2013 MATHEMATICS –MA 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 one of the following symbols: You have not visited the question yet. You have not answered the question. You have answered the question. You have NOT answered the question, but have marked the question for review. You have answered the question, but marked it for review. 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. c. Click on Save and Next to save your answer for the current question and then go to the next 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. MA 1/14 2013 MATHEMATICS –MA 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 ‘0’ 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. MA 2/14 2013 MATHEMATICS –MA 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 – Q.25 carry 1mark each. Questions Q.26 – 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 – Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 – Q.60 carry 1mark each, and questions Q.61 – 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. MA 3/14 2013 MATHEMATICS –MA USEFUL DATA FOR MA: MATHEMATICS MA 4/14 2013 MATHEMATICS –MA Q. 1 – Q. 25 carry one mark each. Q.1 The possible set of eigen values of a 4 × 4 skew-symmetric orthogonal real matrix is (B) {±๐๐, ±1} (A) {±๐๐} Q.2 The coefficient of (๐ง๐ง − ๐๐)2 in the Taylor series expansion of sin ๐ง๐ง ๐๐(๐ง๐ง) = ๏ฟฝ ๐ง๐ง − ๐๐ if ๐ง๐ง ≠ ๐๐ −1 if ๐ง๐ง = ๐๐ around ๐๐ is 1 Q.3 1 Q.5 Q.6 1 6 (A) 2 (B) − 2 (C) P:๐ด๐ด ∪ ๐ต๐ต = ๐ด๐ด ∪ ๐ต๐ต. Q:๐ด๐ด ∩ ๐ต๐ต = ๐ด๐ด ∩ ๐ต๐ต. R:(๐ด๐ด ∪ ๐ต๐ต)ห = ๐ด๐ดห ∪ ๐ต๐ตห. S:(๐ด๐ด ∩ ๐ต๐ต)ห = ๐ด๐ดห ∩ ๐ต๐ตห. (B) P and S only (C) Q and R only (D) {0, ±๐๐} 1 (D) − 6 Consider โ2 with the usual topology. Which of the following statements are TRUE for all ๐ด๐ด, ๐ต๐ต ⊆ โ2 ? (A) P and R only Q.4 (C) {±1} (D) Q and S only 3 Let ๐๐: โ → โ be a continuous function with ๐๐(1) = 5 and ๐๐(3) = 11. If ๐๐(๐ฅ๐ฅ) = ∫1 ๐๐(๐ฅ๐ฅ + ๐ก๐ก)๐๐๐๐ then ๐๐′(0) is equal to ______ Let ๐๐ be a 2 × 2 complex matrix such that trace(๐๐) = 1 anddet(๐๐) = −6. Then, trace(๐๐4 − ๐๐3 ) is ______ Suppose that ๐ ๐ is a unique factorization domain and that ๐๐, ๐๐ ∈ ๐ ๐ are distinct irreducible elements. Which of the following statements is TRUE? Q.7 (A) The ideal 〈1 + ๐๐〉 is a prime ideal (B) The ideal 〈๐๐ + ๐๐〉 is a prime ideal (C) The ideal 〈1 + ๐๐๐๐〉 is a prime ideal (D) The ideal 〈๐๐〉 is not necessarily a maximal ideal Q.8 (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 Let ๐๐ be a compact Hausdorff topological space and let ๐๐ be a topological space. Let ๐๐: ๐๐ → ๐๐ be a bijective continuous mapping. Which of the following is TRUE? Consider the linear programming problem: 3 ๐ฅ๐ฅ + ๐ฆ๐ฆ 2 subject to 2๐ฅ๐ฅ + 3๐ฆ๐ฆ ≤ 16, ๐ฅ๐ฅ + 4๐ฆ๐ฆ ≤ 18, ๐ฅ๐ฅ ≥ 0, ๐ฆ๐ฆ ≥ 0. If ๐๐ denotes the set of all solutions of the above problem, then Maximize MA (A) ๐๐ is empty (C) ๐๐ is a line segment (B) ๐๐ is a singleton (D) ๐๐ has positive area 5/14 2013 MATHEMATICS –MA Q.9 Which of the following groups has a proper subgroup that is NOT cyclic? Q.10 (A) โค15 × โค77 (B) ๐๐3 (C) (โค, +) (D) (โ, +) The value of the integral ∞ ∞ 1 ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๐๐ −๐ฆ๐ฆ/2 ๐๐๐๐๐๐๐๐ ๐ฆ๐ฆ 0 ๐ฅ๐ฅ is ______ Q.11 Suppose the random variable ๐๐ has uniform distribution on [0,1] and ๐๐ = −2 log ๐๐. The density of ๐๐ is (A) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ 0 (B) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ ๐๐ −๐ฅ๐ฅ if ๐ฅ๐ฅ > 0 otherwise 2๐๐ −2๐ฅ๐ฅ if ๐ฅ๐ฅ > 0 0 otherwise (C) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ 0 1 −๐ฅ๐ฅ ๐๐ 2 if ๐ฅ๐ฅ 2 Q.12 1/2 (D) ๐๐(๐ฅ๐ฅ) = ๏ฟฝ 0 Q.13 (A) must be 2 (C) must be ๐๐ Q.14 >0 otherwise if ๐ฅ๐ฅ ∈ [0,2] otherwise Let ๐๐ be an entire function on โ such that |๐๐(๐ง๐ง)| ≤ 100 log|๐ง๐ง|for each ๐ง๐ง with|๐ง๐ง| ≥ 2. If ๐๐(๐๐) = 2๐๐, then ๐๐(1) (B) must be 2๐๐ (D) cannot be determined from the given data The number of group homomorphisms from โค3 to โค9 is ______ Let ๐ข๐ข(๐ฅ๐ฅ, ๐ก๐ก) be the solution to the wave equation ๐๐ 2 ๐ข๐ข ๐๐ 2 ๐ข๐ข (๐ฅ๐ฅ, (๐ฅ๐ฅ, ๐ก๐ก), ๐ก๐ก) = ๐๐๐ฅ๐ฅ 2 ๐๐๐ก๐ก 2 Q.15 Then, the value of ๐ข๐ข(1,1) is ______ Let ๐๐(๐ฅ๐ฅ) = ∑∞ ๐๐=1 sin (๐๐๐๐ ) . ๐๐ 2 ๐๐๐๐ (๐ฅ๐ฅ, 0) = 0. ๐๐๐๐ Then (A) lim๐ฅ๐ฅ→0 ๐๐(๐ฅ๐ฅ) = 0 (C) lim๐ฅ๐ฅ→0 ๐๐(๐ฅ๐ฅ) = ๐๐ 2 /6 MA ๐ข๐ข(๐ฅ๐ฅ, 0) = cos(5๐๐๐๐) , (B) lim๐ฅ๐ฅ→0 ๐๐(๐ฅ๐ฅ) = 1 (D) lim๐ฅ๐ฅ→0 ๐๐(๐ฅ๐ฅ) does not exist 6/14 2013 Q.16 Q.17 Q.18 MATHEMATICS –MA Suppose ๐๐ is a random variable with ๐๐(๐๐ = ๐๐) = (1 − ๐๐)๐๐ ๐๐ for ๐๐ ∈ {0,1,2, … } and some ๐๐ ∈ (0,1). For the hypothesis testing problem 1 1 ๐ป๐ป0 : ๐๐ = ๐ป๐ป1 : ๐๐ ≠ 2 2 consider the test “Reject ๐ป๐ป0 if ๐๐ ≤ ๐ด๐ด or if ๐๐ ≥ ๐ต๐ต”, where๐ด๐ด < ๐ต๐ต are given positive integers. The type-I error of this test is (A) 1 + 2−๐ต๐ต − 2−๐ด๐ด (B) 1 − 2−๐ต๐ต + 2−๐ด๐ด (C) 1 + 2−๐ต๐ต − 2−๐ด๐ด−1 (D) 1 − 2−๐ต๐ต + 2−๐ด๐ด−1 Let G be a group of order 231. The number of elements of order 11 in G is ______ Let ๐๐: โ2 → โ2 be defined by ๐๐(๐ฅ๐ฅ, ๐ฆ๐ฆ) = (๐๐ ๐ฅ๐ฅ+๐ฆ๐ฆ , ๐๐ ๐ฅ๐ฅ−๐ฆ๐ฆ ). The area of the image of the region {(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 : 0 < ๐ฅ๐ฅ, ๐ฆ๐ฆ < 1} under the mapping ๐๐ is (B) ๐๐ − 1 (A) 1 Q.19 Which of the following is a field? (A) โ[๐ฅ๐ฅ]๏ฟฝ〈๐ฅ๐ฅ 2 Q.20 (D) ๐๐ 2 − 1 + 2〉 (B) โค[๐ฅ๐ฅ]๏ฟฝ〈๐ฅ๐ฅ 2 + 2〉 (D) โ[๐ฅ๐ฅ]๏ฟฝ〈๐ฅ๐ฅ 2 − 2〉 (C) โ[๐ฅ๐ฅ]๏ฟฝ〈๐ฅ๐ฅ 2 (C) ๐๐ 2 − 2〉 Let๐ฅ๐ฅ0 = 0. Define ๐ฅ๐ฅ๐๐+1 = cos ๐ฅ๐ฅ๐๐ for every ๐๐ ≥ 0. Then (A) {๐ฅ๐ฅ๐๐ } is increasing and convergent (B) {๐ฅ๐ฅ๐๐ } is decreasing and convergent (C) {๐ฅ๐ฅ๐๐ } is convergent and ๐ฅ๐ฅ2๐๐ < lim๐๐ →∞ ๐ฅ๐ฅ๐๐ < ๐ฅ๐ฅ2๐๐+1 for every ๐๐ ∈ 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 ๐ง๐ง๐๐ 3/๐ง๐ง ๐๐๐๐ is (A) 3๐๐๐๐ Q.22 (B) 5๐๐๐๐ (C) 7๐๐๐๐ (D) 9๐๐๐๐ For each ๐๐ > 0, let ๐๐๐๐ be a random variable with exponential density ๐๐๐๐ −๐๐๐๐ on(0, ∞). 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 MA 7/14 2013 Q.23 MATHEMATICS –MA 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 √๐ฅ๐ฅ = ๐ฅ๐ฅ. Then 1 1 ∞ (A) ∑∞ ๐๐=1 ๐๐ converges but ∑๐๐=1 ๐๐ diverges ๐๐ Q.24 Q.25 1 1 ๐๐ ∞ (C) Both ∑∞ ๐๐=1 ๐๐ and ∑๐๐=1 ๐๐ converge ๐๐ ๐๐ 1 1 ∞ (B) ∑∞ ๐๐=1 ๐๐ diverges but ∑๐๐=1 ๐๐ converges ๐๐ 1 1 ๐๐ ∞ (D) Both ∑∞ ๐๐=1 ๐๐ and ∑๐๐=1 ๐๐ diverge ๐๐ ๐๐ ๏ฟฝ. Let ๐๐ be an analytic function on๐ท๐ท = {๐ง๐ง ∈ โ: |๐ง๐ง| ≤ 1}. Assume that |๐๐(๐ง๐ง)| ≤ 1 for each ๐ง๐ง ∈ ๐ท๐ท ๐๐ ′′ Then, which of the following is NOT a possible value of๏ฟฝ๐๐ ๏ฟฝ (0)? (A) 2 (B) 6 (C) 7 1/9 ๐๐ 9 The number of non-isomorphic abelian groups of order 24 is ______ (D) √2 + ๐๐√2 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 1 ๐๐1 = ๏ฟฝ๐๐ ∈ ๐๐ โถ ๐๐(1) = 0, ๐๐ ๏ฟฝ ๏ฟฝ = 0, ๐๐(5) = 0, ๐๐(7) = 0๏ฟฝ, 2 1 ๐๐(3) = 0, ๐๐(4) = 0, ๐๐(7) = 0๏ฟฝ. ๐๐2 = ๏ฟฝ๐๐ ∈ ๐๐ โถ ๐๐ ๏ฟฝ ๏ฟฝ = 0, 2 Then the dimension of ๐๐1 ∩ ๐๐2 is ______ Q.27 Let ๐๐, ๐๐ โถ [0,1] → โ be defined by ๐๐(๐ฅ๐ฅ) = ๏ฟฝ๐ฅ๐ฅ 0 and 1 if ๐ฅ๐ฅ ∈ โ ∩ [0,1] ๐๐(๐ฅ๐ฅ) = ๏ฟฝ 0 otherwise. Then Q.28 (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 Consider the following linear programming problem: Maximize subject to Then the optimal value is ______ Q.29 ๐ฅ๐ฅ + 3๐ฆ๐ฆ + 6๐ง๐ง − ๐ค๐ค 5๐ฅ๐ฅ + ๐ฆ๐ฆ + 6๐ง๐ง + 7๐ค๐ค ≤ 20, 6๐ฅ๐ฅ + 2๐ฆ๐ฆ + 2๐ง๐ง + 9๐ค๐ค ≤ 40, ๐ฅ๐ฅ ≥ 0, ๐ฆ๐ฆ ≥ 0, ๐ง๐ง ≥ 0, ๐ค๐ค ≥ 0. Suppose ๐๐ is a real-valued random variable. Which of the following values CANNOTbe attained by ๐ธ๐ธ[๐๐] and ๐ธ๐ธ[๐๐ 2 ], respectively? (A) 0 and 1 MA 1 for ๐๐ ∈ โ ๐๐ otherwise if ๐ฅ๐ฅ = (B) 2 and 3 1 1 (C) 2and 3 (D) 2 and 5 8/14 2013 Q.30 MATHEMATICS –MA Which of the following subsets of โ2 is NOT compact? (A) {(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 โถ −1 ≤ ๐ฅ๐ฅ ≤ 1, ๐ฆ๐ฆ = sin ๐ฅ๐ฅ} (B) {(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 โถ −1 ≤ ๐ฆ๐ฆ ≤ 1, ๐ฆ๐ฆ = ๐ฅ๐ฅ 8 − ๐ฅ๐ฅ 3 − 1} (C) {(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 โถ ๐ฆ๐ฆ = 0, sin(๐๐ −๐ฅ๐ฅ ) = 0} 1 Q.31 Let ๐๐be the real vector space of 2 × 3 matrices with real entries. Let ๐๐: ๐๐ → ๐๐ be defined by ๐ฅ๐ฅ1 ๐๐ ๏ฟฝ๏ฟฝ๐ฅ๐ฅ Q.32 1 (D)๏ฟฝ(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 โถ ๐ฅ๐ฅ > 0, ๐ฆ๐ฆ = sin ๏ฟฝ๐ฅ๐ฅ ๏ฟฝ๏ฟฝ โ ๏ฟฝ(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 โถ ๐ฅ๐ฅ > 0, ๐ฆ๐ฆ = ๐ฅ๐ฅ ๏ฟฝ The determinant of ๐๐ is ______ 4 ๐ฅ๐ฅ2 ๐ฅ๐ฅ5 ๐ฅ๐ฅ3 −๐ฅ๐ฅ6 ๐ฅ๐ฅ6 ๏ฟฝ๏ฟฝ = ๏ฟฝ ๐ฅ๐ฅ3 ๐ฅ๐ฅ4 ๐ฅ๐ฅ5 ๐ฅ๐ฅ1 ๐ฅ๐ฅ2 ๏ฟฝ. Let โ be a Hilbert space and let {๐๐๐๐ โถ ๐๐ ≥ 1} be an orthonormal basis of โ. Suppose ๐๐: โ → โ is a bounded linear operator. Which of the following CANNOT be true? (A) ๐๐(๐๐๐๐ ) = ๐๐1 for all ๐๐ ≥ 1 (B) ๐๐(๐๐๐๐ ) = ๐๐๐๐+1 for all ๐๐ ≥ 1 ๐๐+1 ๐๐๐๐ ๐๐ (C) ๐๐(๐๐๐๐ ) = ๏ฟฝ Q.33 for all ๐๐ ≥ 1 (D) ๐๐(๐๐๐๐ ) = ๐๐๐๐−1 for all ๐๐ ≥ 2 and ๐๐(๐๐1 ) = 0 The value of the limit lim ๐๐→∞ ∑∞ ๐๐=๐๐+1 is (B) some ๐๐ ∈ (0,1) (A) 0 Q.34 Q.35 2 (C) 1 2−๐๐ 2 ๐ง๐ง−๐๐ (D) ∞ Let ๐๐: โ\{3๐๐} → โ be defined by ๐๐(๐ง๐ง) = 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 โถ Im(๐ง๐ง) > 0} (D) There is no straight line in โ\{3๐๐} which is mapped onto a straight line in โ by ๐๐ 1 2 0 The matrix ๐ด๐ด = ๏ฟฝ1 3 1๏ฟฝ can be decomposed uniquely into the product ๐ด๐ด = ๐ฟ๐ฟ๐ฟ๐ฟ, where 0 1 3 ๐ข๐ข11 ๐ข๐ข12 ๐ข๐ข13 1 0 0 ๐ฟ๐ฟ = ๏ฟฝ๐๐21 1 0๏ฟฝ and ๐๐ = ๏ฟฝ 0 ๐ข๐ข22 ๐ข๐ข23 ๏ฟฝ. The solution of the system ๐ฟ๐ฟ๐ฟ๐ฟ = [1 2 2]๐ก๐ก is 0 0 ๐ข๐ข33 ๐๐31 ๐๐32 1 (A) [1 MA 2−๐๐ 1 1]๐ก๐ก (B) [1 1 0]๐ก๐ก (C) [0 1 1]๐ก๐ก (D) [1 0 1]๐ก๐ก 9/14 2013 Q.36 MATHEMATICS –MA √๐๐ < ∞๏ฟฝ. Then the supremum of ๐๐ is Let ๐๐ = ๏ฟฝ๐ฅ๐ฅ ∈ โ โถ ๐ฅ๐ฅ ≥ 0, ∑∞ ๐๐=1 ๐ฅ๐ฅ (A) 1 Q.37 1 (B) ๐๐ (D) ∞ 2 The image of the region {๐ง๐ง ∈ โ โถ Re(๐ง๐ง) > Im(๐ง๐ง) > 0} under the mapping ๐ง๐ง โฆ ๐๐ ๐ง๐ง is (A) {๐ค๐ค ∈ โ โถ Re(๐ค๐ค) > 0, Im(๐ค๐ค) > 0} Q.38 (C) 0 (C){๐ค๐ค ∈ โ โถ |๐ค๐ค| > 1} (B){๐ค๐ค ∈ โ โถ Re(๐ค๐ค) > 0, Im(๐ค๐ค) > 0, |๐ค๐ค| > 1} (D) {๐ค๐ค ∈ โ โถ Im(๐ค๐ค) > 0, |๐ค๐ค| > 1} Which of the following groups contains a unique normal subgroup of order four? (A) โค2 โจโค4 (B) The dihedral group, ๐ท๐ท4 , of order eight Q.39 (C) The quaternion group, ๐๐8 Q.40 (A) ๐ต๐ต−1 ๐ด๐ด๐ก๐ก ๐ต๐ต (B) ๐ต๐ต๐ด๐ด๐ก๐ก ๐ต๐ต−1 (C) ๐ต๐ต−1 ๐ด๐ด๐ด๐ด (D) ๐ด๐ด๐ก๐ก (A) 0 and 30 (B) 1 and 30 (C) 0 and 25 (D) 1 and 25 Q.41 Let ๐ต๐ต be a real symmetric positive-definite ๐๐ × ๐๐ matrix. Consider the inner product on โ๐๐ defined by 〈๐๐, ๐๐〉 = ๐๐ ๐ก๐ก ๐ต๐ต๐ต๐ต. Let ๐ด๐ด be an ๐๐ × ๐๐ real matrix and let ๐๐: โ๐๐ → โ๐๐ be the linear operator defined by ๐๐(๐๐) = ๐ด๐ด๐ด๐ด for all ๐๐ ∈ โ๐๐ . If ๐๐ is the adjoint of ๐๐,then ๐๐(๐๐) = ๐ถ๐ถ๐ถ๐ถ for all ๐๐ ∈ โ๐๐ ,where ๐ถ๐ถ is the matrix Let X be an arbitrary random variable that takes values in{0, 1, … , 10}. The minimum and maximum possible values of the variance of X are Let ๐๐ be the space of all4 × 3 matrices with entries in the finite field of three elements. Then the number of matrices of rank three in ๐๐ is (A) (B) (C) (D) Q.42 (D) โค2 โจโค2 โจโค2 (34 − 3)(34 − 32 )(34 − 33 ) (34 − 1)(34 − 2)(34 − 3) (34 − 1)(34 − 3)(34 − 32 ) 34 (34 − 1)(34 − 2) Let ๐๐ be a vector space of dimension ๐๐ ≥ 2. Let ๐๐: ๐๐ → ๐๐ be a linear transformation such that ๐๐ ๐๐+1 = 0 and ๐๐ ๐๐ ≠ 0 for some ๐๐ ≥ 1. Then which of the following is necessarily TRUE? Q.43 (A) Rank (๐๐ ๐๐ ) ≤ Nullity (๐๐ ๐๐ ) (C) ๐๐ is diagonalizable (B) trace (๐๐) ≠ 0 (D) ๐๐ = ๐๐ Q.44 (A) ๐๐ cannot be 5 (C) ๐๐ cannot be 3 (B) ๐๐ can be 2 (D) ๐๐ can be 4 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? Which of the following statements are TRUE? P: If ๐๐ ∈ ๐ฟ๐ฟ1 (โ), then ๐๐ is continuous. Q: If ๐๐ ∈ ๐ฟ๐ฟ1 (โ) and lim|๐ฅ๐ฅ|→∞ ๐๐(๐ฅ๐ฅ) exists, then the limit is zero. R: If ๐๐ ∈ ๐ฟ๐ฟ1 (โ), then ๐๐ is bounded. S: If ๐๐ ∈ ๐ฟ๐ฟ1 (โ) is uniformly continuous, then lim|๐ฅ๐ฅ|→∞ ๐๐(๐ฅ๐ฅ) exists and equals zero. (A) Q and S only MA (B) P and R only (C) P and Q only (D) R and S only 10/14 2013 Q.45 MATHEMATICS –MA Let ๐ข๐ข be a real valued harmonic function on โ. Let ๐๐: โ2 → โ be defined by 2๐๐ ๐๐(๐ฅ๐ฅ, ๐ฆ๐ฆ) = ๏ฟฝ ๐ข๐ข๏ฟฝ๐๐ ๐๐๐๐ (๐ฅ๐ฅ + ๐๐๐๐)๏ฟฝ sin ๐๐ ๐๐๐๐. 0 Which of the following statements is TRUE? Q.46 (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 Let ๐๐ = {๐ง๐ง ∈ โ โถ |๐ง๐ง| = 1} with the induced topology from โ and let ๐๐: [0,2] → ๐๐ be defined as ๐๐(๐ก๐ก) = ๐๐ 2๐๐๐๐๐๐ . Then, which of the following is TRUE? (A) (B) (C) (D) Q.47 ๐พ๐พ is closed in [0,2] โน ๐๐(๐พ๐พ)is closed in ๐๐ ๐๐ is open in [0,2] โน ๐๐(๐๐) is open in ๐๐ ๐๐(๐๐) is closed in ๐๐ โน ๐๐ is closed in [0,2] ๐๐(๐๐) is open in ๐๐ โน ๐๐ is open in [0,2] Assume that all the zeros of the polynomial ๐๐๐๐ ๐ฅ๐ฅ ๐๐ + ๐๐๐๐−1 ๐ฅ๐ฅ ๐๐−1 + โฏ + ๐๐1 ๐ฅ๐ฅ + ๐๐0 have negative real parts. If ๐ข๐ข(๐ก๐ก) is any solution to the ordinary differential equation ๐๐๐๐ then lim๐ก๐ก→∞ ๐ข๐ข(๐ก๐ก) is equal to (A) 0 Common Data Questions ๐๐๐๐ ๐ข๐ข ๐๐๐๐−1 ๐ข๐ข ๐๐๐๐ + ๐๐ + โฏ + ๐๐1 + ๐๐0 ๐ข๐ข = 0, ๐๐−1 ๐๐ ๐๐−1 ๐๐๐ก๐ก ๐๐๐ก๐ก ๐๐๐๐ (B) 1 (C) ๐๐ − 1 (D) ∞ 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 〈๐ฅ๐ฅ, ๐ฆ๐ฆ〉 = ∑∞ ๐๐=1 ๐ฅ๐ฅ๐๐ ๐ฆ๐ฆ๐๐ for all ๐ฅ๐ฅ = (๐ฅ๐ฅ๐๐ ) and ๐ฆ๐ฆ = (๐ฆ๐ฆ๐๐ ) in๐๐00 . Define ๐๐: ๐๐00 → โ by ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ ๐๐ ∞ ∑๐๐=1 . Let ๐๐ be the kernel of ๐๐. Q.48 ๐๐ Which of the following is FALSE? (A) ๐๐ is a continuous linear functional ๐๐ (B) โ๐๐โ ≤ 6 √ Q.49 (C) There does not exist any ๐ฆ๐ฆ ∈ ๐๐00 such that ๐๐(๐ฅ๐ฅ) = 〈๐ฅ๐ฅ, ๐ฆ๐ฆ〉 for all ๐ฅ๐ฅ ∈ ๐๐00 (D) ๐๐ ⊥ ≠ {0} Which of the following is FALSE? (A) ๐๐00 ≠ ๐๐ (B) ๐๐ is closed (C) ๐๐00 is not a complete inner product space (D) ๐๐00 = ๐๐ ⊕ ๐๐ ⊥ MA 11/14 2013 MATHEMATICS –MA Common Data for Questions 50 and 51: Let ๐๐1 , ๐๐2 , … , ๐๐๐๐ be an i.i.d. random sample from exponential distribution with mean ๐๐. In other words, they have density 1 −๐ฅ๐ฅ/๐๐ ๐๐ if ๐ฅ๐ฅ > 0 ๐๐(๐ฅ๐ฅ) = ๏ฟฝ ๐๐ 0 otherwise. Q.50 Q.51 Which of the following is NOT an unbiased estimate of ๐๐? (A) ๐๐1 1 (B) ๐๐−1 (๐๐2 + ๐๐3 + โฏ + ๐๐๐๐ ) (C) ๐๐ ⋅ (min{๐๐1 , ๐๐2 , … , ๐๐๐๐ }) 1 (D) ๐๐ max{๐๐1 , ๐๐2 , … , ๐๐๐๐ } Consider the problem of estimating ๐๐. The m.s.e (mean square error) of the estimate is (A) ๐๐2 1 (B) ๐๐+1 ๐๐2 ๐๐(๐๐) = ๐๐1 + ๐๐2 + โฏ + ๐๐๐๐ ๐๐ + 1 (C) 1 ๐๐2 (๐๐+1)2 (D) ๐๐ 2 ๐๐2 (๐๐+1)2 Linked Answer Questions Statement for Linked Answer Questions 52 and 53: Let ๐๐ = {(๐ฅ๐ฅ, ๐ฆ๐ฆ) ∈ โ2 : ๐ฅ๐ฅ 2 + ๐ฆ๐ฆ 2 = 1} ∪ ([−1,1] × {0}) ∪ ({0} × [−1,1]). Let ๐๐0 = max{๐๐ โถ ๐๐ < ∞, there are ๐๐ distinct points ๐๐1 , … , ๐๐๐๐ ∈ ๐๐ such that ๐๐ โ {๐๐1 , … , ๐๐๐๐ } is connected} Q.52 The value of ๐๐0 is ______ Q.53 Let ๐๐1 , … , ๐๐๐๐ 0 +1 be ๐๐0 + 1 distinct points and ๐๐ = ๐๐ โ ๏ฟฝ๐๐1 , … , ๐๐๐๐ 0 +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 Wrönskian of two linearly independent solutions๐ฆ๐ฆ1 and ๐ฆ๐ฆ2 of the equation ๐ฆ๐ฆ ′′ + ๐๐(๐ฅ๐ฅ)๐ฆ๐ฆ ′ + ๐๐(๐ฅ๐ฅ)๐ฆ๐ฆ = 0. Q.54 The product ๐๐(๐ฆ๐ฆ1 , ๐ฆ๐ฆ2 )๐๐(๐ฅ๐ฅ) equals (A) ๐ฆ๐ฆ2 ๐ฆ๐ฆ1′′ − ๐ฆ๐ฆ1 ๐ฆ๐ฆ2′′ Q.55 (C) ๐ฆ๐ฆ1′ ๐ฆ๐ฆ2′′ − ๐ฆ๐ฆ2′ ๐ฆ๐ฆ1′′ (D) ๐ฆ๐ฆ2′ ๐ฆ๐ฆ1′ − ๐ฆ๐ฆ1′′ ๐ฆ๐ฆ2′′ If ๐ฆ๐ฆ1 = ๐๐ 2๐ฅ๐ฅ and ๐ฆ๐ฆ2 = ๐ฅ๐ฅ๐ฅ๐ฅ 2๐ฅ๐ฅ , then the value of ๐๐(0) is (A) 4 MA (B) ๐ฆ๐ฆ1 ๐ฆ๐ฆ2′ − ๐ฆ๐ฆ2 ๐ฆ๐ฆ1′ (B) −4 (C) 2 (D) −2 12/14 2013 MATHEMATICS –MA General Aptitude (GA) Questions Q. 56 – 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 Q.57 (C) 89 (D) 96 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 Q.58 (B) 93 (B) II (C) III (D) IV Which of the following options is the closest in meaning to the word given below: Primeval (A) Modern (C) Primitive Q.59 (B) Historic (D) Antique 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 (C) Education: Knowledge (B) Wealth: Peace (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 ≤ 6 and ๐๐ + 2๐๐ ≤ 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 − 7| = 5 then the values of 2 |๐๐| − | − ๐๐| is: (A) 2, 1/3 MA (B) 1/2, 3 (C) 3/2, 9 (D) 2/3, 9 13/14 2013 Q.63 MATHEMATICS –MA 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 Q.65 (B) 10 (C) 15 (D) 30 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 MA 14/14