MA: MATI(EMATICS
Duration: Three Hours
Read the following instructions carefully.
Maximum Marks: 100
1. Do not open the seal of the Question Booklet until you are asked to do so by the invigilator. 2. Take out the Optical Response Sheet (ORS) from this Question Booklet without breaking and read the instructions printed on the ORS carefully. the seal
3. On the right half of the ORS, using ONLY a black ink ball point pen, (i) darken the bubble corresponding to your test paper code and the appropriate bubble under each digit of your registration number and (ii) write your registration number, your name and name of the examination centre and put your signature at the specified location. 4. This Question Booklet contains 20 pages including blank pages for rough work. After you are permitted to open the seal, please check all pages and report discrepancies, if any, to the invigilator. 5. There are a total of 65 questions carrying 100 marks. All these questions are of objective type. Each question has only one correct answer. Questions must be answered on the left hand side of the ORS by darkening the appropriate bubble (marked A, B, C, D) using ONLY a black ink ball point pen against the question number. For each question darken the bubble of the correct answer. More than one answer bubbled against a question will be treated as an incorrect response. 6. Since bubbles darkened by the black ink ball point pen cannot be erased, candidates should darken the bubbles in the ORS very carefully. 7. Questions Q.1  Q.25 carry 1 mark each. Questions Q.26  Q.55 carry 2 marks each. The 2 marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is unattempted, then the answer to the second question in the pair will not be evaluated. 8. Questions Q.56  Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56  Q.60 carry 1 mark each, and questions Q.61  Q.65 carry 2 marks each. 9. Unattempted questions will result in zero mark and wrong answers will result in NEGATIVE marks. For all 1 mark questions, ~ mark will be deducted for each wrong answer. For all 2 marks questions, J.j mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question.
10. Calculator is allowed whereas charts, graph sheets or tables are NOT allowed in the examination hall. 11. Rough work can be done on the question paper itself. Blank pages are provided at the end of the question paper for rough work. 12. Before the start of the examination, write your name and registration number in the space provided below using a black ink ball point pen.
Name Registration
MA
AlO\(
Number
GVPTA
MAl
~ I '
10
I
b
I \
I
L
121/20
2012
MATHEMATICS
MA
Notations and Symbols used
: Set of all real numbers : Set of all complex numbers : Set of all integers F : A field : The set of all ntuples of complex numbers : The set of all ntuples over F : Cartesian product of rings RpR2, ... .R; : Partial derivative with respect to x. : Normal distribution with mean E(X) : Expectation of X : Covariance between X and Y : The group of all permutations on n symbols : The set of all polynomials of degree at most n : Cyclic Group of Order n Z(G) : Centre of the Group G
,J ••... \,A,~
f.l
and variance
(j2
Cov(X,Y)
'1
~.
t.
i=ri
MA
MATHEMATICS MA
Q. 1  Q. 25 carry one mark each.
Q.l The straight lines LI: x = 0, L2: Y =
W= Z2 W
V;
r~') _~~~) 1,
/V;::. \.4 +1 .
°
and L3: x + Y = 1 are mapped by the transformation
h&~It2N1)
.
l.4 '2'_ ~ 
=
°
into the curves CI, C2 and C3 respectively. The angle of intersection between the curves at is ' V:::
V
tilt , ~

!A.
(A) 0
(B) 7r / 4
(C) 7r /2
(D) 7r


1
2.
u
~+..l..2
Q.2
In a topological space, which of the following statements is NOT always true : (A) (B) (C) (D) Union Union Union Union of any finite family of compact sets is compact. of any family of closed sets is closed. of any family of connected sets having a non empty intersection is connected. of any family of dense subsets is dense.
Q.3
Consider the following statements: P: The family of subsets {A" =(  :, :). n = 1,2, ..} satisfies the finite intersection property. Q: On an infinite set X, a metric d : X x X ~ R is defined as d(x,y) The metric space (X,d) is compact. R: In a Frechet (~) topological space, every finite set is closed. S: If f: R ~ X is continuous, where R is given the usual topology and (X, r) is a Hausdorff
={O, x = yY . 1,
x::f:.
( T2 ) space, then
(A) P and R Q.4
f is a oneone
function.
Which of the above statements are correct? (B) P and S (C) Rand S (D) Q and S
Let H be a Hilbert space and S1. denote the orthogonal complement of a set S c H . Which of the following is INCORRECT? (A) For S"S2 <;;,H;SI cS2 ~SI1. <;;,S;(C) {O}1. = H
(B) Sc
(S1.)1.
(D) S1. is always closed.
Q.5
Let H be a complex Hilbert space, T:H ) H be a bounded linear operator and let T* denote the adjoint of T. Which of the following statements are always TRUE? P: \/x,Y
E
H,(Tx,y)=(x,T*
y) y)
Q: \/x,y
E
H,(x,TY)=(T* x,y) x,T* y)
(D) P and S Then which of
R: \/x,y E H,(x,Ty)=(x,T* (A) P and Q Q.6 Let X
s. 't/x,y E H,(Tx,Ty)=(T*
(C) Q and S
(B) P and R and let :3={¢,{a},{b},{a,b},X}
= {a,b,c}
be a topology defined onX.
the following statements are TRUE? P: (X,:3) is a Hausdorff space. R: (X,:3) is a normal space. (B) Q andR
Q: (X,:3) is a regular space. S: (X,:3) is a connected space. (C) Rand S (D) P and S
3/20
(A) P and Q
MA
2012
MATHEMATICS  MA
Q.7
Consider the statements P: If X is a normed linear space and M ~ X is a subspace, then the closure of X. Q: If X is a Banach space and
M
is also a subspace
I
Xn
is an absolutely convergent series in X , then
I
Xn
is
convergent. R: Let MI and M2 be subspaces of an inner product space such that MI n Mz = {O} .Then
'IIml EMI' m2 EM2; Ilml +m211 =llmlllz +llm21lz . S: Let f: X ~ Y . be a linear transformation from the Banach Space X into the Banach space Y.
If f is continuous, then the graph of/is always compact. The correct statements amongst the above are:
(A) P and R only
z
(B) Q and R only
(C) P and Q only
(D) Rand S only
Q.8
A continuous random variable X hatS f:obability density function ;he
3 e.~( ~ 1)
f(x)
=
se
0,
5,
X> 0
x ~ O.
. . :::;
t 5
ll~2)
e.
>"
The probability density function of Y = 3X + 2 is
(A) fey)
=
{
se
0,
3
1
(y2)
I 5
,
y>2
(B)
fey)
=
{2 se
0,
4,2) 5 ,
y>2
y~2
(y2)
y~2
~(,2) 5 ,
(C) fey)
=
{se ,
5
,
y>2
(D)
fey)
=
{4 se
0,
y>2
0,
Q.9
y~2
(52)
y~2
A simple random sample of size 10 from N(fJ,
gives 98% confidence interval (20.49, 23.51).
Then the null hypothesis Ho: fJ = 20.5 against H A : fJ (A) (B) (C) (D) Q.lO can be rejected at cannot be rejected can be rejected at cannot be rejected 2% level of significance at 5% level of significance 10% level of significance at any level of significance
* 20.5
For the linear programming problem Maximize z = Xl + 2xz + 3X3 Subject to
2Xl
 4X4
+3xz 
X3 
x4 = 15
3x4
4X4
g t9.:2 :.\~
~Lft3 ~::~j
,
6xI
+ x2 + X3 
= 21
= 30
~
8xI + 2xz + 3X3
~2 tb . :::)0 Q
xl> x2' x3' x4
0,
(A) (B) (C) (D)
MA
an optimal solution a degenerate ~~.~i9Ie~ution a nondegenerate basic feasible solution a nonbasic feasible solution
4120
..
Q.ll
Which one ofthe following statements is TRUE?
(A) A convex set cannot have infinite many extreme points . .fJ
(B) A linear programming problem can have infinite many extreme points. (C) A linear programming problem can have exactly two different optimal solutions.> (D) A linear programming problem can have a nonbasic optimal solution. J Q.l2
J
Let a = e21ri/S and the matrix
,
'",
2
t>(
p( 1. ()( ~
~ ~
1 a a a3 a4,
t)
1( ,( '2
C>(:>
of '1
Y
aa
M=,iOOa2
a2 a3 a4 a3 a4
a4
I .
I
b
/,)
0 p(2.P(?~'1
0
t
tIJ
0(>
1(
0 0 0 a3 a4 0 0 0 0
boo
(A) 5
0
P('1
Then the trace of the matr~ I +
(B) 0
Mild is
(C) 3
t~
.P('lrf;;>J..'1
\
i;:52
'.'11~.)·1
3.J
I
,i IX ,(p(fL,t\{''(
/
/.
~'"'(D)
5"
V
,r
H./~+.(L
ro r
l,.( ..( '1 ~ <><'
~+ .
a given
Q
Q.13
Let V= (2'be the vector space over the field of complex numbers andB={(1,i),(i,I)}be ,<.( (A) .!;(ZI,Z:)=!(ZI iz2),
ordered basis of V. Then for which of the following, B * = U;,h} is a dual basis of B over (?
p(
4~ +~ 11+
'2,
2
h(ZpZ2)=!(ZI
2
+iz2)
(B) .!;(ZpZ2)=!(ZI
2
+iz2),' h(ZpZ2)=!(iZI
2
+ Z2)
+Z2) Z2)
.
(C)
J;(ZpZ2)=!(ZIiZz},
2
1
h(Zp'z2)=!(izl
2
(D) J;(ZpZ2)=(ZI
2
. 1 +iz2), h(ZpZ2)=(izl
2
Q.14
Let R
(A)
= ~x~x~
and I
= ~x~x
{O}. Then which of the following statement is correct?
lA, )
i (~yj .
V\
(B) I is a prime ideal but not a maximal ideal of R. ' '\!' o ~ \: r (C) I is both maximal ideal as well as a prime ideal of R . A Q rC§...) (D) I is neither a maximal ideal nor a prime ideal of R . rZ Q.l5 The function u(r, e) satisfying the Laplace equation
2
I is a maximal ideal but not a prime ideal of R.
i(\(~)
c(e9)~\u~I~)
,~ ./ ~,
,'
/
L~{~)t)
~e.,
~ ,
iiu +! au +~ B u =0 e c r c e' ar2 r ar r2 Be2 ' subj ect to the conditions u( e, e) = 1, u( e2 , e) = 0 is
(A)ln(e/ Q.l6
(A) strong rru.mmaon y =  e 3 (C) weak maxima on y = Q.42
.
(B)"
strong mmima on y =  e 3'
 r
,,1...
/L."';..7;"Xn t;V3 t:
_(IjIp)CvP)
3ex
1
(D) strong maxima on y = 
3ex l'1I~~\
4
_~l.p) eLf)
L . e . . A partie If'0 mass m IS constrame d to move on a circ Ie WIithra di a w hi h i If'IS r 1. 'P)b( ~ J..p·hl IUS IC itse tmga out VI its vertical diameter with a constant angular velocity oi, Assume that the initial angular velocity is ;: I _, V zero and g is the acceleration due to gravity. If B be the inclination of the radius vector of the ~ 1::>/ particle with the axis of rotation and B denotes the derivative of B with respect to t, then the Lagrangian of this system is (A) !ma2((P+(isin2B)+mgacosB 2 < (C) ma2((P+2m2cosB)mgasinB
.
. .
1
(B) ma\e2+2msinB)mgasinB 2 (D) ma
_ 3;;!!,'}.f
g,.~. , .~
1= .
7,~
+~/i e i
<"
:)
'.
1
'
1
2'2
2
MA
f2 d
~ + o e t. . L
(1
\1
~
~'V'J:/
,Zy11
{:j
~:z70e x, j =
2"
(B +msin2B)+mgasinB '1 UA eo ~
'Y1(
r(
I (
1 = y of cc+ 3"
'2 9/20
+_2 e 3
J(
G·f tl::;t
2012
MATHEMA Trcs  MA
Q.43
For the matrix
M
=
which of the following statements are correct? P: M is skewHermitian and iM is Hermitian Q: M is Hermitian and iM is skew Hermitian V· R: eigenvalues of M are real S: eigenvalues of iM are real (A) P and R only Q.44 (B) Q and R only (C) P and S only
=
l
2 32i 4
3+2i4] 5 6i, 6i 3
(D) Q and S only
Let T:~ ~ ~ be the map given by T(p(x»
f p'(t)dt.
\
x
If the matrix of T relative to the
standard bases B\ = B2 = { 1, X2 ,X3} is M and M' denotes the transpose of the matrix M , then X,
M +M' is 0
(A)
0 f
1
(B)
..•1 r
0
.~
0
,I
0
1 1 1
2 0 0 0 2 0 1 0 2 0 0 0 2 1 0
0
0
2 1,~.(j 0 0 1 2 0 0
1
1
0 0 2
0 0 2 0
1 1 1 1
0 2 0 0 2 2 0
1
b+ 6
,
.f
,(!)
i~
f
()
(C)
0 0
1 2
1
(D)
2 1 2 2
1
0
1 0 1 0
Q.45
Using Euler's method taking step size = 0.1, the approximate value ofy obtained corresponding to x = 0.2 for the initial value problem dy = X2 + y2 and y(O) = 1, is dx (A) 1.322
(B) 1.122
(C) 1.222
(D) 1.110
Q.46
The following table gives the unit transportation costs, the supply at each origin and the demand of each destination for a transportation problem. <1 ,60 ti ~ Destination 3 yO D\ D2 D3 D4 Supply 1 310 10' C,,'20 01 60 fO ,<' Origin O2 80 0 bo 03 100 ~o 3'< <7 q <t 3 Demand 40 70 50 80 o 1.0)0 M Q I 0 5.'0 2.0 Let xij denote the number of units to be transported from origin i to destmationj. If the uv metliod is applied to improve the basic feasible solution given by x12= 60, xn = 10, XZ3 = 50, X31 = 40 and X34 = 60, then the variables entering and leaving the basis, respectively, are
X24
= 20,
MA
10/20
2012
MATHEMATICS  MA
Q.47
Consider the system of equations
[~ ;1
Using Jacobi's approximate solution
[X(2)
u.
y(O)
method with the initial guess [x(O)
z(O)Y
=[2.0 3.0 O.Or, the
/2)
Z(2) ] T
after two iterations, is
(A) [2.64 1.70 (B) [2.64 1.70
1.12r l.12r
(C) [2.64 1.70 1.12f (D) [2.64 1.70 1.12f Common Data Questions
Common Data for Questions 48 and 49:
The optimal table for the primal linear programming problem: Maximize Z = 6xI + 12x2 + 12x3  6x4 Subject to
Xl Xl
+ x2 +
+ 4X2
X3 +X4
=4
=8
~
xl> x2' x3' x4
1S
0,
Basic variables (XB)
X3
Xl
X2
X3
X4
RHS Constants (b) 2 2 z=48
3/4 114
j
0 1 0
1
114 1/4
6
x2
0
0
= c
Q.48
6
If YI and Y2 are the qual variables corresponding to the first and second primal constraints, then their values in the optimal solution of the dual problem are, respectively, (A) 0 and 6 (B) 12 and 0 (C) 6 and 3
(D) 4 and 4
Q.49
If the right hand side of the second constraint is changed from 8 to 20, then in the optimal solution of the primal problem, the basic variables will be
MA
11120
~.
2012
MATHEMATICSI
MA
Common Data for Questions 50 and 51: Consider the Fredholm integral equation u (x) Q.50 The resolvent kernel R(X,t;A) (A) Q.5l
= x +A
f x e' u
o
(t) dt.
for this integral equation is (C) ~ 1+2
xe'
IA
Axel (B) I+A
xe'
The solution ofthis integral equation is
(A) x+I
2
x (C) l+xA2
(D) IA
IA Linked Answer Questions
(B)
I~A2
Statement for Linked Answer Questions 52 and 53: The joint probability density function of two random variables X and Y is given as
f (x,y )= { 5 0,
Q.52 E(X)
~(x+/), .
O:s;x:S;I,O:s;y:S;I
elsewhere
and E(Y) are, respectively,
2 3 (A)  and 5 Q.53 Cov(X,Y) (A) 0.01 5 is
3 3 (B)  and 5 5
3 . 6 (C)  and 5 5
(B) 0
(C) 0.01
(D) 0.02
Statement for Linked Answer Questions 54 and 55: Consider the functions fez) Q.54 The residue of fez)
(A) 1
i+az and g(z) (z+ 1)2
= slnh(z),a:;tO.
2a
~(2):/
(D) 3
.;r
at its pole is equal to 1. Then the value of a is
(B) 1 (C) 2
Su,h (,l ~
Q.55
For the value of a obtained in Q.54, the function g(z) is not conformal at a point
(B)
8,zr Ulb~~i)
( W)f "1
pL
.~  I
.
;r(3+i) 6
(C) 2;r 3
(D) i;r
2
MA
12/20
.
2012
MATHEMATICSMA
General Aptitude (GA) Questions (Compulsory) Q. 56  Q. 60 carry one mark each.
Q.56 Choose the most appropriate word from the options given below to complete lhe following sentence:
Given the seriousness of the situation that he had to face, his _ was impressive.
(A) beggary Q.57
(B) nomenclature
(C) jealousy
(D) nonchalance
Choose the most appropriate alternative from the options given below to complete the following sentence:
If the tired soldier wanted to lie down, he _ the mattress out on the balcony.
(A) should take (B) shall take (C) should have taken (D) will have taken Q.58 If(1.001)1259 = 3.52 and (1.001io62 = 7.85, then (1.001i321 = (A) 2.23 Q.59 (B) 4.33 (C) 11.37
(D) 27.64
One of the parts (A, B, C, D) in the sentence given below contains an ERROR. Which one of the following is INCORRECT?
I requested that he should be given the driving test today instead of tomorrow.
(A) requested that (B) should be given (C) the driving test (D) instead of tomorrow Q.60 Which one of the following options is the closest in meaning to the word given below?
Latitude
(A) Eligibility
(B) Freedom
(C) Coercion
CJ
Q. 61 Q. 65 carry two marks each.
Q.6l There are eight bags of rice looking alike, seven of which have equal weight and one is slightly heavier. The weighing balance is of unlimited capacity. Using this balance, the minimum number of weighings required to identify the heavier bag is
(A) 2 (B) 3 (C) 4 (D) 8
230
t!2J
2.70 ~
Q.62
Raju has 14 currency notes in his pocket consisting of only Rs. 20 notes and Rs. 10 notes. The total money value of the notes is Rs. 230. The number of Rs. 10 notes that Raju has is (A) 5
(B) 6 (C) 9
One of the legacies of the Roman legions was discipline. In the legions, military law prevailed and discipline was brutal. Discipline on the battlefield kept units obedient, intact and fighting, even when the odds and conditions were against them. Which one of the following statements best sums up the meaning of the above pass~ge? (A) Thorough regimentation was the main reason for the efficiency of the Roman legions even in adverse circumstances. (B) The legions were treated inhumanly as if the men were animals. (:x.. (C) Discipline was the armies' inheritance from their seniors. /_~ (D) Th~ harsh discipline to which the legions were subjected to led to the odds and conditions against them. \
l;/ ~f
~
<,«:
Kf
;~~
'l/
Q.64
A and B are friends. They decide to meet between 1 PM and 2 PM on a given day. There is a condition that whoever arrives first will not wait for the other for more than 15 minutes. The probability that they will meet on that day is (A) 114
(B) 1116
(C) 7/16
(D) 9/16 d.
Q.65
The data given in the following table summarizes the monthly budget of an avera
~p'y
10~
2{J/~
~7~~
. ~
c/
[;) ~ )o/,o~+: rd~
rn66
&b
~O( G&

((01
0°
rv1{) ~
~
(ft)
0i
°~
The approximate percentage of. the monthly budget NOT spent on savings is (A) 10% (B) 14% (C) 81% l1"tj'ljl (D) 86%