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# Delhi School of Economics Entrance Exam (2009)

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This is the entrance exam paper for the Delhi School of Economics for the year 2009. 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 (2009)

1. 1. A, ~. — ~; Page I U} Delhi School of Economics Department of Economics Entrance Examination for MA Economics, June 27 2009 Option A Series 01 Time 3 Hours Maximum Marks 1 00 I General Instructions: Please read carefully "r [)0 not break the seal on this booklet until lll. 'll'llClC<l tut lt» () by lltt’ mvtgilzttor. Any one breaking ' P Immediately after you ICLCIVC this booklet, lull Ill mtu n tint and [Oil no 1‘ clesigtinted space below: r Check that you have rt bubble sheet (Optical . lzttk Sheet t . lk'L"tllll]l. lll_ mg ll booklet. All questions are to be marked on the (_)[)[l(£ll mail: l1(‘L'l only zunl the entire cxatuinatiott will be Checked by a inttcbtne 'l'lter't-fore it is l't‘l') important that you follow the instrttctious on the bttbblt-—s'ln-t't. r l-‘allowing the instructions on the bubble S'llt‘t‘l, fill in information in boyies 1, 2, 4, 5 and 6 in the l)lll)l)lL'-'llL‘ will sign in box 3. o o In box 5. enter the Cate viz SC (Scheduled Ca the l't‘t[llil't‘(l cl. lbw in'i; _:ilnlm' er. e g, 0123. gory under wlttcli _mtt uh I0 be cmtsidcretl. Ste), ST (Scheduled Tribe), OBC (Other backward Caste); PH(Physically blzttidrcappctl). AF(Dependent 01 Armed Forces Personnel killed or disabled lll ; tctnm)_ SPORTS. or GEN (General Category) O In Box 6 enterol as y I‘ Keep your admission ticket e invtgilators our series number asily accessible to: xettltuttiott by the C L’. ;lll ttnatton, break the pa . We 2 Full Name: Roll Number Series 01
2. 2. Page 2v. ’: irIs to the own 1. Attempt both parts. Part 1: One Mark ll)Sll'll(‘li(lll. S‘. ' Qucstirms l’lrsl check (h at this booklet h lhal the honor NOT on this booklet ° Lach correct Choice will cam you I mark llowcvcr you will lose 1/? m *, 1 / m uic/1[I150/"rec/ clzui(c. If you Sh.1(iC none nflhe bubbles, or moie than (WC lmhblc, you WI“ get 0 for that quesnon. ' You may begin now. Good Luck e 8» E. H1011’ are 4 nmrncd couples in a club 3 c is pan 0! the committee The numl U! mix in Hhlth this cmmmllee can be fanned Ix Al 16 _ I1) H / ’ 3: AI) 56 - and F1 refcrlo the Union um? ents for the sets A, B and C 1. A -- (b’uC): (A— B)u (A—C) / e u. /1—(BuC)= (A—B)n(A—C) ‘ « ni. A—(BuC)= (A;3)nc e / ’ .1) l. is true / krf n. is line C) iii. is true d) none oflhcm is necessarily true. I 3. ': dx is approximately was :1) 0.025 1») J27 u 0.025/‘E (H 0.025 ‘/22: Series ()1
3. 3. x2 — y? subject to the constraintx+ y = l , where X (/71) -; /’= ’*’/ - 3? ’/ x /1%? ' 4. Consider maximizing j'(x, y) = and y are real numbers>This problem has a) no solution. F In) aunique solution. , / c) 2 solutions. —‘ d) an infinity of solutions. 5. Suppose P(x) and Q(x) are real polynomials of degree m and k respectively. where both m and k are less than or equal to the positive integer n. Suppose the equation I’(. -) = Q(. ~:) has at least (n + 1) distinct solutions. Which of the following choices best l‘ describes what this situation implies? I I . , , a) m= k=n r_,1,I—, § : b) m= k<n 7;; - c) P(x) and Q(. x) are identical. v’_»7 - ,1 cl) P(, :) and Q(x) are linear 6. There are three alternative definitions of a consistent estimator ‘ 1. An estimator is consistent if its probability limit equals its true parameter , value as sample size approaches infinity. l - 11. An estimator is consistent if its mean squared error goes to zero as sample size approaches inﬁnity. 111. An estimator is consistent if it is unbiased, and its variance goes to zero as sample size approaches infinity. V Which of the following is correct? i/ tr) Only 1 is Correct ‘ b) Only I] is Correct I c) Only 111 is Correct ‘I d) All three are Correct ‘. 7. Given the data ZXY = 350, EX = 50, ZY = 60, N _—_10,V(x)i= 4, V(Y) = 9_ V ‘I Where V(. ) refers to the population variance. The correlation coefficient between X and Y, the regression (slope) coefficient of Y on X, and the regression (slope) l coefficient of X on Y are. respectively: U . l: D slll a) 35/36. 35/16, 35/8] - t 5/6, 5/4, 5/9 c) 5/6, 35/16. 35/81 (1) J35/36 , 35/16, 35/81 isﬁkysoftj . r
4. 4. lemonade is )0 +. ,2(f— 16): /U0’/9 "0 *5 r" ’ v 3) R520 )o+.9[3o -t). - /0330 - JF-/ a+p vb‘) Rs 15 c) Rs. 10 d) ‘Rs. 5 4- Page 4 of 26 8. In a surprise check in a local bus, 20 passengers were caught without tickets. 'l‘lie sum of squares and the populationstanggard deviation of the amount in their pot'l<etS were Rs. 2000 and Rs. 6, respectively. If the total fine equals the total amount discovered on them, and a uniform fine is imposed, then the fine imposed on an individual is: J) R5-8 t/ aux) ; éxz- zxjz b) Rs. 6 *5 '5‘ c) Rs. 10 / ._ (1) R5_ 12 36 . — Joya; /— [1 2 C9 07% _-8 9 :1! ‘ g p 0. lit a linear regression of Y on X, changing the units of ineasiirgmeiit of the Y variable will affect all of the following except: a) the estiinated intercept parameter b) the C. ‘llll2llC(l slope p'. ir: inicter L‘) the Total Sum of Squares for the regrcssioii J1) R 'qll1ll’L2(l lot‘ the regression ll). Ajuir dice has l)| lll1l)Cl'S I,2.3. 4. S and 6 on its sides. lt is tossed once. I win Rs. I it‘ an odd iiuiiiher sliows up; otherwise lose Rs. l. Let X be the ntiiiiber that shows tip and Y the inoiiey I win. I-Note: Y < 0 ill lose money. ) Wliicli ol'tlie lollotviiig is incoircct’? .; Vt lb’ )<'_ Y _ _ . Prob (X > Y, :1 , ./5 I/5 _ Z5.’/ _€ 5/5; h) Prob(X= 3 I Y = l) = 1/3 "ff _ re) E(Y‘) : 0 _ tl) l’rob(Y = ll X :5) = l l 1. Your budget is such that if you spend your entire income on two goods, x and y. yoti can afford either 4 units of x and 6 units ol y or 12 units of x and 2 units of y. if you spent all your income on x. how mat y units ofx could you buy? ;, F ; fp ; /I) . :7 MR -/ /3 P : 3/"4 c) is _2’/ “(ff (J) e 18 e _ i g , . _— L: I] W —~/2 in 7’ ‘Z/ £== "l r . ~ 12. The demand function for lemonade is Q J = 100- p, and the supply function is Q, = l0 + 2 p , where p is the price in rupees. The govemmcnt levies a sales tax on lemonade after which the volume of sales drops to 60. Then the per. unit tax on
5. 5. Page 5 of 26 13. There are only twoprice taking firms in a market. Their cost functions are C, = . ‘, ' and C. = 2x; , where x. is the output of the i"' firtn. Market supply is stttu of the two {inns output. Then the market supply fuiictioii is . / ’ 3p /40: §z, ‘ I. /"yx= —4_ 32,: D -; >:4>F b) . =i3’—’- 5/I ‘/7 C) oi d) :2,, ‘ / /7 f H. A monopoly faces the demand curve P : 8 — Q. The tnonopoly has ; i coiistant mitt . «. . . . . v r 5 ,2/: ~;' 2-3. : , rust equal to S for Q S 2 and : t cotistaitt -ttiitt cost equal to 3 l0! Q 2 2. lis pioftt ~ / r)’ - inztxiiniziiig output equals: g_°)9 _: M Q . _‘ T - 3 ~: 2 ; :1) 3/2 " g _ : _ / 9 _f g; 11) 2 3,‘ f -l 5/2 ’ ‘ . . W , . -. r . .. ..1 t A , /_t , / 9 / d) Both 3/2 and 5/2 . d. “K _. -I“ 5 w 15. A firm has the production function y = min{1.+2K. 2L + K}, where y is quantity of output. and L & K are the quantities oflabour and Capital inputs respectively. ll the input price of L is Rupee 1 and the input price ofK is Rupees 2. then to produce y , = l2 costs the firm at least ' , . _ Q , er : f _ :1) 10 Rupees V ' * T’ ' . W . . / via’) 12 Rupees it A l‘ ' 3/+55, ‘ c) '14 Rupees [/ wt. ‘V . 5” 5 ( /2 d) 16 Rupees 16. The opportunity cost of holding money (that yields zero nominal return) vis—a—vis some interest bearing bond is: :1) the real interest rate b) the nominal interest rate 9 C) the rtlltal interest rate when measured in real terms and the nominal interest rate when‘ measured in nominal terms d) None of the above‘ F7) in the lS—LM fi'; tniewoi'k. an increase in the expected rate of iiillzttitiii results in at) an increase in the equilibrium value of incoiiie and an iiicreasc in the equilibrium value of real interest rate . b) it decreitse in the eqttilibriuin value of incoine and ‘.1 decrease in the equilibrium value of real interest rate an increases in the equilibrium value of income and a decrease in the C) . eqttilibrittm value of real interest rate 2 S15 kys 0 f t
6. 6. ii Page 6 of 26 d) a decrease in the equilibrium value of income and an increase in the equilibrium value of real interest rate When the nominal wage rate is rigid, the aggregate supply schedule n the output—price space) is: , a) horizontal b) vertical~ c) downward sloping d) upward sloping 19. In the IS-LM framework with an extemal sector i. e., the IS equation now includes a net export term, an appreciation of the (real) exchange rate a) would necessarily result in a decrease in the equilibrium value of income b) would result in a decrease in the equilibrium value of inconieonly if the LM curve is vertical / Q would result in a decrease in the equilibrium value of income only if the . Marshall-Lerner condition is satisfied tl) would result in a decrease in the equilibrium value of income only if the governtnent maintains a balanced budget £1». 9”‘ 20. According to the Baumol-Tobin Model, transactions demand for money should rise by a) Five percent b) Ten-Percent c) Between five and ten percent if income rises by ten percent, the d) None of the above .9 -3 J P T 6’ .3 J 3'8 °Q A’ , it 0., ’ 0 q. _ x. - Q g 0 E5 Ox? I . Q9 ' 0“) 300A / _} _ C’ ? r 9P I 5’ 0‘; l ‘O 4) M’ 0 . l . 0‘ ‘ \$9 w 1 ' ~ * \$0 P: Q Q — ‘ . V 2_ O K I, P Q Q, “)0 ‘Q ' 0 ' O Q ' :3 .9 5? <3 5 ' 0‘ e ~ “ S‘ Ce: ’‘ Q 0 Series ()1 Skysoft . .
7. 7. Page 7 of 26 e) Part II: Two Mark Questions 0 This part of the examination has 4() multiple choice questions. Each question is followed by fotir possible choices, one ol which IS correct. ‘ Indicate the correct choice on the bubble sheet. NOT on this booklet. ‘ 0 Each correct choice will earn you 2 marks. However ymt will [use 2/3 mark _/ "or each iiicm'r_cu choice. lfyou shade none of the bubbles, or more than one bubble, you will get 0 for that question. I — 2]. Consider the following statements: (0 544 > 453 (H) 2100 +3too < 4I0O a) Both (i) and (ii) are false. b) (i) is true, (ii) is false. cl Both (i) and (ii) are true. d) (i) is false. (ii) is true. i2§[The WOW Language has only 2 letters in its alphabet. O and W; the language (5 eys the following niles: (i) deleting successive letters WO from any word which has more than 2 letters, gives another word with the same meaning. (ii) inserting OW or WWOO in any place in a word yields another word with the same meaning. 0, OWOOW, WOO and OWW are 4_words in this language. Which of the following statements is FALSE? a) the vords v@ and O_WW necessarily have the same meaning. 1’) WOO and OWW may not have same in H caning. - V __ A‘ A ; _ . M _ c)’ O and OLVOOW must hav the same meaning. /3.3?»-7 " ‘ 4 ' ‘V J . d) (b) and (c) are true. (3 N D V) 1) / 23. Consider the systein of equations in the tiiikiiowiis x and y: kl V) U‘ I q . ~ (I. ‘+_y: (12 is "K 4 X + (Iy = I n - i I: ~ ‘ - . . ‘ . . . . . - ~ gels of all values ol (1 foi uhich this system has (I) no solution, A (ii) multiple solutions and (iii) a unique solution are respectively a) (i)a<l (ii)ri>] (iii)g: ] b) (i)a= —I (ii)a=1 (iii)a>l (1)11 = ~ 1 (ii) (1 =1 (iii) all other values ofa d) (i) n= -I (ti) (1 =1 (iii) -1 < a < +l kttiskysott
8. 8. /1 it» ‘ Page 8 of 26 24. Consider the system of equations: A x— y+z = I V 3X + Z = 3 - 5x—2y+3z= 5 Th is systetn Inns (1) the unique solution (. '. _t= ,z, )=(I.0.0) /5? ll s I i‘ . t{(t—3_3.- ; an «i re outonse 3 3 I)‘ 6 L7 / L) z 2- 1/ L W C) the solution set {(l——3—. ?‘-, z)| z20} 0 ' 1,) 7., ‘:1- tl) multiple solutions. but not described by (b) or (c). .2 - . . 25x (‘unsicler the function f(x) = “ “nu/ ")' " ¢ 0 ' X: -‘O Then the following is true about the derivative of I‘: ’Y>\$%r'l ‘ '1 ‘ n) / '10) = —| and j" '(. r)is continuous at . t- = 0. .1)‘; /' 't()) = —l and f '(. r) is discontinuous at . ‘ = 0. C) I’ '(0) = 0 and f '(x) is discontinuous at . ‘ = 0, J1) f 't. ') is not defined at x = O. i 26. For all set S . let S2‘ denotes the Cartesian product .5‘ XS. A binary relation R on S is It stthsct of S2 . - R is trainsitivc if (. r._v)e R and (_-, :)e‘ R implies (. '. z)€ R. R is negatively transitive if (. ', y)tE R and (y. :)e R implies (x. z)£ R. o Inverse of R is defined as follows, R" = {(3-'. .x') I (x. y)e R} (‘nnxitlcr the following statements: Statcizient A: R can not be transitive and negatively transitive at the same time. SttIlCttt(-fttl 8: ll R is transitive then R" must he transitive. Statement C: lf R is transitive then R" must be negatively transitive. lltm many of the above statements are true? :1) None V i. ,b) One (1) Two cl) Three 27. A function f(. ', ..r: .T) = (_v, ._v3) is defined as follows. where x, ,.‘3 and T are non negative real numbers and X, + . '2 2 T. Also. for real numbera, _'I = min((z. .', ), _'3 = min(a, .', ) such that y, + y; = T. Find f(5,2,6). :1) (3,3) b) (5.1) e) _ (0,6) iSkMs’O*ft Seri
9. 9. Page 9 of 26 tion f is defined as follows. Here u_b and c are constants. 28. A func (Lr+b 7 ﬂclipa . , I - - . c I , Find values of a and b such that f (c) exists. L1; ; ,- ( 2 ~ ’ . _ ~ . C : an 0 ‘/19’ a’:2C. b——c f gicza b') a= ('. h=—2L2 . i C) a= —3—. b=—l 2 h . 141 2 1 d) a= l.b= lnc [I1-[: 0 A /1* ‘~: ‘/ '(c):1 ‘>u(. ’+? _'~‘), Fil'lQ h/ ./(C)_ ’ “ ‘ l i “) C ‘C/2 ' b) 0 4 ‘ C o'- c K" I f(c>= '7 . , __, ,i’/ '. ’ v” ? gfhiw » V 7'C iZf73;2 always tells the . One of them is u Truth-teller ( (sometimes lies, and the third is a normal person ll know of each others‘ and their own type. ' 30: There are 3 persons, A, B and C truth), another is a Liar (always lies) other times speaks the truth). They a A said: “ I am a normal person. " B said: “A and C sometimes tell the truth. " C said: " B is a normal person. " a) These statements are insufficient to determine who is a Liar. b) A is a normal person. B is it Truth-teller_. C is it Liar. C) Who is normal, or Liar or Truth-teller caimot be ascertained from the SIIIICHICDIS. -pd-) A is a Liar. B is a normal person, C is a Truth—teller. form a nine digit 9 are arranged in random order to at l, 2 and 3 ' 31 . The nine digits 1,2.. . once. Find the probability th number, which uses each digit exactly appear as neigllibours in the increasing order. (’ _V 3) '7 I T I 8) /12 9 [ 7 Q on V s‘ 72 ~ 1 Q 34 d) (%i M/ soft
10. 10. -x Page 10 4 F 16 32. la a stirvey of I02 'l'imarpur residents in 2009, the average income was found i. be Rs. 4635 per niontli. Pl'C'l()l. lS studies show the population vai'iaiice of income in this locality to be Rs. l2342 per month. It is asserted that the average monthly income is Rs. 4650 in this locality. Which conclusion below can be asserted from this information? a) The assertion is rejected at the 10% level. b) The assertion is rejected at the 5% level. yr) The assenioii is not rejected at the 10% level. (I) None oftlic above. 33. Let X denote the absolute value of the difference betwe V / yet] the numbers obtained when two dice are tossed. The expectation ofX is: / i’/ 9 / /7‘ 7 32 Q r- 36 /9 /95 ‘ ——— é ‘. — 33 3 5/26 l " b) l—- _ , ' 36 v, «E7 5: C l_l_Z I _/0[§/ 1/ '8 0 73: « ‘L 5 l6 *’-‘ I l— / ' 4 () I8 :70 34. Let Y denote the number of heads obtained xi’/ henﬂ3 §: oinsj2tLre I0SSffi; variance of Y2 is: O t 2 '* ' _ a) 9.5 F 2/; ‘ 7 b) 8.5 -7- ‘ _ - 1. c) 6.5 3 2' ’ ’ ° J, )/ 7.5 , V 3 .4 V I. <9 . /3/SJ. A coin an_; has 100 employees. 40 men and 60 women. There are 6_ male / - . . w« - andranktobe executive. . How many female executives should there be for gender 7 2. r - / .5? independent‘? I ) i_ ‘ V“ V g’ ‘(Q ' if . »£’»‘~" _ 4,‘, 5.3.. .'_‘. ' "7' V b) 6 f["*"‘ / ”"" 5 ’/ £55’ C) '0 G . é. -+7 d) 8 To "-0 . . _~ 90:10; 2 7 ’ , <>-9 . ~.m~«o’~/ A "“-- . . . - — = 0.7 T) .1’ 'mum 36_ Consider two events A and B with Pr(A) —. 0-,4 ‘md P’(B) '5 m X’ and inmiiiium values of Pl'(/ nB. ) T9-*'l3€C“"5l. "’ ‘"9- My/ (0.4; 0.1) y . t V b) (0.7.-0.4) x ‘ ' C) (0.75 0.1) , (I) / (). -I: ()) / /§, ,4). . /55, f 6’ F 6' * /9 I / IT) / (at: /V7/l ”D/ _ . _ k I ,3», .‘C. ":. "s I 7 = ' ’ ’/
11. 11. V the following possible assumptions about our data. Page II of 26 37. Jai and Vijay are taking a exam in statistics. The exam has only three grades B and C. The probability that Jai gets_a_B is 0.3, the probability that Vtiay gets B is 0.4. the probability that neither gets a_ii A, but at least one gets a B is 0.1. What IS U16 probability that neither gets a C btit at least one gets a B'. ’;}/ . Vt’ﬂ/ ‘W/3) ‘ 9 ’ 7 W : ’D ’ 3 I 3) at am; r»/5.7. 0 v / r‘-/ z= “—- L}r)’O.6 “ NA). /)5) ; rm. /’ /1 _' / ‘ rj‘ / /S. C) 03 I’ , " / f( ' 7 4 /1"! ’ 4 -/2 f 9" d) Insufficient data to answer the question 0 5)‘ 5’) r J D 5 f, 38. You've been toid that a fatnilyhas two children and one of these is A a daughter. What is the probability that the other child is also a daughter. ’ / o/S) / KDLDJ/ KS‘/ q// )) X 1/2 )) 1/3 c) 1/4 d) 3/4 Questions 39 and 40. Sl1)[)0SC. lC(t. 'l squares is used to fit it litie relating _= and . . n; .mc)_- _-, : [3, + [33 :4. + 8.. Asstttite that in our data not all the x‘: ai'c t_tlcntic: il. so that at least some 01' the x‘s are tlil'l'eretit from their sttitiple tiicati . ' . Now t. ‘t)l). |lCl' (1 ) q Eta. ) = 0. (2) C0'(. ‘(i. £;) = 0. (3) Homoskedasticity: Var(e. ) = 62, a constant. ‘ (4) No autocorrelation: Cov(E; .€; ) = 0 for i¢j. Indicate the one best answer to each question below. 39. when are the least-squares estimators unbiased? Va) Only if our data satisfy assumptions (1) and (2). b) Only if otir data satisfy assumptions (1). (2), (3). and (4). c_) Only if otir data satisfy assumptions (1), (2). and (3). d) Only if our data satisfy (1). (2). and (4). 40. When are the least-squares estimators “best” (lowest variance) of any unbiased estimators , :1) Only if ottr data satisfy assumptions (1) and (2). . .b’)” Only if ottr (lata satisfy assumptions (1), (2). (3). and (4). C) Otily ifour data satisfy assumptioiis (1), (2), and (4). . (1) Cannot be detcrniinctl, without additional assumptions. eiskysoft 6 l
12. 12. Page 12 of 26 41. A consumer spends an income of Rs. 100 on two goods. dosas and pizzas. Let x denote the number of dosas and y the number of pi7.7.as consumed (fractions allowed). The consumer's utility function is U = e': “: . lf the price of a dosa is Rs. 5. and the price of a pi7.'/ .a is Rs. l(), then the number of pi7.z: :ts this consumer will buy is {A /90 b)l0 c)5 d)8 42. Romeo and Juliet have 96 chocolates to divide between them. Romeo has the utility function U = Raj‘ and Juliet has the utility function U = where R is Romeo's chocolate consumption and J is . |ttli<: t's chocolate consumption. Which of the following is true :1) Romeo would want to give Jttliet some chocolates if he had more than 62. b) Juliet would want to give Romeo some chocolates if she had more than 60. c) Romeo and Juliet would never disagree abottt how to divide the chocolates . /"(H Juliet would want to give Romeo some chocolates if she had more than 64 chocolates. ‘ 43. A consumer spends an income of Rs. IN) on only two goods. A and I3. Assutne non satiation. i. c.. more of any good is preferred to less. Suppose the price of B is lixed at Rs. 20. When the price of A is Rs. 10. the consumer buys 3;llnilS of B. VVlie_n 4% M750 the price of A is Rs. 20. she buys 5 units oi’ A. From this we can conclude , 2 D _g, g/‘ll /0; 3.7» V , l. A is an inferior good » , , ‘ ll. A is it Giffcn good F” , lll. B is a complement of A F’ " (l)) lztndll c) lundlll d) 1,11 and 111 Questions 44 and 45 fsuraksha’ is the sole producer and supplier of security systems in India and the sole employer of locksmiths in the labour market. The deinaitd for’ 5cc1n'i[y sysle, -ns is ])(p) : 100- p , where /2 is the price. The prqdnction of security systems only requires locksmiths and the production function is given by f(L) = 4L- where L is the number of locksmitlis employed. The supply curve for locksmilhs 15 given by L(w) = max(0,—‘2Z — 20), where w is the wage rate. Q 2 / . 9-9 of . . - ». « av- ” - 44. How many locksmitlts will ‘Sttraltslitf employ? /‘ / 7 f :1) (2,; /WTT7, bi / ’/”/ ‘ / ’’'°'”[ C) d) 1 .
13. 13. Page 13 of 26 45: If the govemment sets the minimum wage is 70, how many locksniiths will Suraksha employ? ' a) 5 b) 10 , c) 15 d) 20 Questions 46 and 97 Suppose that a typical graduate student at the Delhi School of Economics lives in :1 two good world. books (X) and movies (y), with utility I d _ function u(x, y) = x/ Syé. Prices of books and movies are SO and I0 i'espectivel_v. Su ose the University is considering the followine schemes. PP . - . . Scheme 1: 750 is paid as fellowship and additional 250 as book grant. Naturally, book grant can only be spent on books. Scheme 2: i000 as scholarship and gets one movie free on each book they pttrehasc. Believing that books and movies are perl'ectl_-' divisible, L'()t‘l1pt| l€ the optimal consuniption bundle under each scheme. 46. Optimal consuiiiption btiiidlc under scheme 1 is a) (4 books. 80 mo'ies_) (5 books, 75 movies) c) (6.5 books, 57.5 movies) d) (10 books. 50 movies) 47. Optimal consumption bundle under scheme 2 is )( a) (4 books. 80 movies) b) (4 books. 84 movies) -, (c) (5 books. 75 movies) AT) (5 books, 80 movies) 48. Let X staiid for the coiisuiiiptioii set and let R. l. P i'espectivel_y stand for the weak preference relation, indifferetice relation and strict prefei'ence relation of a COllStl£llL‘I'. The weak preference relation R is said to satisfy Qtiasitrziiisitivity if and only if for all X. y. 2 belonging to X. xPy and yPz —+ xPz. Which of the following preference relations over X = (x, y.7.l satisfies Quasitransitivity? zt) xPy & yPz & zPx b) xPy & yP2 & zlx / e) xPy & ylz & zlx d) yPx & ylz & xPz igkysoft
14. 14. V‘ D‘, D( I/ bi, ' l L 'l / ! -. i". < k XX ( C) _ ‘l tl) ‘i Page 14 7 49. (Tonsider an exchange economy with two consumers (A&B) and two goods (x&y). z. 'S| lll)C that total amount of x available is 4 and total amount of y availalvir 2 which is It) be optimally distributed between A & B. A's utility function is U, ‘ = x, :+ 4,', y, + 4y, ( and B’s utility function is U3 = xn + yg. The contract curve for '1. exchange economy will be: the entire boundary ofthe edgeworth box (OM) ’ "" " ’, allocations satisfying (x, (=0, 05y, (:2) and (0£xi; S4, y. ;=O) allocations satisfying (OSXAS4, y, (=O) and (x5=0, 0_<_y3f2) all points inside the edgeworth box. ’ 7- 3 ‘ 3 W) /1*)’ 50. Consider the exchange economy in the above question. Suppose A is endowed ti vitlt 3 units of good 1 and l unit of good 2, and B is endowed with 1 unit of each , good. A competitive equilibrium is described by the followin g prices (of goods X nil ' Y respectively) and allocation of goodg. _ /74’/ ,1 r 5/ M” ’ ’ ">5 ; § M/ " 7 ii) Prices 2 ( L2) and (x, (, yA) = (2.3, 2) , (xn, _’n) = (1.5. 0) l’rices = (2.! ) and (x, (, yA) : (2.5, L5) . (xii. yn) = (L5. 0.5) Prices = (Li) and (x, (, yA) = (2. 2) , (xii. ya) = (2. 0) Prices = (l. l) and (XA, yA) = (2.5, 1.5) . (X3. yg) = (1.5, 0.5) Questions 51 to 54 The following set of questions use a common set of informatirm given below. Read the information carefully and then answer the questions sequentially. Consider an economy which is described by the following two relationship between aggregate income (Y), aggregate price level (P), domestic interest rate (r), and the real exchange rate (e) : (i) GoodsMarketEquilibritm Condition: Y ; C0’) + I(r) + G+ NX(Y, 6) . . . . M (ii)MoneyM arl<etEquilibritm Condition: —P— : L(Y, r) where C= E+o'(Y—T): 0<a<l T= rY: O<r<l G= Z;' I: /—~(51‘: t3>fl , /‘X 2 , _'_{; ,}/ _y; -; ()<; /< l: 7>t) L(Y, ;-) 2 Z +01)’ —ﬂr: (1,, /J >0 r= F M: /7 P= [_’ _ - _ - - , . ’ " Suppose now you draw the IS and the LM relationship in the (Y, e) plane with ) In the horizontal axis and e in the vertical axis. St"l'l€S O!
15. 15. Page 15 of26 SJ. In this case a) the IS curve is upward sloping and the LM cttrve is (l0''n-'; ll"(l sloping the IS curveis downward sloping and the LM curve is vet'tic'. tl c) the IS curve is vertical and the LM curve is upward sloping d) the IS curve is downward sloping and the LM curve is upward sloping 57.. An increase in F shifts ' a) the IS curve to the left and the LM curve to the right b) the IS curve to the right and the LM curve to the left c)" both the IS and the LM curves to the left d) both the IS and the LM curves to the right 53. An increase in F results in a) an increase in the equilibrium value of Y and a decrease in the equilibrium value of e / tr) a decrease in the equilibrium value of I’ and an iiicictise in the cqiiililirium value of e ’ c) at decrease in the eqttilibritint value of Y and '. l tlcci'c; t.~c in the cqtiilibriuni value of e ‘ d) none of the above 54. If the government arbitrarily fixes the real exchange rate at some (7 a) the two markets can be simultaneously in equilibrium only under special parametric restrictions ' b) the two markets can be siinultaneotislv in cquilibriuni ll thc governnicnt follows an accommodating interest rate policy c) the two markets can be simultaneously in equilibrium if the government follows an accommodating tnoncy supply rule d) all of the above 55. A fall in the interest rate a) will reduce savings unambiguously b) will have an ambiguous effect on savings because of an ambiguous substitution effect . ' will reduce savings unambiguously only for a borrower- d) will reduce savings unambiguously only for a lender. Spa An increase in the rate of depreciation. according to the neoclassical theory of investment, will ' / a) lower investment by raising the user cost of capital b) raise investment by lowering the user cost c) raise investment because now more capital is depreciating d) none of the above. 57. An increase in the saving rate in the _Solow model 21) increases the growth rate of the economy permanently . / b) increases the growth rate of the economy in the transition to the steady state but not in the steady state — Sllskys Oft
16. 16. Page 16 of 26 c) reduces the growth rate because aggregate demand falls (I) none of the above. 58. The “golden rule of accumulation" is the a) savings ratio that generates the highest growth rate of the economy I») the savirigs ratio that generates the highest capital-labour ratio c) the savings ratio where consumption (per capita) is maximized both in the transition to the steady state and in the steady state ‘/20 the savings ratio where consumption (per capita) is maximized in the steady state 59. At the “golden rule of accumulation” a) all wages invested , all profits constnned b) all wages as well as profits invested }7) all wages consumed, all profits invested (l) all wages as well as profits consumed ()0. Consider the following three definitions for a cotIntr_v's current account surplus. Which of them is correct’? (i) equal to its trade balance plus net income from abroad (ii) equal to its trade balance plus foreign direct investment (iii) equal to the change in its claims against the rest of the world a) (i) and (ii) b) (ii)and (iii) (i) and (iii) d) None fiskysoft Series 01
17. 17. at Delhi Schlqol of Economics Department of Econoinies Entrance Examination for M. A. Economics Option B I June 27, 2009 Time 3 hours Maximum marks 100 General instructions. l’lea. ~‘. v read the lnllm-. -iiig llI. ll'l1(‘l ions t'au'elully o Check that your t-xaiiiiiitttioii has [)Zgt’. \$ 1 tn (3 éilltl you have been given a l)l:1ltlZ Answer booklet. Do not start writing until lIlSll'll('l. t’ll to «to so by the iiwigilator 0 Fill in your Name and Roll Number on l-ll(' sniaill slip . l [i(‘l| (’tl to the . -nsm. -i lwm. -lzlt-1 Do not write this tiilorinatinii illl_''l1€l't‘ else in this lmnl-: li't - Vlieii you finish. liuntl in this Exﬂlllllltlltntlt aloni; -. -itli the Ansxver booklet to lltt' invigilator - Do not (lii: ~;tiirl> _mtn iieiglilmrs at . 'n_i' tinn- / u_ymi«'- engagilig in illegal t-x: mi— ination pi-aeti'r. es will In: itn1nediat. el_v ('. ’ltIl(: ll Zlllll -that person’s ('il| (lill? llll‘(‘ will l)t. ‘ t'.1)ll(? Bllt‘(_l. l Uo not llLr l-vluw llH> llltv‘ This space is lot Hllltlrll mi -; nl_x' l"it‘tit. iott. s' Roll NlllIIl}(‘: ' EEE 2009 /3 1
18. 18. Instructions. Answer Question 1. Question 1. Each of the ten parts to of this question is follnwo<l by lintr possible, answers ((a) to (dl), one of which is correct. Indicate the (. ‘()t‘t‘('. (‘. L ans“-t: i' for t? :(Zl] part. i ’ 3'01"‘ 'rHtS'Cl booklet. Eacli correct choice will earn you '2 nmrks ll0’(’'(? l’. _-on will lose‘ 2,73 tnark for each incorrect choice. (A) A binary relation R on a set S is a subset of S2‘ R. is said to he t. r:msiti'e if for all J: ,y.2 6 S. (I, y) E R and (y, z) E R implies (1,2) 6 R. R. is said t. o be negatively‘ tr-ansit. iA'e ii’. for all . 'L'. y,z E S, (Ly) Q’ R and (y, z) {J It implies (. r.: _l Q’ If‘ Deﬁne the liiiiarv I‘(’lZl. iOll It’. "' on S by R‘l = 6 S‘ ] (.1/, .r) E I? ) Consider the following Sl. &ll. t:‘t’lit’tll. S: a ll" 1? is transitive, then it is not riegzitii-'el_v I. miisiti-c. a ll" R is transitive, then R” is transitive. - ll H is transitive, then R" is negatix-el_t' ttmisitive. g llm-c nt. iny of the above stateinents are true)’ tn} 0 (ll l (cl 2 (ti) .3 (Bl Cmisidtti‘ : language with only two letters in its iilplialwt. 0 anti W. This lzliiguagé olieys the lollowiiig rules: (i) Deleting successive letters W0 from any word ’lll(: ll has more th:1n two letters yields another word with the SitllI(‘ meaning. (ii) ltiserling OH" or W l/ l/O0 in : tn' place in a word yields another word with the same nieaning. (iii) (7. Olrl/ O0W, v ll'()(_) and 0l’l*'ll’ are words in this laiigtiagtx Wliirli of the lollowing . <tzil.0iiir~itt. s is false? (:1) The worrls WOO and OWVV have the szmte meaning. (ii) li’()() and OWW may not lla. "eitlIC same meaning. ' (C) O and OWOOW have the same meaning. . (tl) Both (b) and (e) are true. (C) Tltere are three persons: .4, B and C. One of them a Truth—t. eller (who alwaysiu H-, i|5 the L, -ui], )_ another is a Liar (who always lies) and the third is a normal person (who soinetiines lies, other times speaks the truth). A said: “I am a normal person. " B said: -' “. -l and C sometimes tell the truth. " C said: “/3 is a normal person. " -A, B and C are 5 av: t.t'e of ever_y person’s nature. iiskysort 2
19. 19. (ll) Tliese slnteenienls are insnflidenl to determine the Liar. (l)) A is the normal person, B is the 'I‘ruth—teller. C is the Liar. (c) These st: teine‘nts are insufficient to determine the Liar, the normal person and the Truth-teller. (d) A is the l. .i: n'. 13 is the normal person. C is the Truth-teller. (D) Let _[(r) = max{. r.l~ ‘2y i 1‘ 2 U. ‘y 2 0. 21- + y = The (lerivati'e of f at r is h0" (b) I) (C) ‘2 (d) 6/? (I3) Consinler the stxm-r'm-nts: (i) 5'” > -153. and (ii) ‘2'”" + 3”" < 41"") (n) llmh (i) znul Iii; are false. (1)) ii} is true. (iii i>: lnlse. (V) liutli (i) nnil i_ii}':11'vLiin'. N) (i) is lnlsv. (ii) is true. ll“) l-l‘l. Y (l(‘ll()l' lll(' innnlun ul lw: u.ls olit. -nim-<l whvn | lll'l‘U Jam <'uin. ~. ; m- li). 't‘tl. ’l'ln'-‘ '; ni; un'v ul Y‘) i> (21) 9.5 (l)) 8,5 (C) 6.5-7 ((l)‘ 7.? ) (C) Consider events .4 and B with l"i= ub(. z1) = U. «l‘: nid I’rol)(B) = l).7. Tlirr inzixiinuni uml minimum values of Prolilyl H B) respe(: t.ircl_y are (:1) 0.4 and 0.1 (b) 0.7 and 0.4 (C) 0.7 and 0.1 (d) 0.4 and 0 (H) The nine digits ], ... ,9 are arranged randomly to form a nine digit number with each digit used exactly once. The probability that 1. 2 and 3 appear as neighbors in the increasing order is (a) 1/12 (b) 1/72 (c)1/84 (d)(? /3V’ BEE 2009 B 3 iskysoft
20. 20. Q? 5-». (I) A hlnml Lest cl<~. t,e<: ts a given ilismw-= with pml. >nl_iilit. _g 8/10 givnn that the t. c_<Lc. l ~ actimll_- has the disease. With probability 2/ it). the Lost, inmrmcily 5],0w5 Um 1,, -i. ,;. -,_ the cli. '~; ease in 21 disease-free person. Siippnse 1/10 of the population l| '«. S the (liseasc. h" is the probability that the person tested acl. ualI_y has the disease if the test intlicv. .3 presence of the disease? (3.) 1 (li) 9/13 (0) 4/13 («ll T/ lii (Ji Suppose X and Y are indepemlniil rnniimn ‘-“li'l5. l)l(‘. S with stmiilimi Noimzil 9. tions The pmbal)ilit. y 0|‘ z' < --1 is Hll1](. ’]) e’ (4). 1). Wlizii’ is the pi‘nlinli| i|_. - of 11., X3 l nml Y‘ < ~17 in‘: 3;: (M 15.2 (L I 2123 (<l- Ilgi‘ Section ll Instructions. Answer : in_i‘ four of the Inllt')i'iII[{ iii-n qnesr. ion. ~ II) the . -nsu'(-: Eacli qIl(‘. "<llOI]. is worth 20 inzirks. Thv ni. -irks [oi various parts ofn qiiosiiuii . -ire .11‘. - if at the curl of each question. Question 2. Let (. , . ) : S? " x 5%" - 31‘ bra : m inner product on 9?". (A) Show that, if my 6 3?". then / .r.2_/ ') S (. "r, .'c)‘/2(y. ,y)l/2. 4 (Bi Show that equality holds in l. =') if and only if y = 0 or rr 2 / y for smne . - ~. tin. Let A be an 71. X 71 real n1i. tl’lX. ;' siilasiinrrr V of? ??" is said to lie in'; ris. int. with i(’. ‘?['f" ' to . 'l il‘ / lV C V (C) If A is symmetric and V C ii? " is : in inVI‘1.I'l-'v1Yl[‘Sl, ll3S[')7(‘C with respect 1.0 7.. 2 there (exists A 3 O and . r, E V. .1’ 7.‘ H. .<n(‘h that A :1‘ —= x.17. iii». 2, 1?: Question 3. Let V be a vector space and lei . C(/ , V) be the set olalllinear transim n; :«. Lio1.. '1‘ ; V _ V_ Let, 1 e £(V, »’) be the idanl. it. _- i. ra. nsformation. P E £(V, V) is Cillliiil pi‘Oject. o1- of V if (3) V : ’R, (P) (§_. :'(]7)_ and (b) P(1i. +1u) 11 for all u E ‘/ .1’/ ' ll EEE 9009 B
21. 21. r. iiskysoft w E . ’(P): where 7Z(P) denotes the range space of P, A/ (P) denotes the null space of I’ and G3. denotes a direct sum, Prove the following statements. « (A) P is a projector of V if and only if it is ideinpotent, i. e., P2 = P. (B) If U is a vector space and X : U —» V is a linear transformation with ‘R(P) = 7?. (X), then P is a projector if and only if PX = X. (C) P is a projector if and only if I — P is a projector. P (s, e, 6) Question 4. Consider a metric space (, ', rI) and a function f : X —- f is said to be upper seniicmitinuous at I 6 X if for every ( > 0, there exists an open iieighliorlioocl U of r. such tlnit f(U) C (-—oo, f(a: ) + c). f is snirl to be upper seinicontinuons if it is so at every IL‘ 6 X. Prox-‘e the following st: il. eniems. (AI _/ is upper scmit-ontinm)ii. < if : Il'l(l unli‘ if _/ ""l((—: ‘. rjl is '! [)(‘H in (. '. ii} in: (, ‘'(‘l‘}' 1' t 3): ‘. 7 ill) ll H", I I E 1} is it fniiiili‘ oi upper . (’H1l(, l)Illllllll)ll. 'fllllCIl()lI>L_]-, : X —— '11‘. llllfll lln. ‘ fuiiclion i; : . ‘ --~ Eli‘ ilefined by g(. r) : : ll]lV(‘/ ‘,(. I i l i if 1} is upper . SCllIl(‘L)lifiilll0Ll>' (C) If {f, | : E I) is 21 fainily of upper . iL‘llli('()l]IlllllUl| .\$ l'un(~tions 4/‘ : X —- ’h‘ unrl I is n ﬁnite Sci. then the function g : X —» 5? defined by _r/ (Jr) = sup{f, (:r) I i E 1} is upper seiiiic-ontiiiuoiis. A ib’, G.8) Question 5. L. 'unsider the . EllCll(l(: ‘illl-lll(3ll'l(‘ . <[). 'u‘e (’Ji""’, Let X . ‘li"‘ and _/ A : .-' —« Ill‘. X is said to in-21 convex set iffor every : I.‘. y 5 . ' znnl I E (tl.1). veh: ive t. 'r—i—(_l —~ my -. =_ f is sziicl to be a convex function at 1' 6 X if for i. -vi. -r_v y E X and L E (U. 1). 11+ ii — 1); ; E X implies f(t: r + (1 — t)y) S tf(a: ) + (1 — ! )j(y). f is said to be a convex function if it is 11 convex function at every '1: E X. Prove the folloxving statements. (A) lf X is H convex set, then f : X —o ll? is a convex function if and only if {(: r.y) 6 X X 3? l f(: z:) 5 y) is ‘a convex set. ’ I I (B) If X is open in ER" and f is convex and differentiable at :1: E X, then f(y) — f(r) Z Df(: v)-(y - 1') fol‘ L''<t: '_«' 1/ G X. where 1)f(:1:) denotes the (li-'rivzili'e of f at : i:. (C) HA’ is open in 9?" and f is convex and twice dil‘l1-rentiable all .1‘ e X. iln-n / )"’_/ ‘(. i:) is positive seiiiideliiiite. (4.—8,5) EEE 2009 B 5
22. 22. . Question 6. Lcl, A/ be the set. of natural numbers. A set. X is said to be’ nnipty or l. ln-= .r(: uxisls 21 l, )i_iect, ion f 2 {I, .. . , 11} —~> X for some n 6 JV. he rlreiiiiinemlile if there is a bi_jecl, ion f : _/ V —o X. If countable. Prove the following stat ‘ (A) Consider an indexed family of sets {X, - l i E i clonumerable for every i E I, then U I (B) If n 6 JV and (C) A set, > In 39 X is m'l. l1r-3r finite or‘ A then it is said to be l ements. I}. ll’ I is (lcnnmer . ~e; X, is denumerable. X is a denumerable set, then X" is denumernble. X is counlzablc if and only if there is an i mection f: X -4 / V”. l ' iskysoft EEE 200.9 B
23. 23. pw1x——Z—— DSE Satwhlowy Pre/ mired AI«w(L{'/ <u«ma4’ doyaé 055- Year 2004 Pan‘-1 M 32 (M I1<'/ I39fVl**/ (dz) 0.0251/(2-rt) (0/l NTWl»WI"l0W (0) P06 a«wl«Q(></ ) MOMMMM (ala ALL+lMee/ are/ wrrwf (la) 5/6, 5/4, 5/9 (W) R2 8 (at) R ; ¢;, u.a. rwbfor H«1xrU_jrbs44". ovv (ax) P(X>Y) = 1 (b5 16 (bl 15 . (cu) xx: 3p/4 (ob) I3a+l~ 3/2 aunts/2 (b«) 12 R: /496%» (C3 H4-lx real, L4'4‘tWb9i" Vwhz wlrvuv vwea», W4'wl/ Luv rwlx ftwvvvy tuwb flint : x.: «.~. ~.. ,c». ..: J; Lvvfuu+rwlvw£~uvmww4»wL(, wwwuxaLfwvw 17. (0) {xv H’W/64/M»ULl9Vt/ M4“/ vﬂ»(Ml/ Ofrﬁa-1/i»Vvl’? zV©\$? l‘Vﬂrl'0 18- (ob) Wﬁvwrﬂl/ -°‘1>P4'M€ 19. (ca woualol/ re9uJ, {—(, wa«oLe<>reaMvLvv+l'~. o:: v.. =1g izffivw Ma»r&l¢aJ. L—Lervwr oovwl»uh. ovv' ' i»\$«sa»h&fwd«‘ ' 20. (az) Fbve/ perouvf H .9.-°9°. l. °‘\$"I: l‘. “?"l“ H 3‘ H N H U H .4‘ H 91 H .5‘ Pa4+—.2 1- M’) (0) WWW (ii/ l WWW 2. (M ﬂ«1xword4«WO0 anolx0/ x/W neowso-riLg lware/ ﬂ«. &sawwomea. m', «g.
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25. 25. 40- (0) (0) wwwwvl