Test code: ME I/ME II, 2007

Transcription

1 Test code: ME I/ME II, 007 Syllabus for ME I, 007 Matrx Algebra: Matrces and Vectors, Matrx Operatons. Permutaton and Combnaton. Calculus: Functons, Lmts, Contnuty, Dfferentaton of functons of one or more varables, Unconstraned optmzaton, Defnte and Indefnte Integrals: ntegraton by parts and ntegraton by substtuton, Constraned optmzaton of functons of not more than two varables. Lnear Programmng: Formulatons, statements of Prmal and Dual problems, Graphcal solutons. Theory of Polynomal Equatons (up to thrd degree). Elementary Statstcs: Measures of central tendency; dsperson, correlaton, Elementary probablty theory, Probablty mass functon, Probablty densty functon and Dstrbuton functon. Sample Questons for ME I (Mathematcs), 007. Let α and β be any two postve real numbers. Then α ( + x) Lm x 0 equals β ( + x) α α + (A) ; (B) ; (C) β β + α ; (D). β. Suppose the number X s odd. Then X s (A) odd; (B) not prme; (C) necessarly postve; (D) none of the above.

2 . The value of k for whch the functon f = kx ( x) ke s a probablty densty functon on the nterval [0, ] s (A) k = log ; (B) k = log ; (C) k = log ; (D) k = log p and q are postve ntegers such that p q s a prme number. Then, p q s (A) a prme number; (B) an even number greater than ; (C) an odd number greater than but not prme; (D) none of these. 5. Any non-decreasng functon defned on the nterval [ a, b] (A) s dfferentable on (a, b); (B) s contnuous n [ a, b] but not dfferentable; (C) has a contnuous nverse; (D) none of these. x 4 6. The equaton = 0 s 8 satsfed by (A) x = ; (B) x = ; (C) x = 4 ; (D) none of these. 7. If f ( x) = x + x + x + x + K,, then f (x) s (A) x ; (B) f ( x) ; (C) f ( x) x ; (D) f ( x). f ( x) + 8. If P = log ( xy) and Q = log ( xy), then P + Q equals x (A) PQ; (B) P/Q; (C) Q/P; (D) (PQ)/. y x + 9. The soluton to dx s 4 x + x 4 x + x (A) 4x + + constant; (B) log x + log x + constant;

3 (C) log x 4 + x + constant; (D) 4 x + x 4x + + constant. 0. The set of all values of x for whch x x + > 0 s (A) (,) ; (B) (, ) ; (C) (, ) (, ) ; (D) (,) (, ).. Consder the functons f ( x) = x and f ( x) = 4x + 7 defned on the real lne. Then (A) f s one-to-one and onto, but not f ; (B) f s one-to-one and onto, but not f ; (C) both f and f are one-to-one and onto; (D) none of the above.. If a+ b+ x a + x f ( x) =, a > 0, b > 0, then f (0) equals b + x (A) b a+ b a b a b ; (B) a b a log ( ) + b ab a b a+ b ; (C) a b a log + b ab b a ; (D). ba. The lnear programmng problem has max z = 0.5x +.5y x, y subject to: x + y 6 x + y 5 x + y 5 x, y 0

4 (A) no soluton; (C) a corner soluton; (B) a unque non-degenerate soluton; (D) nfntely many solutons. 4. Let f ( x; θ ) = θ f ( x;) + ( θ ) f ( x; 0), where θ s a constant satsfyng 0 < θ <. Further, both f (x;) and f (x; 0) are probablty densty functons (p.d.f.). Then (A) f ( x; θ ) s a p.d.f. for all values of θ ; (B) f ( x; θ ) s a p.d.f. only for 0 < θ < ; (C) f ( x; θ ) s a p.d.f. only for θ < ; (D) f ( x; θ ) s not a p.d.f. for any value of θ. 5. The correlaton coeffcent r for the followng fve pars of observatons x 5 4 y satsfes (A) r > 0 ; (B) r < 0. 5 ; (C) 0.5 < r < 0 ; (D) r = An n -coordnated functon f s called homothetc f t can be expressed as an ncreasng transformaton of a homogeneous functon of degree one. Let and n = n r x = f ( x =, f ( x = a x + b, where x > 0 for all, 0 < r <, a > 0 and b are constants. ) Then (A) f s not homothetc but f s; (B) f s not homothetc but f s; (C) both f and f are homothetc; (D) none of the above. ) 7. If h x) = x (, then ( h( h( x) ) h equals 4

5 (A) ; (B) x ; (C) ; (D) - x. x x x 8. The functon x x + s x (A) contnuous but not dfferentable at x = 0; (B) dfferentable at x = 0; (C) not contnuous at x = 0; (D) contnuously dfferentable at x = 0. dx 9. ( x )( x ) x x( x ) (A) log ( x ) equals + constant; ( x ) (B) log + x( x ) constant; x (C) log + ( x )( x ) constant; ( x ) (D) log + x( x ) constant. 0. Experence shows that 0% of the people reservng tables at a certan restaurant never show up. If the restaurant has 50 tables and takes 5 reservatons, then the probablty that t wll be able to accommodate everyone s (A) 09 ; (B) ; (C) ; (D)

6 . For any real number x, defne [x] as the hghest nteger value not greater than x. For example, [0.5] = 0, [] = and [.5] =. Let I = {[ x] + [ x ]}dx. Then I equals 0 (A) ; (B) 5 ; (C) ; (D) none of these.. Every nteger of the form ( n n)( n 4) (for n =, 4, K) s (A) dvsble by 6 but not always dvsble by ; (B) dvsble by but not always dvsble by 4; (C) dvsble by 4 but not always dvsble by 0; (D) dvsble by 0 but not always dvsble by 70.. Two varetes of mango, A and B, are avalable at prces Rs. p and Rs. p per kg, respectvely. One buyer buys 5 kg. of A and 0 kg. of B and another buyer spends Rs. 00 on A and Rs. 50 on B. If the average expendture per mango (rrespectve of varety) s the same for the two buyers, then whch of the followng statements s the most approprate? (A) p = p ; (B) p = p ; 4 p (C) p = p or p = p ; (D) <. 4 4 p 4. For a gven bvarate data set ( x, y ; =,, K, n), the squared correlaton coeffcent ( r ) between x and y s found to be. Whch of the followng statements s the most approprate? 6

7 (A) In the (x, y) scatter dagram, all ponts le on a straght lne. (B) In the (x, y) scatter dagram, all ponts le on the curve y = x. (C) In the (x, y) scatter dagram, all ponts le on the curve y = a + bx, a > 0, b > 0. (D) In the (x, y) scatter dagram, all ponts le on the curve y = a + bx, a, b any real numbers. 5. The number of possble permutatons of the ntegers to 7 such that the numbers and always precede the number and the numbers 6 and 7 always succeed the number s (A) 70; (B) 68; (C) 84; (D) none of these. 6. Suppose the real valued contnuous functon f defned on the set of non-negatve real numbers satsfes the condton f ( x) = xf ( x), then f () equals (A) ; (B) ; (C) ; (D) f (). 7. Suppose a dscrete random varable X takes on the values 0,,, K, n wth frequences proportonal to bnomal coeffcents n, 0 n,, n n respectvely. Then the mean ( µ ) and the varance ( σ ) of the dstrbuton are n n (A) µ = and σ = ; n n (B) µ = and σ = ; 4 4 n n (C) µ = and σ = ; 4 n n (D) µ = and σ =. 4 7

8 8. Consder a square that has sdes of length unts. Fve ponts are placed anywhere nsde ths square. Whch of the followng statements s ncorrect? (A) There cannot be any two ponts whose dstance s more than. (B) The square can be parttoned nto four squares of sde unt each such that at least one unt square has two ponts that les on or nsde t. (C) At least two ponts can be found whose dstance s less than. (D) Statements (A), (B) and (C) are all ncorrect. 9. Gven that f s a real-valued dfferentable functon such that f ( x) f ( x) < 0 for all real x, t follows that (A) f (x) s an ncreasng functon; (B) f (x) s a decreasng functon; (C) f (x) s an ncreasng functon; (D) f (x) s a decreasng functon. 0. Let p, q, r, s be four arbtrary postve numbers. Then the value of ( p + p + ) ( q + q + ) ( r + r + ) ( s + s + ) pqrs (A) 8; (B) 9; (C) 0. (D) None of these. s at least as large as 8

9 Syllabus for ME II (Economcs), 007 Mcroeconomcs: Theory of consumer behavour, Theory of Producton, Market Structures under Perfect Competton, Monopoly, Prce Dscrmnaton, Duopoly wth Cournot and Bertrand Competton (elementary problems) and Welfare economcs. Macroeconomcs: Natonal Income Accountng, Smple Keynesan Model of Income Determnaton and the Multpler, IS-LM Model, Model of Aggregate Demand and Aggregate Supply, Harrod-Domar and Solow Models of Growth, Money, Bankng and Inflaton. Sample questons for ME II (Economcs), 007. (a) There s a cake of sze to be dvded between two persons, and. Person s gong to cut the cake nto two peces, but person wll select one of the two peces for hmself frst. The remanng pece wll go to person. What s the optmal cuttng decson for player? Justfy your answer. (b) Kamal has been gven a free tcket to attend a classcal musc concert. If Kamal had to pay for the tcket, he would have pad up to Rs. 00/- to attend the concert. On the same evenng, Kamal s alternatve entertanment opton s a flm musc and dance event for whch tckets are prced at Rs. 00/- each. Suppose also that Kamal s wllng to pay up to Rs. X to attend the flm musc and dance event. What does Kamal do,.e., does he attend the classcal musc concert, or does he attend the flm musc and dance show, or does he do nether? Justfy your answer.. Suppose market demand s descrbed by the equaton P = 00 Q and compettve condtons preval. The short-run supply curve s P = Q. Fnd the ntal short-run equlbrum prce and quantty. Let the long-run supply curve be P = 60 + Q. Verfy whether the market s also n the long-run equlbrum at the ntal short-run equlbrum that you have worked out. Now suppose that the market demand at every prce s 9

10 doubled. What s the new market demand curve? What happens to the equlbrum n the very short-run? What s the new short-run equlbrum? What s the new long-run equlbrum? If a prce celng s mposed at the old equlbrum, estmate the perceved shortage. Show all your results n a dagram.. (a) Suppose n year economc actvtes n a country consttute only producton of wheat worth Rs Of ths, wheat worth Rs. 50 s exported and the rest remans unsold. Suppose further that n year no producton takes place, but the unsold wheat of year s sold domestcally and resdents of the country mport shrts worth Rs. 50. Fll n, wth adequate explanaton, the followng chart : Year GDP = Consumpton + Investment + Export - Import (b) Consder an IS-LM model for a closed economy wth government where nvestment ( I ) s a functon of rate of nterest ( r ) only. An ncrease n government expendture s found to crowd out 50 unts of prvate nvestment. The government wants to prevent ths di by a mnmum change n the supply of real money balance. It s gven that = 50, dr slope of the LM curve, dr ( LM ) = dy 50, slope of the IS curve, dr dy ( IS) = 5, and all relatons are lnear. Compute the change n y from the ntal to the fnal equlbrum when all adjustments have been made. 4. (a) Consder a consumer wth ncome W who consumes three goods, whch we denote as =,,. Let the amount of good that the consumer consumes be x and the prce 0

11 of good be p. Suppose that the consumer s preference s descrbed by the utlty functon U ( x, x, x) = x x x. () Set up the utlty maxmzaton problem and wrte down the Lagrangan. () Wrte down the frst order necessary condtons for an nteror maxmum and then obtan the Marshallan (or uncompensated) demand functons. (b) The producton functon, Y = F( K, L), satsfes the followng propertes: () CRS, () symmetrc n terms of nputs and () F (,) =. The prce of each nput s Rs. /- per unt and the prce of the product s Rs. /- per unt. Wthout usng calculus fnd the frm s optmal level of producton. 5.(a) A monopolst has contracted to sell as much of hs output as he lkes to the government at Rs.00/- per unt. Hs sale to the government s postve. He also sells to prvate buyers at Rs 50/- per unt. What s the prce elastcty of demand for the monopolst s products n the prvate market? (b) Mrs. Pathak s very partcular about her consumpton of tea. She always takes 50 grams of sugar wth 0 grams of ground tea. She has allocated Rs 55 for her spendng on tea and sugar per month. (Assume that she doesn t offer tea to her guests or anybody else and she doesn t consume sugar for any other purpose). Sugar and tea are sold at pasa per 0 grams and 50 pasa per 0 grams respectvely. Determne how much of tea and sugar she demands per month. (c) Consder the IS-LM model wth government expendture and taxaton. A change n the ncome tax rate changes the equlbrum from ( = 000, r = 4% ) ( = 500, r = 6% ) y to y, where y, r denote ncome and rate of nterest, respectvely. It s gven that a unt ncrease n y ncreases demand for real money balance by 0.5 of a unt. Compute the change n real money demand that results from a % ncrease n the rate of nterest. (Assume that all relatonshps are lnear.)

12 6. (a) An economy produces two goods, corn and machne, usng for ther producton only labor and some of the goods themselves. Producton of one unt of corn requres 0. unts of corn, 0. machnes and 5 man-hours of labor. Smlarly, producton of one machne requres 0.4 unts of corn, 0.6 machnes and 0 man-hours of labor. () If the economy requres 48 unts of corn but no machne for fnal consumpton, how much of each of the two commodtes s to be produced? How much labor wll be requred? () If the wage rate s Rs. /- per man-hour, what are the prces of corn and machnes, f prce of each commodty s equated to ts average cost of producton? (b) Consder two consumers A and B, each wth ncome W. They spend ther entre budget over the two commodtes, X and Y. Compare the demand curves of the two consumers under the assumpton that ther utlty functons are U B x + y = respectvely. U A = x + y and 7. Consder a Smple Keynesan Model wthout government for an open economy, where both consumpton and mport are proportonal functons of ncome (Y ). Suppose that average propenstes to consume and mport are 0.8 and 0., respectvely. The nvestment ( I ) functon and the level of export ( X ) are gven by I = Y and X = 00. () Compute the aggregate demand functon f the maxmum possble level of mports s 450. Can there be an equlbrum for ths model? Show your result graphcally. () How does your answer to part () change f the lmt to mport s rased to 65? What can you say about the stablty of equlbrum f t exsts?

13 8. Suppose an economc agent s lfe s dvded nto two perods, the frst perod consttutes her youth and the second her old age. There s a sngle consumpton good, C, avalable n both perods and the agent s utlty functon s gven by C C θ θ u ( C, C ) = θ +, 0 < θ <, ρ > 0, + ρ θ where the frst term represents utlty from consumpton durng youth. The second term represents dscounted utlty from consumpton n old age, /(+ ρ ) beng the dscount factor. Durng the perod, the agent has a unt of labour whch she supples nelastcally for a wage rate w. Any savngs (.e., ncome mnus consumpton durng the frst perod) earns a rate of nterest r, the proceeds from whch are avalable n old age n unts of the only consumpton good avalable n the economy. Denote savngs by s. The agent maxmzes utlty subjects to her budget constrant. ) Show that θ represents the elastcty of margnal utlty wth respect to consumpton n each perod. ) Wrte down the agent s optmzaton problem,.e., her problem of maxmzng utlty subject to the budget constrant. ) Fnd an expresson for s as a functon of w and r. (v)how does s change n response to a change n r? In partcular, show that ths change depends on whether θ exceeds or falls short of unty. (v)gve an ntutve explanaton of your fndng n (v) 9. Consder a neo-classcal one-sector growth model wth the producton functon Y = KL. If 0% of ncome s nvested and captal stock deprecates at the rate of 7% and labour force grows at the rate of %, fnd out the level of per capta ncome n the steady-state equlbrum.