DSE 2011 question paper

All 60 questions from the DSE 2011 Delhi School of Economics MA entrance paper. Open any question for the worked solution.

1. Q1For the real-valued function f(x)=x⁴-4x³+6x²-4x+1 (defined for all real numbers x), the point x=1 is
2. Q2Consider the function f mapping points of the plane into the plane, defined by f(x,y)=(x-y, x+y). The range of this function is
3. Q3Suppose {v₁,v₂,…,vₙ} is a set of linearly dependent vectors, none of them being the zero vector. Suppose c₁,c₂,…,cₙ are scalars, not all zero, such…
4. Q4Suppose the non-zero n× 1 column vector x solves the system of equations Ax=b, where A is an m× n matrix whose columns are the vectors a₁,a₂,…,aₙ…
5. Q5For a system of linear equations Ax=b with m equations and n variables, where m>n and b is a given vector, the following is true.
6. Q6Suppose we are interested in estimating the mean height of M.A. students in Delhi University. An average height estimated from a random sample of…
7. Q7A doctor testing a diagnostic tool for a rare disease wants to minimize the chance that the test will find a patient to be healthy when she is in…
8. Q8Suppose you have the following estimated regression equation: ln Y=1.2+0.5X. Which of the following is a good interpretation of the estimated…
9. Q9Let A₁,A₂,A₃ be independent events with probabilities p₁,p₂,p₃ respectively. The probability that none of these events occurs equals
10. Q10The market for good X has the demand function D(p)=100-15p where p is the price of good X. There are ten price taking firms, each having a cost…
11. Q11The market for good X has the demand function D(p)=100-p and the supply function S(p)=20+3p where p is the price of good X. If the government…
12. Q12A monopolist, who can produce output at a cost of Rs. 20 per unit, faces an inverse demand curve given by p=60-2q. The profit maximizing price for…
13. Q13Consider a duopoly market in which both firms choose quantities. Suppose we have the reaction curve of each firm, i.e., the curve that yields the…
14. Q14A society has 3 individuals and 3 alternatives A,B and C. Individuals 1 and 2 strictly prefer A to B and B to C. Individual 3 strictly prefers C to…
15. Q15A natural monopoly is said to occur when
16. Q16If everybody in the economy decides to consume a higher proportion of their income, the corresponding IS curve
17. Q17An increase in the expected rate of inflation will
18. Q18A devaluation of domestic currency will improve the country's trade balance only if the following is true for price elasticities of demand:
19. Q19Under nominal wage rigidity, the short run aggregate supply schedule will be
20. Q20Okun's Law refers to
21. Q21Suppose the function f:ℜ₊₊→ℜ is given by f(x)=∫₁^xt⁻¹ dt. Consider the following statements: for x,y∈ℜ₊₊: (i) f(x+y)=f(x)+f(y); (ii)…
22. Q22Suppose a real valued function f is defined for all real numbers excepting 0, and satisfies the following condition: f(xy)=f(x)+f(y) for all x,y in…
23. Q23The closest point on the parabola y=1/4x² from a given point (0,b) on the vertical axis, with b>0, is the origin if and only if
24. Q24Let f be a real valued differentiable function defined for all x≥ a. Consider the function F defined by F(x)=∫ₐ^xf(t) dt. If f is increasing on any…
25. Q25Consider the function f:ℜ→ℜ defined by f(x)={ x², if x∈ℚ; 0, otherwise } where ℚ is the set of rational numbers. Then, f is
26. Q26Suppose A and B are square matrices that satisfy AB+BA=0, where 0 is a square matrix of zeros. Then it must be that
27. Q27Consider a Society of individuals who may belong to sets called Clubs and Tribes, satisfying: the entire Society is a Tribe; the empty subset is a…
28. Q28(Same Clubs-and-Tribes setup as the previous question.) The union of a collection of Clubs is necessarily
29. Q29(Same Clubs-and-Tribes setup.) Which of the following statements is necessarily true?
30. Q30(Same Clubs-and-Tribes setup.) Given a Club and a Tribe, the set of individuals who belong to the given Tribe but not to the given Club necessarily…
31. Q31Suppose X and Y are random variables. X takes values -1 and 1; Y takes integer values 1 to 6. The joint distribution has P(X=-1) row [0.1, a, 0.3…
32. Q32(Same joint distribution as the previous question.) The value ' b ' in the table equals
33. Q33Suppose X and Y are independent random variables that follow the uniform distribution on the interval [0,1]. Let Z=min {X,Y}. Pr (Z<0.5) (the…
34. Q34(Same setup: X,Y i.i.d. U[0,1], Z=min {X,Y} .) Pr (Z<0.5| X=0.75) is
35. Q35Two players, A and B, play a best of seven table tennis match (the first to win 4 games wins the match, at most 7 games). The two players are…
36. Q36In order to join a club, Mr. A must choose one of two identical boxes and draw one ball from it; he is admitted if the drawn ball is green. You are…
37. Q37Consider the equation estimated using ordinary least squares: Y=a₀+a₁X₁+a₂X₂+a₃(X₂-X₁)+a₄X₁X₂+u. Which of the following statements is true?
38. Q38A researcher wants to test whether there are gender differences in the rates of immunization for boys and girls, after controlling for parental…
39. Q39Instead of estimating a production function y=a₀+a₁K+a₂L+u (where y is output, K is capital input and L is labour input), a researcher estimates…
40. Q40A student may take a test at most thrice. Each time, his score is an independent draw from the uniform distribution on [0,100]. After learning a…
41. Q41A consumer has utility function u(x₁,x₂)=min {2x₁+x₂, x₁+2x₂}. Her income is y=100, the prices are p₁=20 and p₂=30. The amount of x₁ in the utility…
42. Q42Consider the same utility function and income as above, u=min {2x₁+x₂, x₁+2x₂}, y=100, but suppose the prices are p₁=10 and p₂=30. Then the amount…
43. Q43A consumer spends Rs. 100 on only two goods, A and B, with non-satiation. The price of B is fixed at Rs. 20. When the price of A is Rs. 10, the…
44. Q44A firm uses two inputs to produce output. Greater use of both inputs increases output. Moreover, for any combination of positive input prices, the…
45. Q45Consider an exchange economy with two agents, 1 and 2, and two goods, X and Y. There are 6 units of X and 4 units of Y available. An allocation is…
46. Q46Consider an exchange economy with two agents and two goods; the agents have the same Leontief preferences min {x,y} as in the previous question.…
47. Q47Consider an exchange economy with two agents and two goods. Agent 2's utility is Leontief min {x,y} as above. The equilibrium allocation is…
48. Q48A firm has an order to supply 20 units of output. It can divide production across two plants, 1 and 2, with cost functions c₁(q₁)=q₁² and…
49. Q49Consider an exchange economy with two agents and aggregate endowment (10,10). The agents' utility functions are u₁(x₁,y₁)=x₁+2y₁ and…
50. Q50Consider two games in strategic form. Game 1: the row player chooses Enter or Not enter and the column player's only option is Hawk, with payoffs…
51. Q51Suppose the following equation holds for an economy at every time t: Pₜ=(1+a)WₜNₜ/Yₜ. That is, the price of output is a fixed markup over unit…
52. Q52Consider the fixed-price model: C=C₀+c₁(Y-T) with 0<c₁<1; I=i₀+i₁r with i₁<0; Y=C+I+G; money demand M^d=m₀+m₁Y+m₂r with m₂<0<m₁; and M^d=M. In this…
53. Q53(Same fixed-price model.) The more negative is m₂, ceteris paribus,
54. Q54(Same fixed-price model.) If c₁ is very close to 1, then an increase in M has
55. Q55Consider an economy with aggregate production Yₜ=A(Kₜ)^α(Lₜ)^(1-α), A>0, 0<α<1, both factors fully employed. A constant fraction s of output is…
56. Q56(Same Solow economy.) The rate of growth of aggregate output in the steady state is
57. Q57(Same Solow economy.) The golden rule value of capital per worker is
58. Q58(Same Solow economy.) The golden rule level of consumption per worker is
59. Q59(Same Solow economy.) An increase in the value of the parameter A from its current value
60. Q60(Same Solow economy.) An increase in the value of the parameter n from its current value