Delhi School of Economics Entrance Exam (2012)

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This is the entrance exam paper for the Delhi School of Economics for the year 2012. It contains both options A and B. Exam papers for other years are available as well here. Much more information on the DSE Entrance Exam and DSE Entrance preparation help available on www.crackdse.com

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Delhi School of Economics Entrance Exam (2012)

  1. 1. Entrance Examination for NI. A. Economics, 2012 Option A (Series 01) Time. 3 hours Maximum marks. 100 General instructions. Please read the following instructions carefully. 0 Check that you have a bubbl‘e'—‘s‘h'eétaeconipaiiying this examination book- let. Do not break the seal on this booklet until instructed to do so by the invigilator. s Immediately on receipt of this booklet, fill in your Signature, Name, Roll number and Booklet number (see the top left—hand-side of the bubb1e—sheet) in the space provided below. _ o This examination will be checked by a machine. Therefore, it is very important that you follow the instructions on the bubb1e—sheet. 0 Fill in the required information in Boxes 1, 2, 4, 5 and 6 on the bubble—sheet. The invigilator will sign in Box 3. a Do not disturb your neighbours at any time. o 1/ lake sure you do not have calculators, mobile telephones, papers, books, etc. , on your person. Anyone engaging in illegal examination practices will be immediately evicted and that person’s candidature will be cancelled. ‘ 0 When you finish the examination, hand in this booklet and the bubble- sheet to the invigilator. Signature Name Roll number Booklet number
  2. 2. EEE 2012 A 01 1 Before you start 0 Check that this booklet has pages 1 through 27. Also check that the top of each page is marked with EEE 2012 A 01. Bring any inconsistency to the attention of the invigilator. , 0 You may use the blank pages at the end of this booklet, marked Rough work, to do your calculations and drawings. No ot_h_e_r_p_aper, _wi_ll, b,e provided for this purpose. Your “Rough work” will not be read or checked. You may begin now. Enjoy! ___m_ Part I o This part of the examination consists of 20 multiple—choice-"questions. Each question is followed by four possible aiiswers, at least one of which is correct. If more than one choice is correct, choose only the ‘best one’. Among the correct answers, the ‘best answer’ is the one that implies (or includes) the other correct answer(s). Indicate your chosen best answer on the bubble-sheet by shading the appropriate bubble. o For each question, you will get 1 mark if you choose only the best answer. If you choose none of the answers, then you will get 0 for that question. However, if you choose something other than the best answer or multiple answers, then you will get —1/3 mark for that question. QUESTION 1. A rural landowner can deposit his savings in a commercial bank and receive an annual interest rate of 8%. Alternatively he can lend to villagers who need credit. If all loans are of the same size and only sixty per cent of them are repaid, the interest rate that'would make his earnings the same as from depositing his savings in a bank is (a) 8% (b) 48% (c) 80% (d) 120% . Answer: (a) _. *iSkyso‘ft ‘
  3. 3. I583 2012 A 01 2 QUESTION 2. Consider a t. wo—person two-good exchange economy, where agents are denoted by A, B and goods are denoted by X, Y. A Pareto optimal . allocation of this economy may not remain Pareto optimal if (a) Everything else remaining the same, Agent A transfers a part of her endowment to Agent B (b) Everything else remaining the same, Agent A gets additional endow- ment . .. - . . . .. . (c) Everything else remaining the same, Agent A’s utility function is mono- tonically transformed (d) All of the above Answer: (b) QUESTION 3. Given the same production function and market demand, :1 monopolist earns at least as much profit as a competitive firm in the short run, because (a) Monopolist is free to charge the competitive market price (b) Short run profit of a competitive firm is zero (c) Marginal cost of a monopolist is smaller than that of a competitive firm . . (d) Average cost of a monopolist is smaller than that of a competitive firm Answer: (a) QUESTION 4. Mr. B thinks cheese is addictive — the more you eat, the more you want. Suppose 2: denotes the quantity of cheese. Mr. B’s utility function can be represented by (3) u($. y) -= $2 + y (b) u(x, y) = lnx + lny (C) “(-'5»? /) = 90 + y (d) u(= I=, 3/) = minfmy 1/} Answer: (a) QUESTION 5. Sania and Saina are bargaining over how to split 10 Rupees. Both claimants simultaneously name shares they would like to have, 31 and 32, where 0 3 s1,s2 3 10. If 31 + .92 S 10 then the claimants receive the ’ 5 K31”
  4. 4. EEE 2012 A 01 3 shares they named; otherwise both receive zero. Find all pure strategy Nash equilibria of this game (a) S1=5,S2 = 5 {(31,532) I 31 + 82 = (c) {(s1,sg) I 31 + 82 3 10} (d) There is no pure strategy Nash equilibrium Answer: (b) QUESTION 6. “Developing new antibiotics is"e>zp‘e‘rrs‘ive. "I‘t is also known that the more frequently bacteria are exposed to antibiotics, the more quickly the bacteria will develop resistance to the antibiotics. Yet, usually, each parent will press a doctor for an antibiotic if there is any chance it’ will heal a child quicker than that without drugs. ” _ The above statement can be best understood as a problem of (a) Public good: Bacteria are public bad (b) Adverse selection: Doctors have information that is not available to parents (c) Risk aversion: Parents are risk averse (d) Short—run utility maximization: Most parents are myopic; they fail to see the long run effect Answer: (a) QUESTION 7. If the correlation between variables X and Y is 0, then (a) The regressions of Y on X and X on Y intersect at right angles, and pass tlnongli (X, (b) The regressions of Y on X and X on Y do not intersect at right angles, but do pass through (X, Y) (c) The regressions of Y on X and X on Y intersect at right angles, but do not pass through (X, Y) 1 (d) The regicssiuiis of Y on X and X on Y do not intersect at right angles, and do not pass through (X, Y) Answer: (a)
  5. 5. EEE 2012 A 01 .1 QUESTION 8. One reason why the sample median is used as an estimator of the population mean is that (a) The average of all sample medians equals the population mean (b) The sample median equals the population mean (C) The sample median . is unaffected by extreme values (d) The sample median occurs more often than the mode or the mean _I Answer: QUESTION 9. Suppose X is a random variable, which follows U rm’ f orm[= «-171]. » Find the covariance between X and X 2 (a) 1 (b) ‘/4 (C) ‘/8 (d) 0 Answer: (d) QUESTION 10. Which of the following statements is not an indicator that multicollinearity among two or more variables is present in a multiple regression model estimated using OLS: (a) Small changes to the data can cause large changes to estimated coef- ficients (b) Coefficients are estimated with bias (c) The estimated coefficients have large standard errors even though the (d) A test ofjoint significance of two or more coefficients is significant even though individually they are not significant Answer: (b) QUESTION 11. Given the linear regression Y 2 a + flX; a very high correlation between variables X and Y necessarily implies that (a_) The slope coefficient B is statistically significant (B) The observations (1:, y) lie along a straight line (c) Small changes in X cause large changes in Y 7 S3?’
  6. 6. BEE 2012 A 01 01 (d) The regression line is steep Answer: (b) QUESTION 12. What would happen (other things being equal) to a confi- dence interval if you calculated a 99% confidence interval rather than a 95% confidence interval? (a) It will be narrower (b) It will not change (c) The . sample size will increase (d) It will become wider Answer: (d) QUESTION 13. An n — gon, is a regular polygon with n equal sides. Find the number of diagonals (edges of an n — gon are not considered as diagonals) of a 10 — gen. " (a) 20 diagonals (b) 25 diagonals (C) 35 ‘diagonals (d) 45 diagonals Answer: (e) QUESTION 14. f01a: "sin(a: )da: (a) Does not exist (b) Is necessarily greater than 1 - (c) Is greater than 1/(n + 1) (d) Is less than 1/(n +1) V Answer: (d) QUESTION 15. lim, ,_, oo (, /(n — 1) — /77,) (a) Equals 1 (b) Equals 0 (c) Does not exist (d) Depends on n iskysoft
  7. 7. EEE ‘.7012 A 01 6 Answer: (b) QUESTION 16. The equation 3:7 = :2: + 1 (a) Has no real solution (b) Has no positive real solution (c) Has a real solution in the interval (0,2) (d) Hasa-real--solution: but not within (0, 2) Answer: (c) QUESTION 17. The cumulative distribution function F(2:) of a random variable has a slope of 1 for :5 in the interval [0, 1] and takes a constant value thereafter. Which of the following statements most accurately defines the probability density function of X 7 (a) It is zero in the interval [0, 1] and 1 for all higher values of 3; (b) It is 1 in the interval [0, 1] and zero for all higher values of fl: (c) It is increasing in the interval [0, 1] and constant and positive for all higher values of x (d) It is increasing in the interval [0, 1] and zero for all higher values of :1: Answer: QUESTION 18. A and B are two non empty sets. A—B= {:z: eA[: r¢B} and A+B= i(A . - B)LJ(B—A) Consider the following statements; Statement 1: A + B = B implies A Q B Statement 2: A + B = 0 implies A = B Statement 3: A + B = A U B implies A D B = (fl How many of the above statements are correct? (a) 0 (b) 1 (c) 2 id) 3 Answer: (d) 33‘?
  8. 8. EEE 2012 A 01 7 QUESTION 19. The short—run aggregate supply curve is upward sloping because (a) A lower price level creates a wealth effect (b) Lower taxes motivate people to work [more (C) Money wages do not immediately change when the price level changes (d) Most business firms operate with long-term contracts for output but not labour Answer; (c) QUESTION 20. The investment demand curve shifts rightward if (a) The expected profit rate_ingreases (b) The real interest rate falls . (C) Savers increase their thriftiness (d) The economy moves into a recession Answer: (a) End of Part I. . . Proceed to Part II of the examination on the next page. .9 iskysoft 10
  9. 9. EEE 2012 A 01 8 _ Part II ‘ o This part of the examination consists of 40 multiple-choice questions. Each question is followed by four possible answers, a. t least one of which is correct. If more than one choice is correct, choose only the ‘best one’. Among the correct answers, the ‘best answer’ is the one that jmplies (or includes")"tf1e"'othér"correct answer(s). Indicate your chosen best answer on the bubble-sheet by shading the appropriate bubble. 3 For each question, you will get choose only the best answer. If you choose none of the answers, then you will get 0 for that question. However, if you choose something other than the best answer or multiple answers, then you will get ——2/3 mark for that question. ‘ . C o The following notational conventions apply wherever the following sym- bols are used. ER denotes the set of real numbers. _, §R++ denotes the set of positive real numbers. 8?" denotes the n—dimensional vector space. QUESTION 21. A firm producing hockey sticks in Punjab has a production function ‘given by Q = 2/ K L where K stands for capital, L stands for labour and Q stands for output. The rental rate of capital is‘Rs 1 and the wage rate ’ is Rs 4. What will be the firm’s cost function? ' (a) 2Q (b) Q (0) Q” (d) 2Q2 Answer: (a) QUESTION 22. Antony and Cleopatra run a food stall together. Their A--joint profit is 10 Rupees per month, from vi1ic_h, Antony gets 10,; Rupees and Cleopatra gets Io Rupees. Utility functions of Antony and Cleopatra are UA = 3/IAIC and U0 = 10, respectively. Consider the following divisions, Division 1: IA : 7,10 = 3, Division 2: IA = 3, [C = 7 (a) Both the divisions are Pareto optimal " k" ft *5 yso H 3,“
  10. 10. EEE 2012 A 01 9 (b) None of the divisions is Pareto optimal (c) Only Division 1 is Pareto optimal (d) Only Division 2 is Pareto optimal Answer: (d) “QUESTION 23. Consider a homogenous goods market with the demand function Q = 30 — P, where Q and P denote quantity and price respectively. There are two firms playing a price game in the following manner: firm 1 rice and then firm 2 chooses a price. When they charge the same price they share the market equally and otherwise the market demand goes to the firm charging lower price. Firm 1 has a capacity constraint at the output level 5 units such that upto five units the marginal cost of production is Rs 3 per unit of output, however beyond 5 units it cannot produce any output. Firm 2 does not have any capacity constraint, it can produce any amount with the marginal cost Rs 6. What would be the equilibrium price in the market? (a) 3 (b) 5 (c) 6f 5, where e is very small positive number (d) 3+6, where e is very small positive number Answer: (b) QUESTION 24. Consider the following two—player game. The players si- multaneously draw one sample each from a continuous random variable X, which follows Uniform [0, 100]. After observing the value of her own sample, which is private information (that is, opponent does not observe it), play- ers simultaneously and independently choose one of the following: SWAP, RETAIN. If both the players choose SWAP then they exchange their ini- tially drawn nu_mbers. Otherwise, if at least one person chooses RETAIN, both of them retain their numbers. A player earns as many Rupees-as the number she is holding at the end of the game. Find the probability that the players will exchange their initially drawn num- bers »&ia‘)_=1»kysoft '12
  11. 11. .- . ..»~; c.r. v.. ... ,.. .2.-4-. :»: . EEE 2012 A 01 (b) ‘/2 (C) ‘/8 (d) 0 Answer: 10 QUESTION 25. The productivity of a labourer depends on his daily wage. In some range of wage, the more he is more work he is able to do in a given between productivity. and daily wage is as follows. No work is done for wage below Rs. 20 per day. Each rupee earned above 20 increases productivity by 5 units until the daily wage is Rs. 140 per day. Beyond this level of wage, productivity is constant. If a ‘farmer needs to hire labour for a total of 6000 units of work per‘ day, how many labourers is heflikely to hire? (a) 4 (b) 5 (C) 6 (d) 10 Answer: (d) QUESTION 26. Consider a society in which half the population earns 100 rupees per day and" the other half earns 200 rupees per day. The Gini ' QUESTION 27. A and B live in an exchange economy. There are two goods X and Y. utility functions of A and B are U"(: z:, y) = :3 + gij and UB(a: ,y) = :52 + y2, respectively. A’s endowment is 2 units of X and 1 unit of Y, while B’s endowment is 1 unit of X and 2 units of Y. Consider the following allocations Allocation 1: A gets 1.5 units of X and zero unit of Y, B gets 1.5 units of X iskysoft Z M 13 _ , paid, the better his health andithei I
  12. 12. EEE 2012 A 01 11 and 3 units of Y Allocation 2: A gets 1.5 units of X and 1.5 unit of Y, B gets 1.5 units of X and 1.5 units of Y (a) Both the allocations are Pareto optimal (b) None of the allocations is Pareto optimal (6) Only allocation 1 is Pareto optimal (d) Only allocation 2 is Pareto optimal Answer: (c) QUESTION 28. ‘Fill in the blanks: Given a downward sloping linear de- mand curve and constant marginal cost curves, if a per unit tax is imposed on a monopoly, a monopoly will the quantity of its good and its revenue after tax will H (a) Increase; Decrease (b) Increase; Increase (c) Decrease; Increase (d) Decrease; Decrease Answer: (d) QUESTION 29. The demandcurve for electricity is D(p) = 120 — p. The marginal cost of electricity production is ll/ [C1‘(q) = 20 + q. The marginal cost of pollution due to electricity production is MC2(q) : 3q. Find the competitive equilibrium output and the social optimum output. (a) 50; 20 (b) 50; 30 (c) 60; 20 (d) 60; 30 Answer: (a) QUESTION 30. Suppose that there are two agents in an economy with income 1:1 and 152. If the richer person transfers a portion of her income to. the poorer person without changing . the income ranking (that is the rich remains richer even after the transfer) then it is Called a progressive transfer. Welfare of this economy is measured by a function W(:1:1, :52). W is a ‘good’ 14
  13. 13. EEE 2012 A 01 12 measure of social welfare if the social welfare increases due to a progressive transfer. Consider the following candidates for W; W1 (5121, 12) = 21:1 + :1:2 I/ V2(: I:1,2:2) = rc1:z:2 (a) Only W1 is a ‘good’ measure (b) Only W2 is a ‘good’ measure (c) Both W1 and I/ V2 are ‘good’ measures (d) Neither W1 nor W2 is a ‘good’ measure A‘rrswer': ‘"(b')' QUESTION 31. Consider the following model, estimated using OLS ‘Y, =[fX, +€, ; i:1,'2,: .., n where there is no intercept, and'Var(e,7) : 02. Which of the following state— ments is not true? (a) The R2 from this regression can be large even if X and Y have low correlation. (b) The least squares residuals need not sum to zero. (C) The mean square error is given by Ej(Y, - — ~ 1). ((1) The least squares estimator of the slope coefficient is given by: nflxiyi *- ZXiXYr "XX? ‘ (ZXi)2 Answer: (d) QUESTION 32. An analyst trying to estimate the demand for rice has ' estimated the following two models Model 1: D = 50 + 0.3)’ + 0.1P + 12N; R2 = 0.7. Where D is the demand for rice per household, Y is income per household, P is the price of rice and N is household size. The standard errors associated with the coefficient Y is 0.1, that associated with P is 0.05, and of N is 3. Model 2: D/ N = 50 + 0.2Y/ N — 0".5P; R2 = 0.9 where the standard errors of the estimated slope coefficients are‘0.1 and 0.2 respectively. Which of the following statements is true? 3 M5’ . - 15
  14. 14. EEE 2012 A 01 13 (a) Model 2 is preferred to Model 1 because it has a higher R2. (b) Model 1 is preferred to Model 2 because the coefficients are all signif- icant (c) The two models are not comparable in terms of fit (d) None of the above Answer: (c) QUE§TIQN 33. Consider the regressions Y, -‘ = 31* + fl~2*XZ-* + 12,-’, and Y, ~ = ,3; + fi2X, - + ‘L2,’, where Yi* = ml)’; and Xi‘ = w2X, -; ’lU1,'U)2 are constants; Is it true that (a) Bi‘ = mm} (b) B; = 332 A . 2 - (c) Var(B2 ) = Var(Bg) (d) riy 75 r§. y., where 7' denotes correlation coefficient Answer: (C) QUESTION 34. For variables X and Y we have the data ZXY_—: .350, EX = 50, EY=60, 76:5, a§( =4,a$ = 9 where X denotes the mean of X and 0} denotes the variance of X. VVhich of the following holds (a) A one unit change in X causes a 1.25 unit change in Y, and a one unit change in Y causes a 0.6 unit change in X (b) A one unit change in X causes a 0.6 unit change in Y, and a one unit change in Y causes a 1.25 unit change in X T (c) A 10% change in X causes a 15% change in Y (d) The regression of Y on X passes through the origin Answer: (a) QUESTION 35. X is a normally distributed random variable with unknown mean u and standard deviation equal to 2. The value of the sample mean from a random sample of size 25 is 10. Which of the following values lie . within the 95% confidence interval for , u? iskysoft ‘T 16 . .
  15. 15. EEE 20.12 A 01 14 (a) 9.3 (b) 9.8 (C) 10.6 (d) All of the above Answer: (d) QUESTION 36. An econon1etricia. n uses data'frorr1‘"tlie" Consumer Expen- diture Survey conducted by the National Sample Survey Organization for the years 1991 and 2001 and plots the cumulative distribution function for real consumer expenditure per capita for these years. He finds that cumulative distribution function for 2001 is everywhere to the right of that for 1991. Consider the following, Conclusion 1: The Gini coefficient for 2001 is higher than for 1991 Conclusion 2: Consumption expenditure of every individual has increased in 2001 compared to 1991 Conclusion 3: Real consumption expenditure per capita has increased in 2001 compared to 1991 Which of the above conclusions are correct (a) Only is correct (b) and are correct (0) (ii) and are correct (d) None of the conclusions is correct Answer: (a) QUESTION 37. Let X1,X2,. .., X,, be random samples from a normal distribution with parameters /1. and 02. Then the random variable (11 —1)S2 ‘2 i _%<X. « — X)? U U2 :1 has a chi—squared (X2) distribution with 71 ~ 1 degrees of freedom. Here X denotes the mean of X1, X2, . . . ,X, ,. Suppose area under a chi—squared curve with n — 1 degrees of freedom to the right of X3’, ,_1 is v. A 100(1 — Oz)% confidence interval for the variance 02 is 3%? ’ 17
  16. 16. EEE 2012 A 01 15 (n—l)S2 n—1)S2 (a) lxa n——l ’Xl—°‘n—1 2: 7» —1)S2 (n——1 S2 (b) 7 TL‘ X'l--g, n—1, Xg, n—1 Answer: (a) QUESTION 38. Suppose you have 500 observations and you regress wage (measured in rupees per hour) on experience in the labour market, exper (measures in years), and on experience in the labour market squared, (experz). Your estimated OLS equation is 'wT. §e= 3.73+ 0.298 exper— 0.0061 e: t;per2 (0.35) (0.041) (00009) where the standard errors are in brackets. The estimated equation implies (a) The returns to experience is strictly increasing (b) The returns to experience is strictly diminishing (c) The returns to experience is constant (d) Experience has no statistically significant effect on wage Answer: (b) QUESTION 39. Suppose you have a sample of size one from one of the following densities Ho: f(x)=2:1: 032331 H1: f(1:)=2—2:r 039331 Let Oz and /3’ denote type I error and type II error, respectively. Find the test procedure of the form “Reject Ho if 2: < It” with or = 0.09. Find fl for this test. 18-
  17. 17. EEE 2012 A 01 15 (Mk=0&fi=0% (mkzeafizew (qk=0a5=0m (mk=0a5:0w Answer: (d) QUESTION 40. An urn contains‘ equal ‘n'1‘1mb‘er"'of' green and red balls. Suppose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the um, and the selection process is repeated. Each time a green ball is picked you get 1 Rupee. The first time you pick a red ball, you pay 1 Rupee and the game ends. Your expected income from this game is (al 00 Positive but finite (0) Zero (d) Negative Answer: (C) QUESTION 41. Two women and four men are to be seated randomly around a circular table. Find the probability that the women are not seated next to each other. (a) ‘/2 (b) 1/3 (C) 2/5 (<1) 3/5 Answer: (d) QUESTION 42. A fair coin is tossed until a head comes up for the first time. The probability of this happening on an odd—numbered toss is (a) 1/2 <b>.1/s (c) 2/3 (<0 3/4 , , stli A" l
  18. 18. EEE 2012 A 01 17 Answer: (c) QUESTION 43. An experiment has 10 equally likely outcomes. Let A and B be two non—empty events of the experiment. If A consists of 4 outcomes, then the number of outcomes B must have so that A and B are independent, is (a) 4 (b) 3 or 9 (c) 6 (d) 5 or 10 Answer: (d) QUESTION 44. Consider the system of equations (135 + fly = 0 ‘its; + 1/y = 0 a, fi,; t and u are i. i.d random variable. Each of them takes value 1 and 0 with equal probability. Statement A: The probability that the system of equations has a unique solution is %. Statement B: The probability that the system of equations has at least one solution is 1. ' (a) Both the statements are correct (b) Both the statements are false (c) Statement A is correct but B is false (d) Statement B is correct but A is false Answer: (a) , QUESTION 45. f(2:, y) = :t + y + my where 113,1; 6 §R++. For c E §R++, let us define, L = {(r. y)€§R2|f(x. y)Sc} U = {(x, y)€5R2|f(r, y)ZC} 2:, y)€ 319 | f(m/1: C} iskyslozflt ‘ 20
  19. 19. EEE 2012 A 01 18 Which of the above sets are convex? (a) L (b) U (C) 1 (d) All of them . Answer: (b) . .. - , _ . .. . QUESTION 46. What is the total number of local maxima and local min- ima of the following function - __ (2+: z:)3 if —3<: r§—]. f($)—{x’/3 if—1<: z:<2 (501 (b)2 (0)3 ((1)4 Answer: (b) QUESTION 47. Suppose that f(x) is twice differentiable and strictly con- cave in ac. Define, mm) = "° where c is a constant. Then g(: r) is (a) Decreasing function (b) Increasing function (e) Decreasing function when c > 0 and increasing function when c < 0 (d) Increasing function when c > 0 and decreasing function when c < 0 Answer: (a) QUESTION 48. A rectangle has its lower left hand corner at the origin and its upper right hand corner on the graph of f = $2 + (1/12:2). For which x is the area of the rectangle minimized? (a)x=0 . (b): c=oo iskysoft “ - 2. ~ 357
  20. 20. EEE 2012 A 01 19 (C) 1: : %)l/1 (d) x = 2'/3 Answer: (c) QUESTION 49. The real valued function f(a: ) = :t: + + (st: — 1) + |2: — 1|, where | a:| , | :r: — 1| stand for absolute values (a) Is differentiable everywhere except at :2: = 0 (b) Is not continuous at 0 (o) Is not differentiable at 1 (d) Is not continuous at 1 Answer: (c) QUESTION 50. Let 1: : (: e1,2:2,‘. .., :z, ,) and y = (y1,y2,. .., y,, ) be two vectors in ER". Define, (I: (8) y = ‘_, j,’. ’=, :;, -|y, -|, where | y,-| stands for absolute value. How many of the following statements are correct Statement 1: m (82 y : y (8) 1: Statement 2: 2 (8) II: = 0 implies :5 = (0, 0, . . . ,0) Statement 3: x®(c. y) 2 c(a: ®y), where_c E §R++ and c. y = (cy1,cy2,. . . ,cy, ,) (a) 0 (5) 1 (c) 2 (d) 3 Answer: (b) The following set of information is relevant for the next gig ques- tions. Read the information carefully before answering the ques- ti_ons below. Consider an economy where the nominal wage rate is set by a process of wage bargaining between the workers and the producers before actual production takes place. Thus at any period t, the nominal wage rate (I/ Vt) is a function of the expected price level (Pg), the rate of unemployment (u)) and the average productivity of the workers (A). The exact functional relationship is given below: Wt -—: -PfF(‘u. ¢,At); Fu < FA > 0 22
  21. 21. EEE 2012 A 01 20 Once the nominal wage is determined, the producers set the actual price level (P, ) as a constant mark up ()1) over the nominal wage rate: P, = (1 + ; i)lVt. Define the actual rate of inflation as 7r; = P‘; :‘1" and the expected rate of P . . = _p inflation as 7rf : —1—Pt——| :1 QUESTION 51. Given the above wage and price setting equations, derive the relationship between expected rate of -infla-tiorrand ‘actual rate of inflation. Which of the following equations represents this relationship? (3-) 71¢ = (1 ‘l’7’cc)lF(uta /1:)’ fill "1 (13) 70. = (1 +7"f)(1 + /1)F(Ut»/12) "1 (C) 70 = 7Tf(1+ #)F(Ut»/1:) (d) None of the above Answer: (b) QUESTION 52. Suppose the average productivity of workers remains con- stant at a level A. Given the relationship in previous question, which of the following equations defines the ‘natural rate of unemployment’? (EL) : /1 (b) F(ut, A) = fi (C) F’(u: ,Z) = L 1+}: (d) None of the above Answer: (b) QUESTION 53. A one-shot increase in the average productivity of worker, ceteris paribus, leads to (a) An increasein the natural rate of = _:n-: mplo}, 'ment. (b) A decrease in the natural rate of unemployment (c) No change in the natural rate of unemployment (d) Some change in the natural rate of unemployment but the direction is ambiguous Answer: (a) QUESTION 54. A one—shot increase in the producer’s mark up, ceteris paribus, leadsto — Z53 23
  22. 22. EEE 2012 A 01 21 (a) An increase in the natural rate of unemployment (b) A decrease in the natural rate of unemployment (c) No change in the natural rate of unemployment (d) Some change in the natural rate of unemployment but the direction is ambiguous Answer: (a) QUESTION 55. Go back to the relationship between 7r¢, and 7rf above (first question of this list). Suppose expectations are static, i. e., 1rf——-711-. -r-Also’ let F(u, , A, ) = {:1 — 1 and A; = A (a constant). The corresponding value 0 the non-accelerating inflation rate of unemployment is given by " (a) u, = A 1+ (d) = 2 (c) u, = A (d) None of the above Answer: (c) QUESTION 56. Now suppose the workers productivity increases at a con- stant rate 7, that is, A‘, ;"_l| “ = 'y . - To maintain a non-accelerating inflation rate, rate of unemployment has to increase at the rate (<3) 7 (b) /1 (C) M + 7 (d) 7 Answer: (a) The following set of information is relevant for the next §9_u_r ques- tions. Read the information carefully before answering the ques- tions below. I Consider a small open economy with fixed nominal exchange rate (E), fixed domestic~priee level (P) and fixed foreign pricerlevel (P‘). Let 5 be the corresponding real exchange rate. The goods market equilibrium condition Iiskysoft ‘ 24
  23. 23. EEE 2012 A 01 22 is given by the following IS equation: 1w Y= C+I+G+X——: — where C = co + C1)" represents domestic consumption 1 = d1Y —— dgr. represents. domestic investment G represents government expenditure X = q;1Y* — 1:25 represents export Y‘ represents income of the foreign country I M = m1Y + mge represents import QUESTION 57. Suppose the rate of interest r) as exogenously given. Then a unit increase in the foreign price level, ceteris paribus, increases domestic output by T3-Y—e: t2 1 __e____m_ __* (3) (1-C1-d1+: l) P T—nJY—e: c2 _. _s_______ (b) (z-: (1—c1—d1)+m1) ‘- ".3 e: c2— Y 1 . .___5_m_ _* (C) (l—c1—d1-F4-) P E (d) None of the above Answer: (c) QUESTION 58. Write the goods market clearing level of output as a func- tion of the real exchange rate (5) and other If you plot this relationship- between Y and 5 in the Y, e plane (with 5 on the vertical axis), you will get (a) A positively sloped schedule if the Marshall—Ler. ner condition is satisfied (b) A negatively sloped schedule if the Marshall—Lerner condition is satis- fied (c) A positively sloped schedule irrespective of the Marshall—Ijerner--eon-di—~ tion ° iskysoft 1 . 25 Z 3;? r
  24. 24. EEE 2012 A 01 23 (d) A negatively sloped schedule irrespective of the Marsl1all—Lerner con- dition Answer: (a) QUESTION 59. Let us now bring inan asset market and a foreign ex-' change market into the picture. Let the asset_market equilibrium condition be represented by the following LM eiqifatioflriz — Z27". Also, let the foreign exchange market equilibrium condition be represented by the follow- ing interest rate parity condition 7" = 7"‘ +«: (-eL—~€—), —-r—* and 5° being the foreign interest rate and the expected future exchange rate respectively. For any given value of 1” and 5° , derive the level of output which will clear both the asset market and the foreign exchange market as a function of the real“ exchange rate (5) ‘and other parameters. If you plot this relationship between Y and 5 in the Y, 5 plane (with 5 on the vertical axis), you will get (a) A positively sloped schedule if the Marshall—Lerner condition is satisfied (b) A negatively sloped schedule if the Marshall—Lerner condition is satis—' fied (c) A positively sloped schedule irrespective of the Marshall—Lerner condi« tion _ _ (d) Aiinegatively sloped schedule irrespective of the Marshall—Lerner con- dition Answer: (d) QUESTION 60. Let the equilibrium output and the equilibrium exchange rate be simultaneously determined by the intersection of the above two sched- ules (derived in. the previous two questions). Suppose Marshall—Lerner con- dition is satisfied. Then (a) An increase in G leads to an increase in the equilibrium value of 5, while an increase in M leads to a decrease in E (b) An increase in G leads to a decrease in the equilibrium value of 5, while‘ ‘ an increase in 17 leads to an increase in 5 (c) Increase in either G’ or W leads to an increase in the equilibrium value of 5 *iSkysoft » ‘L 26
  25. 25. EEE 2012 A 01 of5 Answer: (b) End of Part II 27 24 (d) Increase in either G or TVT leads to a decrease in the equilibrium value
  26. 26. EEE 2012 .4 01 Rough Work 28
  27. 27. ~ ~ Anyone brealdug the seal prematurely will be evie Entrance Examination for M. A. Option 33 June 23, 2012 20001? ( Time 3 hours Maximum marks 100 Instructions Please read the following instructions _ a Do not break the seal on this booklet instructed to do so by the invigilator. bed £rc-m’ the examination hall and his/ her camlidature will be cancelled. ?~ 2 pm in ymzr Name and Roll Number on the cietamable slip below. a When yeu finish, hand in this examinatirm booklet ta the 0 Use of any electronic device (e. g., telephone, ealciilzrtor) is strictiy prohibited during this exazninatioa. Please leave these devices in your bag and away fiem your person. ’ ’ ‘ ' 0 Do not disturb your neighbours for any reason at any time. . o Anyone enmging in Illegal examlnatiora practices wiil be immediately evicted and that person’s candidature will be cancelled. - ' _______ ‘ Do rx>j: __v_. £iia below this line. ______ . ~ This space is for official use only.
  28. 28. Partl Instructions. 4- Check that this examination has pages 1 through 22. o This part of the examination consists of 10 multiple-choice questions. Each question is followed by four possible answers, at least one of which is correct. If more than one choice is correct, choose only the best one. Among the correct answers, the best answer is the one that implies (or includes) the other correct answer(s). Indicate your chosen answer by (a), (b), (c) or (d). c For each question, you will get 2 marks if you choose only the best a. nswer_. If you choose none of the "answers, then you will get 0 for that question. However, if you choose something other than the best answer or multiple answers, then you will get -2/3 mark for that question. You may begin now. Good luck! QUESTION ‘1. Two women and four men are to be seated randomly around a circular table. Find the probability that the women are not seated next to each other. (a) 1/2 (b) 1/3 (c) 2/5. (d) 3/5 QUESTION 2. A fair coin is tossed until 3 head come up for the first time. The probability of this happening on an odd—numbered we is A (a) 1/2 (b) 1/3 (c) 2/3 (d) 3/4 QUESTION 3. Let f(a: ) = :1:+ [ml 4- (x — 1) + | :x: — II for: e 32. (a) f diflerentiable everywhere except at 0. (b) f is not continuous at 0. (c) f is not differentiable at 1. (d) I is not continuous at 1. EEE 2012 B 2
  29. 29. _ 2: , 2 es -— — function (2 + )3 If ( 3 1 f(fl7) "‘ {$2/8’ ‘f3. 6 2] (8.) 1 (b) 2 (C) 3 (d) 4 QUESTION 5. Let f 33+. .. -—r 8? 15 differentiable and f(1) = L Moreover, for every 1: hm ma) — em) = 1 t-M: t-Z Then f(: z:) is (a) 1/32: + 2x’/3 (b) ——1/5:c+ 4x’/5 (c) -1/:1: + 2/2’ (d) 1/:5 QUESTION 6. An n-gun is & regldar diagonals (edges of an n-gon are not (a) 20 diagonals ‘ (b) 25 diagonals (c) 35 diagonals (<1) 45 diagonals polygon with 12 equal sides. Find the number of considered as diagonals) of a 10-gon. QUESTION 7. The equation 2:7 (a) has no real solution. (b) has a real solution in the interval (0,2). (c) has no positive real solution. (d) has a real solution but not within (0,2). =1’-E-1 QUESTION 8. lim. .., ,, (/ Tl. -— 1 — / F») (a) equals 1. (b) equals 0. (c) does not exist. (d) depends on :1. EEE 2012 B’ 4 ~
  30. 30. QUESTION 9. A rectangle has its lower left hand corner at the origin and its upper right hand comer on the graph of f (at) = 3:“ +22". For which x is the area of the rectangle minimiwd? (a) :2: = 0 (5) 3 = 0° (c) m = (%)”“ (d) 37 = 2‘/3 QUESTION 10. Consider the system of equations a:2+fiy=0 ; w+vz/ =0 .41, fi, p and u are i. i.d. random variables, each taking value _1 or O with equal probability. Consider the following propositions. (A) 'I‘he probability that the system of equations has a unique solution is 3/8. (B) The probability that the system of equations has at least one solution is 1. ' (a) Proposition A is correct but B is false. (b) Proposition 13 is correct but A is false. (o) Both Propositions are correct. (d) Both Propositions are false. Part II Instructions. 0 Answer any four of the following five questions in the space following the relevant question. No other paper will be provided for this purpose. b You may use the blank pages at the end of this booklet, marked Rough work, to do calculations, drawings, etc. Your “Rough work‘ will not be read or checked. _- Each question is worth 20 marks. EEE 2012 B 4
  31. 31. QUESTION 11. Let I’ be the see of sequaoes of real numbers z = (2,. ),. €N such that 2320:: 1% < ‘X’- (A) With 3? as the field of scaiam, show that’ I2 is a vector space. {8} Given :5 6 I’, let fiizli = = ___, ::1 £31/2. Verify that is a norm on I’. (C) Given 3,3: 6 1'‘, Eat (any) = ,"’_‘: ,,. 't, ;u. .. Verify that (. , . ) is auinner product (i. e., dot product) on P. ‘ (D) Given 2.31 6 £2, let‘; d(: z:, 1;} = Ha: —~. yfl. Vi; rii‘y thug d is a. distance function on 12. (E) Let B= {:: :€- (3 3 §|2:§§ -: 1}. Slxowflzatif is acouverset; V ' -' - (F) Define c : N x I" -> 3% 193' e(n, :) = 2:. .. Show that e(n, .) is continuous for every n E N'. ' (G) Show that the normed space (13.3.11) is complote, i. e., every Cauchy sequence in I’ converges to a limit point in I“; ANSWER. 9 EEE M12 B 5
  32. 32. X Skysoft 0N 12. Consider the system of d1'§'eremi9.l equations . o;, (:; : ms) w'r1ez~<»: . -= "3 23] whem a: ,fl 6 3! and (a, ,8) 71- (6,0). (A) If the roots of 13‘ am puzek, ’ imagina_ry, ,,verEt_'y that £_: ha solution of this s'ys1.em 9! = (31 , gm) is 0.’. ‘%he_ form 1;; = ='? :.; + and fig-(3) = = c; ai_! _1(og + fit). V (B) C{§a: 'actariea the orbit: of 1;, :na: ‘.fse§=3r A-¢, t{’3‘v. ".; ,). ti-? 're. p1': the orb_it- in SE3. _ (C) (3o§i1me}z§aa the stabifity propexti-2 V 5 of gfwlzan roots of B we guzely imagimuy. {D} 1? that mots of . "3"are naa‘. ' iauzagin. m«, '«z«iPy' him: fine so1z*zti: or. n o£_Tt'!1.is systegm gr = —. (y, , 3:3) is of the form 3; {:3} = -. . :;e, “'. ' (‘: Ofi(¢‘»; v, + / iii}. tswi‘ 3;t; ;{t} = = c1e°“ (‘.2 +_ 3:). ‘A (E) Eerive and graph Liza orbit of y thce mate of B are not purely imaginary. (F) Commeafen. 1;}: t: stabifit r gimp-s: .”s*. ic1: of y when tbr: rcofaxfcf B are not pure! -y imagiaz. -y. (G) '&5:". >.aai data-‘mzines the -zfiremticxn of rotation of: that orbit as t T 00? .5;. I'13W". :‘§R. " EEE 2012 3
  33. 33. QUESTION 13. (1ansidervectqrspanesV, W'andU. LetA: V--)WandB: I'V-—)U be linear mappings. A‘ : W’ —) V is the linear mapping such that AA'A = A and B': U-+Wisthelinea. rmapping suchtha. tBB“B= B. ‘ _ Given a. linear mapping 1), let R(L) denote its range space, p(L) its rank and u(L) its nullity. Prove the following pmopoétions. (A) /104") 2 9(4)- (B) n(-4A‘) = /I(A‘A) = p(A)- (C) u(AA“) = I/ (A. "A‘) »= 2/(A). (D) AA’ projects W on ': 'Z(.4) and A14 projects V on R(A“A). (E) If p(BA) = p(B), then ALBA)‘ = -B‘. (F) If p(BA) = p(A), then (BA)“B = 21". (G) If p(BA) = p(A), then A(BA. )‘B projects W on 1?. (A). ANSWER. EEE 2012 12 11
  34. 34. QUESTION 14. Given 3,3; 6‘ 32", define (2,1)) = {tz-4— (1 —- t)y I t E (0, 1)}. We any that OCR” is a. oonvexaetifz, y€C' implies (x, )('. C‘. Let X c 32" and let {'c. — | 1' e I} be the family or all convex subsets of R“ such that X C C’; for every 1‘. E I. The oonveoehull ofX is cox = O. -€10’. -. Prove the following propositions. (A) X is a. oonvexeet ifand only ifX= ooX. (B)_Let Am = {peRZ; ‘|2;;1p, ~=l} fic¢rm€JV. Then, Xisoonvex. i£a. nd~only~if. - foreveryme. /V, peA, ,.and {a:1,. .., :l: "'}CX, wehave‘Si";1p. -2:‘EX. (c)coxe_—u, ,,€, ,{§; ,!'; ,p, -.~, * | (p, ,.. .,p. ,.)eA. ,, / : ::1,. .., :c”‘eX} (D) ooX = U,'1f. _"1{E; ';1p, -2;" I (p1,. .., p,, )EAm A : c", ... ,z'" EX} (Hint: Bye-(C), every 2: 6 ooX can be represented as the convex combination of a. selection‘ {$1, . . . ,z'“} C X. If m > n + 1, then the vectozs in this selection are linearly dependent. Use this fact-to ehovrtliat 3 can be represented as ‘a convex combination of m—1 vectors from {: c‘, .. . ,x"‘}. ) ANSWER. EEE 20123 , ‘ _ 14
  35. 35. QUESTION 15. Prove the foilowing proposi event and E is the expectation opezatcr. (A) If X is a ra. udom varitxhle and X . _>_ 0, then 130:‘ every tions, where denotes the probability of P 8?-! -2 . , / '°° I-SA-" = .- J; d:2:p. 'c"“‘P[X > 3:} 0 . (B) Let X be 2:. real-valued random variable and let f : 8% f(b)P(X 3* S EfoX 113.‘: every 37.45 R. ' " ‘ W" ' ‘I (C) L64. U be a random vurlabie with the unfiozm distributian —c“ In U , vzhcre c > O. Show chm; C. -: ~ 53+ be imzreasing. Then, on (0,1) and let X = X has the exmnenlzlal distribution with scale parameter (D) Let X and Y be iIh! §1‘. p€: I1r1e1it atandaré Gaussian Nannai) random variables. Show that the diercribufiion _u of Z 2: X/ }’ has the form . ~ ' xsicizfi = #5: -- 1 1? (1 +2’) for z E 5:‘, ANSWEP. ..

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