DSE 2021 question paper

All 47 questions from the DSE 2021 Delhi School of Economics MA entrance paper. Open any question for the worked solution.

1. Q1Consider the problem of maximizing f(x,y) = x² + 2xy + by² with respect to x, y subject to the constraint x + y = 100, where b is a parameter taking…
2. Q2Let f be a real-valued, continuous function defined on the set of all real numbers, satisfying f(f(x)) = x for all x ∈ ℝ. If f is not the identity…
3. Q4Find the value of Σᵣ₌₀⁵⁰ C(100, 2r). (Notation: C(a, b) denotes a choose b .)
4. Q5Suppose that n participants with distinct names arrive for a conference. The organizers have prepared name-tags for all of them, but by mistake…
5. Q6A right-angled triangle has two equal sides of 1 inch. One of these sides, drawn horizontally, is divided into (n+1) equal portions. On each portion…
6. Q7If |x| < 1 and y = Σₖ₌₁^∞ (-1)^(k+1) x^k/k, then which of the following is true? (Notation: |x| denotes the modulus of x and k! is the factorial of…
7. Q8Consider the vector space ℝⁿ for some n ≥ 2, over the field of real numbers, and the following two statements. Statement A: {v₁, v₂, …, vₙ₋₁, vₙ}…
8. Q9Consider the following system of equations in the unknowns x₁, x₂, x₃, with λ a real parameter: x₁ + 2x₂ - 2x₃ = 0, 2x₁ - x₂ + λ x₃ = 0, 3x₁ + x₂…
9. Q10There are four statements. Which of these are true? Statement 1: {y is a rational number | -200 ≤ y ≤ 200} is a countable set. Statement 2: {y = m/n…
10. Q11If x² + y² + z² = 1, then what is the maximum value of (-x + z)?
11. Q12Consider a triangle with sides AC = 2, AB = √3 and BC = 1. Choose a random point P inside the triangle. The minimum distances from P to AC, AB and…
12. Q13One girl is training for a half marathon in Delhi. She runs xᵢ kilometres on the i -th day for the next 12 days (x₁ on day 1, x₂ on day 2, and so…
13. Q14Three boxes are presented to a contestant. Exactly one box has an expensive gift and the other two are empty. Each box has a declaration imprinted…
14. Q15Let a₁, a₂, a₃, …, aₙ be real numbers such that a₁ + a₂ + a₃ + … + aₙ = 0. Suppose n ≥ 5. Which of the following statements is true about the roots…
15. Q16Which of the following statements is true?
16. Q17Ms A likes grapes (G) and strawberries (S). With every two pieces of grapes she consumes three pieces of strawberries, and every piece of grape…
17. Q18Suppose that commodity prices are strictly positive. Consider the following utility functions. One of these utility functions has no income effect…
18. Q19A Rotating Credit and Savings Association (ROSCA) is made up of a group of individuals who pool their money into a common fund, generally structured…
19. Q20An economy has two agents (R and P) and two states (pandemic and normal). If pandemic, R earns 10 and P earns 5; in normal state R earns 20 and P…
20. Q21Suppose that the market for pizza is perfectly competitive with constant long-run marginal cost. If there is a per-unit sales tax on pizza, then…
21. Q22Suppose a monopolist produces oil in one factory but sells it in two cities. In each city she faces the same inverse demand function p = f(q), where…
22. Q23Two buyers bid for a single object in an auction. Each buyer's valuation is private information, drawn independently from the Uniform[0,1]…
23. Q24The competitive market for good X has demand x^d = 100 - p and supply x^s = 20 + 3p. If the government imposes a sales tax of Rs 10 per unit of the…
24. Q25A monopolist sells hand-embroidered Pashmina shawls to two types of customers: 'Rich' and 'Middle Income' (values in thousands of Rs). Rich…
25. Q26'ABC Cinemas' is the monopolist in a town. Demand to see a movie is Q = 100 - 10p in the afternoon and Q = 200 - 10p in the evening, where p is the…
26. Q27In an exchange economy there are two goods X and Y and two agents A and B. A owns 10 units of good Y while B owns 10 units of good X. A has…
27. Q28Which of the following assumptions is not required for the 'First Welfare Theorem' in an exchange economy?
28. Q29Two players independently and simultaneously decide how much to donate to a charity. Each has a budget of Rs 200. Whenever the charity receives at…
29. Q30Consider the same donation game (each budget Rs 200; each gets an extra Rs 60 if aggregate donation ≥ 100; player 1's payoff is [(200 - x) + 60] if…
30. Q31Consider a hypothetical economy with aggregate production function F(L,K) = L^(2/3)K^(1/3) and a 10 percent annual rate of depreciation. Find the…
31. Q32Given below are two statements. Statement I: 'Balanced Growth' in the Solow model implies identical rates of growth in different sectors of the…
32. Q33Suppose an economy can produce e^k amount of output per capita if it uses k amount of capital per capita. Which of the following is correct?
33. Q34Consider a small open economy where domestic and foreign prices are normalised, with M = L(i,Y), Y = A(i,Y) + G + X(Y,E), and B = 0 = KI(i - i*) +…
34. Q35The aggregate wage bill in an economy is equal to 40 and output (produced by a Cobb-Douglas production function) is equal to 100. The output growth…
35. Q36The Phillips curve relation for an economy is πₜ - πₜ^e = 0.1 - 3uₜ where πₜ^e = πₜ₋₁. Suppose inflation in year (t-1) is 2 percent. In year t, the…
36. Q37If government finances its expenditure with taxes, then which of the following is correct?
37. Q38Consider the economy: C = 0.8(1-τ)Y, τ = 0.25, I = 900 - 50r, Ḡ = 800, L = 0.25Y - 62.5r, M̄/P = 500. (Aggregate consumption C; tax rate τ; income…
38. Q39Consider the same economy (C = 0.8(1-τ)Y, τ = 0.25, I = 900 - 50r, Ḡ = 800, L = 0.25Y - 62.5r, M̄/P = 500). Find the equilibrium aggregate income…
39. Q40There are two countries X and Y with the same production function. Both start with the same levels of capital, labour and technology, and the…
40. Q41Suppose X is a random variable which takes values in the interval [-1/2, 1/2]. The probability distribution function is denoted by Φ(x). Which of…
41. Q42Which of the following statements is correct?
42. Q43You have three variables Y, X₁, X₂. First you regress (OLS) Y on X₁ and X₂ to get the regression coefficients corresponding to X₁ and X₂ as b₁ and…
43. Q44Suppose we run the OLS regression Yᵢ = α + β Xᵢ + ϵᵢ, where notations have their usual meaning. After this we get an estimate Ŷᵢ = α̂ + β̂Xᵢ. Now…
44. Q45Let X₁, X₂ and X₃ be three (pairwise) uncorrelated random variables. The mean and variance of each variable is 0 and 3, respectively. Find the…
45. Q46[STEM RECONSTRUCTED - not from source; see flags] Suppose X and Y are independent random variables with E[Y] = 0. Which of the following is true?
46. Q48A fair coin is tossed until a head comes up for the first time. Find the probability of this happening on an odd-numbered toss.
47. Q49Which of the following statements is true about the Exponential probability distribution?