GATE Math 2013 Question Paper
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GATE Math 2013 Question Paper | Sourav Sir's Classes
1. 2013 MATHEMATICS โMA
MA 1/14
MA:MATHEMATICS
Duration: Three Hours Maximum Marks:100
Please read the following instructions carefully:
General Instructions:
1. Total duration of examination is 180 minutes (3 hours).
2. The clock will be set at the server. The countdown timer in the top right corner of screen will display
the remaining time available for you to complete the examination. When the timer reaches zero, the
examination will end by itself. You will not be required to end or submit your examination.
3. The Question Palette displayed on the right side of screen will show the status of each question using
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The Marked for Review status for a question simply indicates that you would like to look at that
question again. If a question is answered and Marked for Review, your answer for that question
will be considered in the evaluation.
Navigating to a Question
4. To answer a question, do the following:
a. Click on the question number in the Question Palette to go to that question directly.
b. Select an answer for a multiple choice type question. Use the virtual numeric keypad to enter
a number as answer for a numerical type question.
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question.
d. Click on Mark for Review and Next to save your answer for the current question, mark it
for review, and then go to the next question.
e. Caution: Note that your answer for the current question will not be saved, if you
navigate to another question directly by clicking on its question number.
5. You can view all the questions by clicking on the Question Paper button. Note that the options for
multiple choice type questions will not be shown.
2. 2013 MATHEMATICS โMA
MA 2/14
Answering a Question
6. Procedure for answering a multiple choice type question:
a. To select your answer, click on the button of one of the options
b. To deselect your chosen answer, click on the button of the chosen option again or click on the
Clear Response button
c. To change your chosen answer, click on the button of another option
d. To save your answer, you MUST click on the Save and Next button
e. To mark the question for review, click on the Mark for Review and Next button. If an
answer is selected for a question that is Marked for Review, that answer will be considered
in the evaluation.
7. Procedure for answering a numerical answer type question:
a. To enter a number as your answer, use the virtual numerical keypad
b. A fraction (eg.,-0.3 or -.3) can be entered as an answer with or without โ0โ before the decimal
point
c. To clear your answer, click on the Clear Response button
d. To save your answer, you MUST click on the Save and Next button
e. To mark the question for review, click on the Mark for Review and Next button. If an
answer is entered for a question that is Marked for Review, that answer will be considered
in the evaluation.
8. To change your answer to a question that has already been answered, first select that question for
answering and then follow the procedure for answering that type of question.
9. Note that ONLY Questions for which answers are saved or marked for review after answering will be
considered for evaluation.
3. 2013 MATHEMATICS โMA
MA 3/14
Paper specific instructions:
1. There are a total of 65 questions carrying 100 marks. Questions are of multiple choice type or
numerical answer type. A multiple choice type question will have four choices for the answer with
only one correct choice. For numerical answer type questions, the answer is a number and no choices
will be given. A number as the answer should be entered using the virtual keyboard on the monitor.
2. Questions Q.1 โ Q.25 carry 1mark each. Questions Q.26 โ Q.55 carry 2marks each. The 2marks
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 not attempted,
then the answer to the second question in the pair will not be evaluated.
3. Questions Q.56 โ Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks.
Questions Q.56 โ Q.60 carry 1mark each, and questions Q.61 โ Q.65 carry 2marks each.
4. Questions not attempted will result in zero mark. Wrong answers for multiple choice type questions
will result in NEGATIVE marks. For all 1 mark questions, โ mark will be deducted for each wrong
answer. For all 2 marks questions, โ 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. There is no negative
marking for questions of numerical answer type.
5. Calculator is allowed. Charts, graph sheets or tables are NOT allowed in the examination hall.
6. Do the rough work in the Scribble Pad provided.
5. 2013 MATHEMATICS โMA
MA 5/14
Q. 1 โ Q. 25 carry one mark each.
Q.1 The possible set of eigen values of a 4 ร 4 skew-symmetric orthogonal real matrix is
(A) {ยฑ๐๐} (B) {ยฑ๐๐, ยฑ1} (C) {ยฑ1} (D) {0, ยฑ๐๐}
Q.2 The coefficient of (๐ง๐ง โ ๐๐)2
in the Taylor series expansion of
๐๐(๐ง๐ง) = ๏ฟฝ
sin ๐ง๐ง
๐ง๐ง โ ๐๐
if ๐ง๐ง โ ๐๐
โ1 if ๐ง๐ง = ๐๐
around ๐๐ is
(A)
1
2
(B) โ
1
2
(C)
1
6
(D) โ
1
6
Q.3 Consider โ2
with the usual topology. Which of the following statements are TRUE for all ๐ด๐ด, ๐ต๐ต โ
โ2
?
P:๐ด๐ด โช ๐ต๐ต = ๐ด๐ด โช ๐ต๐ต.
Q:๐ด๐ด โฉ ๐ต๐ต = ๐ด๐ด โฉ ๐ต๐ต.
R:(๐ด๐ด โช ๐ต๐ต)ห = ๐ด๐ดห โช ๐ต๐ตห.
S:(๐ด๐ด โฉ ๐ต๐ต)ห = ๐ด๐ดห โฉ ๐ต๐ตห.
(A) P and R only (B) P and S only (C) Q and R only (D) Q and S only
Q.4 Let ๐๐: โ โ โ be a continuous function with ๐๐(1) = 5 and ๐๐(3) = 11. If ๐๐(๐ฅ๐ฅ) = โซ ๐๐(๐ฅ๐ฅ + ๐ก๐ก)๐๐๐๐
3
1
then ๐๐โฒ(0) is equal to ______
Q.5 Let ๐๐ be a 2 ร 2 complex matrix such that trace(๐๐) = 1 anddet(๐๐) = โ6. Then, trace(๐๐4
โ ๐๐3) is
______
Q.6 Suppose that ๐ ๐ is a unique factorization domain and that ๐๐, ๐๐ โ ๐ ๐ are distinct irreducible elements.
Which of the following statements is TRUE?
(A) The ideal โฉ1 + ๐๐โช is a prime ideal
(B) The ideal โฉ๐๐ + ๐๐โช is a prime ideal
(C) The ideal โฉ1 + ๐๐๐๐โช is a prime ideal
(D) The ideal โฉ๐๐โช is not necessarily a maximal ideal
Q.7 Let ๐๐ be a compact Hausdorff topological space and let ๐๐ be a topological space. Let ๐๐: ๐๐ โ ๐๐ be a
bijective continuous mapping. Which of the following is TRUE?
(A) ๐๐ is a closed mapbut not necessarily an open map
(B) ๐๐ is an open map but not necessarily a closed map
(C) ๐๐ is both an open map and a closed map
(D) ๐๐ need not be an open map or a closed map
Q.8 Consider the linear programming problem:
Maximize
๐ฅ๐ฅ +
3
2
๐ฆ๐ฆ
subject to 2๐ฅ๐ฅ + 3๐ฆ๐ฆ โค 16,
๐ฅ๐ฅ + 4๐ฆ๐ฆ โค 18,
๐ฅ๐ฅ โฅ 0, ๐ฆ๐ฆ โฅ 0.
If ๐๐ denotes the set of all solutions of the above problem, then
(A) ๐๐ is empty (B) ๐๐ is a singleton
(C) ๐๐ is a line segment (D) ๐๐ has positive area
6. 2013 MATHEMATICS โMA
MA 6/14
Q.9 Which of the following groups has a proper subgroup that is NOT cyclic?
(A) โค15 ร โค77
(B) ๐๐3
(C) (โค, +)
(D) (โ, +)
Q.10 The value of the integral
๏ฟฝ ๏ฟฝ ๏ฟฝ
1
๐ฆ๐ฆ
๏ฟฝ ๐๐โ๐ฆ๐ฆ/2
โ
๐ฅ๐ฅ
โ
0
๐๐๐๐๐๐๐๐
is ______
Q.11 Suppose the random variable ๐๐ has uniform distribution on [0,1] and ๐๐ = โ2 log ๐๐. The density of
๐๐ is
(A) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ
๐๐โ๐ฅ๐ฅ
if ๐ฅ๐ฅ > 0
0 otherwise
(B) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ 2๐๐โ2๐ฅ๐ฅ
if ๐ฅ๐ฅ > 0
0 otherwise
(C) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ
1
2
๐๐โ
๐ฅ๐ฅ
2if ๐ฅ๐ฅ > 0
0 otherwise
(D) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ
1/2 if ๐ฅ๐ฅ โ [0,2]
0 otherwise
Q.12 Let ๐๐ be an entire function on โ such that |๐๐(๐ง๐ง)| โค 100 log|๐ง๐ง|for each ๐ง๐ง with|๐ง๐ง| โฅ 2. If ๐๐(๐๐) = 2๐๐,
then ๐๐(1)
(A) must be 2 (B) must be 2๐๐
(C) must be ๐๐ (D) cannot be determined from the given data
Q.13 The number of group homomorphisms from โค3 to โค9 is ______
Q.14 Let ๐ข๐ข(๐ฅ๐ฅ, ๐ก๐ก) be the solution to the wave equation
๐๐2
๐ข๐ข
๐๐๐ฅ๐ฅ2
(๐ฅ๐ฅ, ๐ก๐ก) =
๐๐2
๐ข๐ข
๐๐๐ก๐ก2
(๐ฅ๐ฅ, ๐ก๐ก), ๐ข๐ข(๐ฅ๐ฅ, 0) = cos(5๐๐๐๐) ,
๐๐๐๐
๐๐๐๐
(๐ฅ๐ฅ, 0) = 0.
Then, the value of ๐ข๐ข(1,1) is ______
Q.15 Let ๐๐(๐ฅ๐ฅ) = โ
sin (๐๐๐๐ )
๐๐2
โ
๐๐=1 . Then
(A) lim๐ฅ๐ฅโ0 ๐๐(๐ฅ๐ฅ) = 0 (B) lim๐ฅ๐ฅโ0 ๐๐(๐ฅ๐ฅ) = 1
(C) lim๐ฅ๐ฅโ0 ๐๐(๐ฅ๐ฅ) = ๐๐2
/6 (D) lim๐ฅ๐ฅโ0 ๐๐(๐ฅ๐ฅ) does not exist
7. 2013 MATHEMATICS โMA
MA 7/14
Q.16 Suppose ๐๐ is a random variable with ๐๐(๐๐ = ๐๐) = (1 โ ๐๐)๐๐
๐๐ for ๐๐ โ {0,1,2, โฆ } and some ๐๐ โ
(0,1). For the hypothesis testing problem
๐ป๐ป0: ๐๐ =
1
2
๐ป๐ป1: ๐๐ โ
1
2
consider the test โReject ๐ป๐ป0 if ๐๐ โค ๐ด๐ด or if ๐๐ โฅ ๐ต๐ตโ, where๐ด๐ด < ๐ต๐ต are given positive integers. The
type-I error of this test is
(A) 1 + 2โ๐ต๐ต
โ 2โ๐ด๐ด
(B) 1 โ 2โ๐ต๐ต
+ 2โ๐ด๐ด
(C) 1 + 2โ๐ต๐ต
โ 2โ๐ด๐ดโ1
(D) 1 โ 2โ๐ต๐ต
+ 2โ๐ด๐ดโ1
Q.17 Let G be a group of order 231. The number of elements of order 11 in G is ______
Q.18 Let ๐๐: โ2
โ โ2
be defined by ๐๐(๐ฅ๐ฅ, ๐ฆ๐ฆ) = (๐๐๐ฅ๐ฅ+๐ฆ๐ฆ
, ๐๐๐ฅ๐ฅโ๐ฆ๐ฆ ). The area of the image of the region
{(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
:0 < ๐ฅ๐ฅ, ๐ฆ๐ฆ < 1} under the mapping ๐๐ is
(A) 1 (B) ๐๐ โ 1 (C) ๐๐2
(D) ๐๐2
โ 1
Q.19 Which of the following is a field?
(A) โ[๐ฅ๐ฅ]
โฉ๐ฅ๐ฅ2 + 2โช
๏ฟฝ
(B) โค[๐ฅ๐ฅ]
โฉ๐ฅ๐ฅ2 + 2โช
๏ฟฝ
(C) โ[๐ฅ๐ฅ]
โฉ๐ฅ๐ฅ2 โ 2โช
๏ฟฝ
(D) โ[๐ฅ๐ฅ]
โฉ๐ฅ๐ฅ2 โ 2โช
๏ฟฝ
Q.20 Let๐ฅ๐ฅ0 = 0. Define ๐ฅ๐ฅ๐๐+1 = cos ๐ฅ๐ฅ๐๐ for every ๐๐ โฅ 0. Then
(A) {๐ฅ๐ฅ๐๐ } is increasing and convergent
(B) {๐ฅ๐ฅ๐๐ } is decreasing and convergent
(C) {๐ฅ๐ฅ๐๐ } is convergent and ๐ฅ๐ฅ2๐๐ < lim๐๐โโ ๐ฅ๐ฅ๐๐ < ๐ฅ๐ฅ2๐๐+1 for every ๐๐ โ โ
(D) {๐ฅ๐ฅ๐๐ } is not convergent
Q.21 Let ๐ถ๐ถ be the contour |๐ง๐ง| = 2 oriented in the anti-clockwise direction. The value of the integral
โฎ๐ถ๐ถ ๐ง๐ง๐๐3/๐ง๐ง
๐๐๐๐ is
(A) 3๐๐๐๐ (B) 5๐๐๐๐ (C) 7๐๐๐๐ (D) 9๐๐๐๐
Q.22 For each ๐๐ > 0, let ๐๐๐๐ be a random variable with exponential density ๐๐๐๐โ๐๐๐๐
on(0, โ). Then,
Var(log ๐๐๐๐)
(A) is strictly increasing in ๐๐
(B) is strictly decreasing in ๐๐
(C) does not depend on ๐๐
(D) first increases and then decreases in ๐๐
8. 2013 MATHEMATICS โMA
MA 8/14
Q.23 Let {๐๐๐๐ } be the sequence of consecutive positive solutions of the equation tan ๐ฅ๐ฅ = ๐ฅ๐ฅ and let {๐๐๐๐}
be the sequence of consecutive positive solutions of the equationtan โ๐ฅ๐ฅ = ๐ฅ๐ฅ. Then
(A) โ
1
๐๐๐๐
โ
๐๐=1 converges but โ
1
๐๐๐๐
โ
๐๐=1 diverges (B) โ
1
๐๐๐๐
โ
๐๐=1 diverges but โ
1
๐๐๐๐
โ
๐๐=1 converges
(C) Both โ
1
๐๐๐๐
โ
๐๐=1 and โ
1
๐๐๐๐
โ
๐๐=1 converge (D) Both โ
1
๐๐๐๐
โ
๐๐=1 and โ
1
๐๐๐๐
โ
๐๐=1 diverge
Q.24 Let ๐๐ be an analytic function on๐ท๐ท = {๐ง๐ง โ โ: |๐ง๐ง| โค 1}. Assume that |๐๐(๐ง๐ง)| โค 1 for each ๐ง๐ง โ ๐ท๐ท
๏ฟฝ.
Then, which of the following is NOT a possible value of๏ฟฝ๐๐๐๐
๏ฟฝ
โฒโฒ
(0)?
(A) 2 (B) 6 (C)
7
9
๐๐1/9 (D) โ2 + ๐๐โ2
Q.25 The number of non-isomorphic abelian groups of order 24 is ______
Q. 26 to Q. 55 carry two marks each.
Q.26 Let V be the real vector space of all polynomials in one variable with real coefficients and having
degree at most 20. Define the subspaces
๐๐1 = ๏ฟฝ๐๐ โ ๐๐ โถ ๐๐(1) = 0, ๐๐ ๏ฟฝ
1
2
๏ฟฝ = 0, ๐๐(5) = 0, ๐๐(7) = 0๏ฟฝ,
๐๐2 = ๏ฟฝ๐๐ โ ๐๐ โถ ๐๐ ๏ฟฝ
1
2
๏ฟฝ = 0, ๐๐(3) = 0, ๐๐(4) = 0, ๐๐(7) = 0๏ฟฝ.
Then the dimension of ๐๐1 โฉ ๐๐2 is ______
Q.27 Let ๐๐, ๐๐ โถ [0,1] โ โ be defined by
๐๐(๐ฅ๐ฅ) = ๏ฟฝ๐ฅ๐ฅ if ๐ฅ๐ฅ =
1
๐๐
for ๐๐ โ โ
0 otherwise
and
๐๐(๐ฅ๐ฅ) = ๏ฟฝ
1 if ๐ฅ๐ฅ โ โ โฉ [0,1]
0 otherwise.
Then
(A) Both ๐๐ and ๐๐ are Riemann integrable
(B) ๐๐ is Riemann integrable and ๐๐ is Lebesgue integrable
(C) ๐๐ is Riemann integrable and ๐๐ is Lebesgue integrable
(D) Neither ๐๐ nor ๐๐ is Riemann integrable
Q.28 Consider the following linear programming problem:
Maximize ๐ฅ๐ฅ + 3๐ฆ๐ฆ + 6๐ง๐ง โ ๐ค๐ค
subject to 5๐ฅ๐ฅ + ๐ฆ๐ฆ + 6๐ง๐ง + 7๐ค๐ค โค 20,
6๐ฅ๐ฅ + 2๐ฆ๐ฆ + 2๐ง๐ง + 9๐ค๐ค โค 40,
๐ฅ๐ฅ โฅ 0, ๐ฆ๐ฆ โฅ 0, ๐ง๐ง โฅ 0, ๐ค๐ค โฅ 0.
Then the optimal value is ______
Q.29 Suppose ๐๐ is a real-valued random variable. Which of the following values CANNOTbe attained
by ๐ธ๐ธ[๐๐] and ๐ธ๐ธ[๐๐2], respectively?
(A) 0 and 1 (B) 2 and 3 (C)
1
2
and
1
3
(D) 2 and 5
9. 2013 MATHEMATICS โMA
MA 9/14
Q.30 Which of the following subsets of โ2
is NOT compact?
(A) {(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
โถ โ1 โค ๐ฅ๐ฅ โค 1, ๐ฆ๐ฆ = sin๐ฅ๐ฅ}
(B) {(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
โถ โ1 โค ๐ฆ๐ฆ โค 1, ๐ฆ๐ฆ = ๐ฅ๐ฅ8
โ ๐ฅ๐ฅ3
โ 1}
(C) {(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
โถ ๐ฆ๐ฆ = 0, sin(๐๐โ๐ฅ๐ฅ) = 0}
(D)๏ฟฝ(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
โถ ๐ฅ๐ฅ > 0, ๐ฆ๐ฆ = sin ๏ฟฝ
1
๐ฅ๐ฅ
๏ฟฝ๏ฟฝ โ ๏ฟฝ(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
โถ ๐ฅ๐ฅ > 0, ๐ฆ๐ฆ =
1
๐ฅ๐ฅ
๏ฟฝ
Q.31 Let ๐๐be the real vector space of 2 ร 3 matrices with real entries. Let ๐๐: ๐๐ โ ๐๐ be defined by
๐๐ ๏ฟฝ๏ฟฝ
๐ฅ๐ฅ1 ๐ฅ๐ฅ2 ๐ฅ๐ฅ3
๐ฅ๐ฅ4 ๐ฅ๐ฅ5 ๐ฅ๐ฅ6
๏ฟฝ๏ฟฝ = ๏ฟฝ
โ๐ฅ๐ฅ6 ๐ฅ๐ฅ4 ๐ฅ๐ฅ1
๐ฅ๐ฅ3 ๐ฅ๐ฅ5 ๐ฅ๐ฅ2
๏ฟฝ.
The determinant of ๐๐ is ______
Q.32 Let โ be a Hilbert space and let {๐๐๐๐ โถ ๐๐ โฅ 1} be an orthonormal basis of โ. Suppose ๐๐: โ โ โ
is a bounded linear operator. Which of the following CANNOT be true?
(A) ๐๐(๐๐๐๐ ) = ๐๐1 for all ๐๐ โฅ 1
(B) ๐๐(๐๐๐๐ ) = ๐๐๐๐+1 for all ๐๐ โฅ 1
(C) ๐๐(๐๐๐๐ ) = ๏ฟฝ
๐๐+1
๐๐
๐๐๐๐ for all ๐๐ โฅ 1
(D) ๐๐(๐๐๐๐ ) = ๐๐๐๐โ1 for all ๐๐ โฅ 2 and ๐๐(๐๐1) = 0
Q.33 The value of the limit
lim
๐๐โโ
2โ๐๐2
โ 2โ๐๐2
โ
๐๐=๐๐+1
is
(A) 0 (B) some ๐๐ โ (0,1) (C) 1 (D) โ
Q.34 Let ๐๐: โ{3๐๐} โ โ be defined by ๐๐(๐ง๐ง) =
๐ง๐งโ๐๐
๐๐๐๐+3
. Which of the following statements about ๐๐ is
FALSE?
(A) ๐๐ is conformal on โ{3๐๐}
(B) ๐๐ maps circles in โ{3๐๐} onto circles in โ
(C) All the fixed points of ๐๐ are in the region {๐ง๐ง โ โ โถ Im(๐ง๐ง) > 0}
(D) There is no straight line in โ{3๐๐} which is mapped onto a straight line in โ by ๐๐
Q.35
The matrix ๐ด๐ด = ๏ฟฝ
1 2 0
1 3 1
0 1 3
๏ฟฝ can be decomposed uniquely into the product ๐ด๐ด = ๐ฟ๐ฟ๐ฟ๐ฟ, where
๐ฟ๐ฟ = ๏ฟฝ
1 0 0
๐๐21 1 0
๐๐31 ๐๐32 1
๏ฟฝ and ๐๐ = ๏ฟฝ
๐ข๐ข11 ๐ข๐ข12 ๐ข๐ข13
0 ๐ข๐ข22 ๐ข๐ข23
0 0 ๐ข๐ข33
๏ฟฝ. The solution of the system ๐ฟ๐ฟ๐ฟ๐ฟ = [1 2 2]๐ก๐ก
is
(A) [1 1 1]๐ก๐ก
(B) [1 1 0]๐ก๐ก
(C) [0 1 1]๐ก๐ก
(D) [1 0 1]๐ก๐ก
10. 2013 MATHEMATICS โMA
MA 10/14
Q.36 Let ๐๐ = ๏ฟฝ๐ฅ๐ฅ โ โ โถ ๐ฅ๐ฅ โฅ 0, โ ๐ฅ๐ฅโ๐๐
< โ
โ
๐๐=1 ๏ฟฝ. Then the supremum of ๐๐ is
(A) 1 (B)
1
๐๐
(C) 0 (D) โ
Q.37 The image of the region {๐ง๐ง โ โ โถ Re(๐ง๐ง) > Im(๐ง๐ง) > 0} under the mapping ๐ง๐ง โฆ ๐๐๐ง๐ง2
is
(A) {๐ค๐ค โ โ โถ Re(๐ค๐ค) > 0, Im(๐ค๐ค) > 0} (B){๐ค๐ค โ โ โถ Re(๐ค๐ค) > 0, Im(๐ค๐ค) > 0, |๐ค๐ค| > 1}
(C){๐ค๐ค โ โ โถ |๐ค๐ค| > 1} (D) {๐ค๐ค โ โ โถ Im(๐ค๐ค) > 0, |๐ค๐ค| > 1}
Q.38 Which of the following groups contains a unique normal subgroup of order four?
(A) โค2โจโค4 (B) The dihedral group, ๐ท๐ท4, of order eight
(C) The quaternion group, ๐๐8 (D) โค2โจโค2โจโค2
Q.39 Let ๐ต๐ต be a real symmetric positive-definite ๐๐ ร ๐๐ matrix. Consider the inner product on โ๐๐
defined
by โฉ๐๐, ๐๐โช = ๐๐๐ก๐ก
๐ต๐ต๐ต๐ต. Let ๐ด๐ด be an ๐๐ ร ๐๐ real matrix and let ๐๐: โ๐๐
โ โ๐๐
be the linear operator defined
by ๐๐(๐๐) = ๐ด๐ด๐ด๐ด for all ๐๐ โ โ๐๐
. If ๐๐ is the adjoint of ๐๐,then ๐๐(๐๐) = ๐ถ๐ถ๐ถ๐ถ for all ๐๐ โ โ๐๐
,where ๐ถ๐ถ is
the matrix
(A) ๐ต๐ตโ1
๐ด๐ด๐ก๐ก
๐ต๐ต (B) ๐ต๐ต๐ด๐ด๐ก๐ก
๐ต๐ตโ1
(C) ๐ต๐ตโ1
๐ด๐ด๐ด๐ด (D) ๐ด๐ด๐ก๐ก
Q.40 Let X be an arbitrary random variable that takes values in{0, 1, โฆ , 10}. The minimum and
maximum possible values of the variance of X are
(A) 0 and 30 (B) 1 and 30 (C) 0 and 25 (D) 1 and 25
Q.41 Let ๐๐ be the space of all4 ร 3 matrices with entries in the finite field of three elements. Then the
number of matrices of rank three in ๐๐ is
(A) (34
โ 3)(34
โ 32
)(34
โ 33
)
(B) (34
โ 1)(34
โ 2)(34
โ 3)
(C) (34
โ 1)(34
โ 3)(34
โ 32
)
(D) 34
(34
โ 1)(34
โ 2)
Q.43 Let ๐๐ be a convex region in the plane bounded by straight lines. Let ๐๐ have 7 vertices.
Suppose๐๐(๐ฅ๐ฅ, ๐ฆ๐ฆ) = ๐๐๐๐ + ๐๐๐๐ + ๐๐ has maximum value๐๐ and minimum value๐๐ on ๐๐ and ๐๐ < ๐๐. Let
๐๐ = {๐๐ โถ ๐๐ is a vertex of๐๐ and ๐๐ < ๐๐(๐๐) < ๐๐}. If ๐๐ has ๐๐ elements, then which of the following
statements is TRUE?
(A) ๐๐ cannot be 5 (B) ๐๐ can be 2
(C) ๐๐ cannot be 3 (D) ๐๐ can be 4
Q.44 Which of the following statements are TRUE?
P: If ๐๐ โ ๐ฟ๐ฟ1(โ), then ๐๐ is continuous.
Q: If ๐๐ โ ๐ฟ๐ฟ1(โ) and lim|๐ฅ๐ฅ|โโ ๐๐(๐ฅ๐ฅ) exists, then the limit is zero.
R: If ๐๐ โ ๐ฟ๐ฟ1(โ), then ๐๐ is bounded.
S: If ๐๐ โ ๐ฟ๐ฟ1(โ) is uniformly continuous, then lim|๐ฅ๐ฅ|โโ ๐๐(๐ฅ๐ฅ) exists and equals zero.
(A) Q and S only (B) P and R only (C) P and Q only (D) R and S only
Q.42 Let ๐๐ be a vector space of dimension ๐๐ โฅ 2. Let ๐๐: ๐๐ โ ๐๐ be a linear transformation such that
๐๐๐๐+1
= 0 and ๐๐๐๐
โ 0 for some ๐๐ โฅ 1. Then which of the following is necessarily TRUE?
(A) Rank (๐๐๐๐
) โค Nullity (๐๐๐๐
) (B) trace (๐๐) โ 0
(C) ๐๐ is diagonalizable (D) ๐๐ = ๐๐
11. 2013 MATHEMATICS โMA
MA 11/14
Q.45 Let ๐ข๐ข be a real valued harmonic function on โ. Let ๐๐: โ2
โ โ be defined by
๐๐(๐ฅ๐ฅ, ๐ฆ๐ฆ) = ๏ฟฝ ๐ข๐ข๏ฟฝ๐๐๐๐๐๐
(๐ฅ๐ฅ + ๐๐๐๐)๏ฟฝ sin๐๐ ๐๐๐๐.
2๐๐
0
Which of the following statements is TRUE?
(A) ๐๐ is a harmonic polynomial
(B) ๐๐ is a polynomial but not harmonic
(C) ๐๐ is harmonic but not a polynomial
(D) ๐๐ is neither harmonic nor a polynomial
Q.46 Let ๐๐ = {๐ง๐ง โ โ โถ |๐ง๐ง| = 1} with the induced topology from โ and let ๐๐: [0,2] โ ๐๐ be defined
as ๐๐(๐ก๐ก) = ๐๐2๐๐๐๐๐๐
. Then, which of the following is TRUE?
(A) ๐พ๐พ is closed in [0,2] โน ๐๐(๐พ๐พ)is closed in ๐๐
(B) ๐๐ is open in [0,2] โน ๐๐(๐๐) is open in ๐๐
(C) ๐๐(๐๐) is closed in ๐๐ โน ๐๐ is closed in [0,2]
(D) ๐๐(๐๐) is open in ๐๐ โน ๐๐ is open in [0,2]
Q.47 Assume that all the zeros of the polynomial ๐๐๐๐ ๐ฅ๐ฅ๐๐
+ ๐๐๐๐โ1๐ฅ๐ฅ๐๐โ1
+ โฏ + ๐๐1๐ฅ๐ฅ + ๐๐0 have negative real
parts. If ๐ข๐ข(๐ก๐ก) is any solution to the ordinary differential equation
๐๐๐๐
๐๐๐๐
๐ข๐ข
๐๐๐ก๐ก๐๐
+ ๐๐๐๐โ1
๐๐๐๐โ1
๐ข๐ข
๐๐๐ก๐ก๐๐โ1
+ โฏ + ๐๐1
๐๐๐๐
๐๐๐๐
+ ๐๐0๐ข๐ข = 0,
then lim๐ก๐กโโ ๐ข๐ข(๐ก๐ก) is equal to
(A) 0 (B) 1 (C) ๐๐ โ 1 (D) โ
Common Data Questions
Common Data for Questions 48 and 49:
Let ๐๐00 be the vector space of all complex sequences having finitely many non-zero terms. Equip ๐๐00 with
the inner product โฉ๐ฅ๐ฅ, ๐ฆ๐ฆโช = โ ๐ฅ๐ฅ๐๐ ๐ฆ๐ฆ๐๐
โ
๐๐=1 for all ๐ฅ๐ฅ = (๐ฅ๐ฅ๐๐ ) and ๐ฆ๐ฆ = (๐ฆ๐ฆ๐๐ ) in๐๐00. Define ๐๐: ๐๐00 โ โ by ๐๐(๐ฅ๐ฅ) =
โ
๐ฅ๐ฅ๐๐
๐๐
โ
๐๐=1 . Let ๐๐ be the kernel of ๐๐.
Q.48 Which of the following is FALSE?
(A) ๐๐ is a continuous linear functional
(B) โ๐๐โ โค
๐๐
โ6
(C) There does not exist any ๐ฆ๐ฆ โ ๐๐00 such that ๐๐(๐ฅ๐ฅ) = โฉ๐ฅ๐ฅ, ๐ฆ๐ฆโช for all ๐ฅ๐ฅ โ ๐๐00
(D) ๐๐โฅ
โ {0}
Q.49 Which of the following is FALSE?
(A) ๐๐00 โ ๐๐
(B) ๐๐ is closed
(C) ๐๐00 is not a complete inner product space
(D) ๐๐00 = ๐๐ โ ๐๐โฅ
12. 2013 MATHEMATICS โMA
MA 12/14
Common Data for Questions 50 and 51:
Let ๐๐1, ๐๐2, โฆ , ๐๐๐๐ be an i.i.d. random sample from exponential distribution with mean ๐๐. In other words,
they have density
๐๐(๐ฅ๐ฅ) = ๏ฟฝ
1
๐๐
๐๐โ๐ฅ๐ฅ/๐๐
if ๐ฅ๐ฅ > 0
0 otherwise.
Q.50 Which of the following is NOT an unbiased estimate of ๐๐?
(A) ๐๐1
(B)
1
๐๐โ1
(๐๐2 + ๐๐3 + โฏ + ๐๐๐๐ )
(C) ๐๐ โ (min{๐๐1, ๐๐2, โฆ , ๐๐๐๐ })
(D)
1
๐๐
max{๐๐1, ๐๐2, โฆ , ๐๐๐๐ }
Q.51 Consider the problem of estimating ๐๐. The m.s.e (mean square error) of the estimate
๐๐(๐๐) =
๐๐1 + ๐๐2 + โฏ + ๐๐๐๐
๐๐ + 1
is
(A) ๐๐2
(B)
1
๐๐+1
๐๐2
(C)
1
(๐๐+1)2 ๐๐2
(D)
๐๐2
(๐๐+1)2 ๐๐2
Linked Answer Questions
Statement for Linked Answer Questions 52 and 53:
Let ๐๐ = {(๐ฅ๐ฅ, ๐ฆ๐ฆ) โ โ2
: ๐ฅ๐ฅ2
+ ๐ฆ๐ฆ2
= 1} โช ([โ1,1] ร {0}) โช ({0} ร [โ1,1]).
Let ๐๐0 = max{๐๐ โถ ๐๐ < โ, there are ๐๐ distinct points ๐๐1, โฆ , ๐๐๐๐ โ ๐๐ such that ๐๐ โ {๐๐1, โฆ , ๐๐๐๐} is connected}
Q.52 The value of ๐๐0 is ______
Q.53 Let ๐๐1, โฆ , ๐๐๐๐0+1 be ๐๐0 + 1 distinct points and ๐๐ = ๐๐ โ ๏ฟฝ๐๐1, โฆ , ๐๐๐๐0+1๏ฟฝ. Let ๐๐ be the number of
connected components of ๐๐. The maximum possible value of ๐๐ is ______
Statement for Linked Answer Questions 54 and 55:
Let ๐๐(๐ฆ๐ฆ1, ๐ฆ๐ฆ2) be the Wrรถnskian of two linearly independent solutions๐ฆ๐ฆ1 and ๐ฆ๐ฆ2 of the equation ๐ฆ๐ฆโฒโฒ
+
๐๐(๐ฅ๐ฅ)๐ฆ๐ฆโฒ
+ ๐๐(๐ฅ๐ฅ)๐ฆ๐ฆ = 0.
Q.54 The product ๐๐(๐ฆ๐ฆ1, ๐ฆ๐ฆ2)๐๐(๐ฅ๐ฅ) equals
(A) ๐ฆ๐ฆ2๐ฆ๐ฆ1
โฒโฒ
โ ๐ฆ๐ฆ1๐ฆ๐ฆ2
โฒโฒ
(B) ๐ฆ๐ฆ1๐ฆ๐ฆ2
โฒ
โ ๐ฆ๐ฆ2๐ฆ๐ฆ1
โฒ
(C) ๐ฆ๐ฆ1
โฒ
๐ฆ๐ฆ2
โฒโฒ
โ ๐ฆ๐ฆ2
โฒ
๐ฆ๐ฆ1
โฒโฒ
(D) ๐ฆ๐ฆ2
โฒ
๐ฆ๐ฆ1
โฒ
โ ๐ฆ๐ฆ1
โฒโฒ
๐ฆ๐ฆ2
โฒโฒ
Q.55 If ๐ฆ๐ฆ1 = ๐๐2๐ฅ๐ฅ
and ๐ฆ๐ฆ2 = ๐ฅ๐ฅ๐ฅ๐ฅ2๐ฅ๐ฅ
, then the value of ๐๐(0) is
(A) 4 (B) โ4 (C) 2 (D) โ2
13. 2013 MATHEMATICS โMA
MA 13/14
General Aptitude (GA) Questions
Q. 56 โ Q. 60 carry one mark each.
Q.56 A number is as much greater than 75 as it is smaller than 117. The number is:
(A) 91 (B) 93 (C) 89 (D) 96
Q.57 The professor ordered to the students to go out of the class.
I II III IV
Which of the above underlined parts of the sentence is grammatically incorrect?
(A) I (B) II (C) III (D) IV
Q.58 Which of the following options is the closest in meaning to the word given below:
Primeval
(A) Modern (B) Historic
(C) Primitive (D) Antique
Q.59 Friendship, no matter how _________it is, has its limitations.
(A) cordial
(B) intimate
(C) secret
(D) pleasant
Q.60 Select the pair that best expresses a relationship similar to that expressed in the pair:
Medicine: Health
(A) Science: Experiment (B) Wealth: Peace
(C) Education: Knowledge (D) Money: Happiness
Q. 61 to Q. 65 carry two marks each.
Q.61 X and Y are two positive real numbers such that 2๐๐ + ๐๐ โค 6 and ๐๐ + 2๐๐ โค 8. For which of the
following values of (๐๐, ๐๐) the function ๐๐(๐๐, ๐๐) = 3๐๐ + 6๐๐ will give maximum value?
(A) (4/3, 10/3)
(B) (8/3, 20/3)
(C) (8/3, 10/3)
(D) (4/3, 20/3)
Q.62 If |4๐๐ โ 7| = 5 then the values of 2 |๐๐| โ | โ ๐๐| is:
(A) 2, 1/3 (B) 1/2, 3 (C) 3/2, 9 (D) 2/3, 9
14. 2013 MATHEMATICS โMA
MA 14/14
Q.63 Following table provides figures (in rupees) on annual expenditure of a firm for two years - 2010
and 2011.
Category 2010 2011
Raw material 5200 6240
Power & fuel 7000 9450
Salary & wages 9000 12600
Plant & machinery 20000 25000
Advertising 15000 19500
Research & Development 22000 26400
In 2011, which of the following two categories have registered increase by same percentage?
(A) Raw material and Salary & wages
(B) Salary & wages and Advertising
(C) Power & fuel and Advertising
(D) Raw material and Research & Development
Q.64 A firm is selling its product at Rs. 60 per unit. The total cost of production is Rs. 100 and firm is
earning total profit of Rs. 500. Later, the total cost increased by 30%. By what percentage the price
should be increased to maintained the same profit level.
(A) 5 (B) 10 (C) 15 (D) 30
Q.65 Abhishek is elder to Savar.
Savar is younger to Anshul.
Which of the given conclusions is logically valid and is inferred from the above
statements?
(A) Abhishek is elder to Anshul
(B) Anshul is elder to Abhishek
(C) Abhishek and Anshul are of the same age
(D) No conclusion follows
END OF THE QUESTION PAPER