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Questions › DSE › DSE 2020

DSE 2020 question paper

All 50 questions from the DSE 2020 Delhi School of Economics MA entrance paper. Open any question for the worked solution.

  1. Q1Consider independently and identically distributed random variables X₁, …, Xₙ with values in [0,2], each uniformly distributed on [0,2]. If Y=max…
  2. Q2A coin has outcomes H and T with probabilities 3/4 and 1/4. A gambler tosses until the first H. If the first H occurs on the n -th toss and n is…
  3. Q3Two fair dice are tossed simultaneously. The probability that the total number of spots on the upper faces is not divisible by 2, 3, or 5 is
  4. Q4A question has m possible answers. A student knows the correct answer with probability p; if she knows it she picks it, otherwise she guesses…
  5. Q5Y is uniformly distributed on [0,2]. For X=e^Y, the value of the cumulative distribution function of X at x∈[1,e²] is
  6. Q6An economy produces its final commodity with labour only. The price-setting firm charges a 25% mark-up over its per-unit nominal wage cost. Workers…
  7. Q7The aggregate production function is Yₜ=(Kₜ Lₜ)^(1/2). Capital evolves as Kₜ₊₁=(1-δ)Kₜ+Sₜ with Sₜ=sYₜ, Lₜ=Γ (constant), s the saving rate and δ the…
  8. Q8With technology Y=AL (A productivity, L labour), a firm sets price at a constant mark-up μ over effective unit wage cost W/A. Workers' expected real…
  9. Q9A household with an endowment of 1 unit of time maximises u=ln c+bln (1-l), where c is consumption and l∈[0,1] is time spent working, financing…
  10. Q10In an IS-LM model with a given price level P: investment is a decreasing function of the interest rate, saving is an increasing function of income…
  11. Q11Consider the Solow growth model with a given savings ratio, constant population growth, zero capital depreciation and no technical progress. Let k*…
  12. Q12A consumer lives for periods 1 and 2 with utility U=ln c₁+(1+ρ)⁻¹ln c₂, incomes w₁,w₂, and lifetime budget constraint c₁+(1+r)⁻¹c₂=w₁+(1+r)⁻¹w₂…
  13. Q13A consumer lives for periods 1 and 2 with lifetime utility U(c₁,c₂)=(c₁^γ+c₂^γ)/γ, where 0<γ<1 and cᵢ is consumption in period i. The elasticity of…
  14. Q14A and B play a best-of-seven match (first to win four games wins), and each is equally likely to win any game. The probability that the match ends…
  15. Q15Let X and Y be jointly normal, (X,Y)~ N(μ_X,μ_Y,σ_X²,σ_Y²,ρ). If σ_X²=σ_Y² and 0<ρ<1, then
  16. Q16Let a binary relation ≻ on a set X be asymmetric (for all x,y: x≻ y implies ¬(y≻ x)) and negatively transitive (for all x,y,z: x≻ y implies x≻ z or…
  17. Q17Let ≻ on X be asymmetric and negatively transitive. Define x⪰ y iff ¬(y≻ x). The relation ⪰ is said to be total if, for all x,y, x⪰ y or y⪰ x. Then ⪰
  18. Q18Let ≻ on X be asymmetric and negatively transitive, and define x⪰ y iff ¬(y≻ x). The relation ⪰ is said to be transitive if x⪰ y and y⪰ z imply x⪰…
  19. Q19Let ≻ on X be asymmetric and negatively transitive, and define x~ y iff ¬(x≻ y) and ¬(y≻ x). The relation ~ is said to be transitive if x~ y and y~…
  20. Q20Let ≻ on X be asymmetric and negatively transitive, and define x~ y iff ¬(x≻ y) and ¬(y≻ x). The relation ~ is said to be symmetric if x~ y implies…
  21. Q21Player 1 moves first and chooses L or R. L ends the game with payoffs (1,0) (payoffs are (player 1, player 2)). If R, player 2 chooses U or D: U…
  22. Q22For the same game (player 1 chooses L / R; L→(1,0); R→ player 2 chooses U→(0,2) or D→ player 1 chooses L→(4,0) or R→(3,3)), the number of subgame…
  23. Q23In a non-cooperative game, if a profile of strategies
  24. Q24Let G be a 2× 2 game (player 1 chooses rows U,D; player 2 chooses columns L,R), and let G′ be obtained from G by giving player 2 an additional…
  25. Q25An exchange economy has agents 1,2 and goods x,y. Agent 1 lexicographically prefers x to y; agent 2 's utility is u₂(x,y)=x+y. Endowments are…
  26. Q26In the same exchange economy (agent 2 's utility u₂=x+y, endowments (0,10) and (10,0)), suppose agent 1 instead lexicographically prefers y to x.…
  27. Q27An exchange economy has goods x,y and agents 1,2 with endowments (ω¹ₓ,ω¹_y)=(0,9) and (ω²ₓ,ω²_y)=(10,0), and utilities u₁(x,y)=min {x,y} and…
  28. Q28For the same economy (endowments (0,9) and (10,0); u₁=u₂=min {x,y}), the competitive equilibrium allocations are
  29. Q29For the same economy (endowments (0,9) and (10,0); u₁=u₂=min {x,y}), the allocation that gives (10,9) to agent 1 and (0,0) to agent 2 is
  30. Q30Given a non-empty set C⊆ℝⁿ, for every p∈ℝⁿ₊ let c(p)∈ C satisfy p· c(p)≤ p· c for every c∈ C. Then the function e:ℝⁿ₊→ℝ given by e(p)=p· c(p) is
  31. Q31Given a non-empty set C⊆ℝⁿ, for every p∈ℝⁿ₊ let c(p)∈ C satisfy p· c(p)≤ p· c for every c∈ C. Then the function e(p)=p· c(p) is
  32. Q32Suppose u:ℝ→ℝ₊ is strictly increasing with supremum (least upper bound) α∈ℝ. Then the function x↦ u(x)/(α-u(x)) is
  33. Q33The interval [0,∞) can be expressed as
  34. Q34If f:ℝ²→ℝ is given by f(x,y)=xy/(x²+y²) for (x,y)≠(0,0) and f(0,0)=0, then
  35. Q35Suppose f:ℝ→ℝ is twice differentiable, solves f′′(x)-f′(x)-f(x)-1=0 for all x, and satisfies f(0)=0=f(k) for some k>0. Then
  36. Q36Let B be the collection of sets E⊆ℝ such that for every x∈ E there are real numbers a<b with x∈(a,b)⊆ E. Let C be the collection of sets E⊆ℝ such…
  37. Q37The set {(x,y)∈ℝ²| x>0 and y≤ ln x-e^x} is
  38. Q38Suppose the distance between x,y∈ℝ is |x-y| and f:ℝ→ℝ is continuous. If E is an open subset of ℝ, then {x∈ℝ| f(x)∈ E} is
  39. Q39For k≠ 0, which of e^(kπ) and π^(ke) is larger?
  40. Q40Consider A=[cos θ -sin θ; sin θ cos θ] with θ∈[0,2π), and the inner product ⟨ v,w⟩=v₁w₁+v₂w₂ on ℝ². For vectors v,w∈ℝ²,
  41. Q41Let ⌊ x⌋ be the greatest integer ≤ x. The function f:ℝ→ℝ defined by f(x)=x-⌊ x⌋ is
  42. Q42Let ⌈ x⌉ be the smallest integer ≥ x. The function f:ℝ→ℝ defined by f(x)=⌈ x⌉-x is
  43. Q43Let ⌊ x⌋ be the greatest integer ≤ x and ⌈ x⌉ the smallest integer ≥ x. The function f:ℝ→ℝ defined by f(x)=⌈ x⌉-⌊ x⌋ is
  44. Q44Let A={(x,y)∈ℝ²| x≥ 0, y≥ 0, xy≥ 1} and B={(x,y)∈ℝ²| x≤ 0, y≥ 0, xy≤ -1}, and let C={a+b| a∈ A, b∈ B}. Then
  45. Q45Consider a 4× 4 matrix A. Obtain B from A by, in sequence, (1) interchanging the first and fourth columns, then (2) interchanging the second and…
  46. Q46The maximum value of f(x,y)=xy subject to |x|≥ |y| and |x|+|y|≤ 1 is
  47. Q47Consider a decreasing differentiable function g:ℝ₊→ℝ₊ and an increasing continuous function f:ℝ₊→ℝ₊. If F:ℝ₊→ℝ₊ satisfies F(x)=∫₀^(g(x))f(t) dt for…
  48. Q48If f,g:ℝ→ℝ are defined by f(x)=0 if x is rational and f(x)=x if x is irrational, and g(x)=0 if x is irrational and g(x)=x if x is rational, then…
  49. Q49Given nonempty subsets Y₁,…,Yₙ of ℝ², let Y*={Σⱼ₌₁ⁿyⱼ| y₁∈ Y₁,…,yₙ∈ Yₙ}. Fix p∈ℝ², and for a nonempty X⊆ℝ² let v(p,X)=sup {p· x| x∈ X}. Suppose…
  50. Q50If A=[1 0 0 0; 2 2 0 0; 3 2 -2 0; 4 3 2 -1] and Aᵀ is the transpose of A, then det (AᵀA) is