentiable C ª¬ 5iº¼
(B) |
everywhere (C) Q[i]
(D) analytic at the origin | [x]
(D) C
11 [P.T.O.
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Or 33. Let R, S be rings and R S be the
For any two events A and B, which direct product of R and S. Then
of the following are always true ? which of the following is not true ?
(A) P(A or B) = P(A) + P(B) (A) If R and S are commutative
(B) P(A or B) = P(A) + P(B) – then so is R S
P(A and B) (B) If R and S are non-commutative
(C) P(A or B) = P(A) . P(B) then so is R S
(D) P(A and B) = P(A) . P(B) (C) If R and S are with identity then
32. Let A, B be ideals of a ring R : so is R S
(A) AB = A B (D) If R and S are integral domains
then so is R S
(B) AB A B
Or
(C) AB A B
(D) AB = A B if A = B The least squares regression line is
the line :
Or
(A) for which the sum of the
Given that we have collected pairs
residuals about the line is zero
of observations on two variables X
and Y, we would consider fitting a (B) which has the largest sum of the
straight line with X as an squared residuals of any line
explanatory variable if : through the data values
(A) the change in Y is an additive (C) which is determined by use of
constant
a function of the distance
(B) the change in Y is a constant between the observed Y’s and
for each unit change in X the predicted Y’s
(C) the change in Y is a fixed (D) which has the smallest sum of
percent of Y the squared residuals of any line
(D) the change in Y is exponential through the data values
12
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34. Let G be a group, a G a fixed 35. Let I be an ideal of a ring R, such
element and f : G o G be a mapping R
that # Z2, Then which of the
I
given by f(x) = axa–1. Which of the
following is true ?
following is not true ?
(A) Both R and I are finite
(A) f is an automorphism
(B) R is finite iff I is finite
(B) f is not a homomorphism
(C) Both R and I are infinite
(C) f is onto
(D) If R is infinite, then I is finite
(D) f is one-to-one
Or
Or
A survey of college students taking
In a single-factor ANOVA problem
the professional exam to be certified
involving five treatments, with a
as public school teacher shows that
random sample of four observations
15 percent fail. On a national exam
from each one, it is found that :
day, 12,000 students take the test.
SSTr = 16.1408 and SSE = 37.3801. Let X denote the number who fail.
Then the value of test statistic is : The mean value of X is :
(A) 1.522 (A) 12,000
(B) 1.619 (B) 1,800
(C) 2.316 (C) 1,500
(D) 0.432 (D) 10,200
13 [P.T.O.
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37. Let v and w be eigen-vectors of
36. Let A, B be ideals of a ring R.
T corresponding to two distinct
Suppose that A + B = A B. Then eigen-values O1 and O2 respectively.
Then :
which of the following is true ?
(A) for non-zero scalars D1, D2, the
(A) A + B = A vector D1v + D2w is not an
eigen-vector of T
(B) AB = A B
(B) for all scalars D1, D2, the vector
D1v + D2w is not an eigen-vector
(C) AB = A B
of T
(D) Either A B or B A (C) D1v + D2w is an eigen-vector of
T if D1 = D2
Or
(D) D1v + D2w is an eigen-vector of
T if D1 = D2
Which of the following is not
Or
example of a variable ?
The initial solution of a
(A) Gender of a high school Transportation problem can be
obtained by applying any known
graduate method. However, the only condition
is that :
(B) Number of major credit cards a
(A) The rim conditions are satisfied
person has
(B) The solution must be optimum
(C) Type of automobile transmission (C) The solution should be non-
degenerate
(D) Capital city of a country (D) All of the above
14
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38. Let S and T be two diagonalizable 39. Let W be a subspace of a finite
linear operator that are similar.
dimensional vector space V. Then :
Then :
(A) Eigen-values of S and T are (A) Linearly independent subset
conjugates of W need not be Linearly
(B) S and T have same
independent in V
characteristic polynomial
(C) Eigen-vectors of S and T are (B) W need not be finite
conjugates dimensional
(D) S and T are triangulable
Or (C) dim W = dim W0, where W0 is
You are given a two server queueing the annihilator of W
system, in a steady-state condition
(D) dim V = dim W + dim W0, where
where the number of customers in
the system varies between 0 to 4. W0 is the annihilator of W
The probability that there are
Or
exactly n customers in the system
is P n and the values of P n are The problem of replacement in the
1 4 replacement theory is not concerned
P0 = = P4; P1 = = P3 and
16 16
about the :
6
P2 = then Expected number of
16 (A) items that deteriorate
customers in the queue Lq is : graphically
(A) 1 (B) items that fail suddenly
3 (C) determination of optimum
(B)
8
replacement interval
3
(C)
4 (D) maintenance of an item to work
(D) 2 out profitability
15 [P.T.O.
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40. Let N be a nilpotent matrix : Or
(A) I-N is invertible Probability sampling is the
(B) Eigen-values of N are non-zero procedure that gives all units :
(C) N is diagonalizable
(A) an equal, calculable and non-
(D) Eigen-vectors are orthogonal zero chance to be selected
Or
(B) a chance to be included in the
A sequencing problem involving six study
jobs and three machines requires
(C) an equal chance to be selected
evaluation of :
(D) an equal chance to be selected
(A) (6!)3 sequences
or not to be selected
(B) (6! + 6! + 6!) sequences
(C) (6 × 6 × 6) sequences 42. Let T be a linear operator on a finite
dimensional vector space V and let
(D) (6 + 6 + 6) sequences
W be a subspace of V. Then :
41. Let D be the derivative d/dt,
f(t) = 3sin t + 5cos t. The co-ordinates (A) dim W = dim V = dim V/W
of Df(t) with respect to the basis
(B) every basis of V/W can be
{sin t, cos t} are :
constituted with only elements
(A) (–5, 3)
of W
(B) (5, 3)
(C) dim V t dim W + dim V/W
(C) (3, 5)
W
(D) # T(W)
(D) (3, –5) W ker T
16
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Or Or
A researcher chose the respondents
In stratified sampling, the strata :
of his study by interviewing a few
(A) are equal in size to each other available couples and by obtaining
names of new couples from the
(B) are proportionate to the units previous respondents. This
in the target population procedure is called :
(A) Systematic Sampling
(C) are disproportionate to the units
(B) Convenient Sampling
in the target population
(C) Quota Sampling
(D) can be proportionate or
(D) Snowball Sampling
disproportionate to the units in 44. If the number of constants to be
the target population eliminated from the given relation
is greater than the number of
43. Let V be an inner product space
independent variables, then the
over F and u, v V, then which one equation obtained is a :
of the following is false ? (A) linear equation of first order
(B) non-linear equation of second
(A) __u + v__2 + __u – v__2 = 2__u__2 + 2__v__2
order
(B) _<u, v>_ d __u__ . __v__ (C) second order partial differential
equation
(C) __u__ – __v__ d __u – v__
(D) non-linear partial differential
(D) __u – v__ t __u__ + __v__ equation of first order
17 [P.T.O.
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Or Or
The sampling distribution of the
A wholesale distributor has found mean refers to :
that the amount of a customer’s (A) the distribution of various
sample sizes which might be
order is a normal random variable
used in a given study
with a mean of Rs. 2,000 and a
(B) the distribution of different
standard deviation of Rs. 500. What possible values of the sample
mean together with their
is the probability that the total
respective probabilities of
amount in a random sample of 20 occurrence
orders is greater than Rs. 45,000 ? (C) the distribution of the values of
the items in the population
(A) 0.1915
(D) the distribution of the values of
the items actually selected in a
(B) 0.3085
given sample
(C) 0.0125 46. Let W(I1, I2) (x) be the Wronskian
of two linearly independent
(D) 0.0228
solution I1, I2 of constant coefficient
equation L(y) = 0 on I. Then :
45. The order of the differential equation
(A) W(I1, I2) (x) = 0
of the family of all ellipses is :
W(I1, I2) (x 0) = 0 for some
(A) 1 x0 I
(B) W(I1, I2) (x0) = 0
(B) 2 W(I1, I2) (x) = 0 x I
(C) W(I1, I2) (x) = 0
(C) 3
W(I1, I2) (x0) = 0
(D) 4 (D) (A), (B) and (C) are not true
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Or 48. Let y = x2 sin x be a solution of a
Principal component analysis will homogeneous initial value problem.
not be affected if :
Then the least possible order of the
(A) variables are standardized
differential equation is :
(B) variables are studentized
(C) there is a change of origin but (A) 3
not a change of scale
(B) 4
(D) variables are subjected to a
linear transformation (C) 5
47. The solution of the differential (D) 6
equation ycc + 4yc + 3y = 0 is :
Or
(A) continuous and infinitely
differentiable Which of the following statements
(B) continuous and unbounded is not true about a Poisson
(C) continuous but not differentiable Probability Distribution with
(D) unbounded but infinitely parameter O ?
differentiable
(A) The mean of the distribution
Or
is O
Hotelling T2 test can be used for
testing : (B) The standard deviation of the
(A) a general linear hypothesis in distribution is the positive
the mean vector
square root of O
(B) an arbitrary hypothesis about
the mean vector (C) The parameter O must be
(C) simultaneous hypotheses about greater than zero
the mean vectors
(D) The parameter O is coefficient
(D) two-sided hypothesis in the
mean vector of variation
19 [P.T.O.
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49. Let f(x, y, z, p, q) = 0 be a first order 50. For the differential equation :
partial differential equation. Then
1 x2 ycc 2 xyc y 0
the relation between the variables
(A) x = 1 is a regular singular point
involving as many arbitrary
and x = –1 is irregular singular
constants as there are independent
point
variables is called :
(A) general integral (B) x = 1 and x = –1 are regular
singular points
(B) singular integral
(C) complete integral (C) x = –1 is a regular singular
(D) particular solution point and x = 1 is irregular
Or singular point
Suppose the sequences {Xn} and {Yn} (D) x = 1 and x = –1 are ordinary
d points
of r.v.s are such that Xn o X,
P
Yn o Y as n o f and suppose Or
P[Y = 3] = 1. Which of the following The mean and variance of 25 is :
is correct ?
(A) Mean = 25, Variance = 1
P
(A) Xn + Yn o X + 3 as n o f
(B) Mean = 25, Variance = 0
P
(B) XnYn o 3X as n o f
1
d (C) Mean = 25, Variance =
(C) Yn cos(Xn) o 3cos(X) as 25
n o f (D) Mean = 25 and Variance cannot
d be found
(D) Yn X 2n o 3X2 as n o f
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NOV - 30211/II
ROUGH WORK
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NOV - 30211/II
ROUGH WORK
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Test Booklet No.
F
Signature of Invigilators
1. .........................................
2. ......................................... Seat No.
MATHEMATICAL SCIENCES (In figures as in Admit Card)
Paper II Seat No. (In words) ............................
NOV- 30211 ..............................................................
Answer Sheet No.
Time Allowed : 75 Minutes] [Maximum Marks : 100
Number of Pages in this Booklet : 24
Instructions for the Candidates
1. Write your Seat Number in the space
1.
provided on the top of this page. Write
your Answer Sheet No. in the space
provided for Answer Sheet No. on the
top of this page. 2. OMR
2. Write and darken Test Booklet No. on
OMR Answer Sheet.
3. This paper consists of Fifty (50) 3.
multiple choice type of questions. 4. (A), (B), (C) (D)
4. Each item has four alternative
responses marked (A), (B), (C) and (D).
You have to darken the responses as
indicated below on the correct
response against each item. A B C D
Example : A B C D
(C)
Where (C) is the correct response.
5. Your responses to the items for this 5.
paper are to be indicated on the ( ×) ( )(/)
Answer Sheet only. Responses like ( × )
( ) ( / ) and light shaded responses
will not be considered/evaluated.
6. Read instructions given inside 6.
carefully. 7.
7. One Sheet is attached at the end of the
booklet for rough work.
8. You should return the test booklet and 8.
answer sheet both to the invigilator
at the end of the paper and should not
carry any paper with you outside the
examination hall.
9. Answers marked on the body of the 9.
question paper will not be evaluated.Official key not published for this paper.
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Source: Savitribai Phule Pune University. Question paper published by Savitribai Phule Pune University.