DSE 2014 question paper

All 60 questions from the DSE 2014 Delhi School of Economics MA entrance paper. Open any question for the worked solution.

Q1Let aₙ=(1+1/n)ⁿ⁺¹, n=1,2,…. Then the sequence (aₙ)ₙ₌₁^∞
Q2Let M=[4 7; 7 9], A=[1 1; 1 1], B=[1 2; 3 4], C=[1 1; 4 5]. Then M=xA+yB+zC,
Q3Let f be a continuous function from [a,b] to [a,b], differentiable on (a,b). A point y∈[a,b] is a fixed point if y=f(y). If f′(x)≠ 1 for any…
Q4Which of the following statements is true for all real numbers a,b with a<b?
Q5Let O(0,0), P(3,4) and Q(6,0) be the vertices of triangle OPQ. If a point S in the interior is such that triangles OPS, PQS and OQS have equal area…
Q6Consider an exchange economy with two agents, 1 and 2, and two goods X and Y. Agent 1's endowment is (0,10) and Agent 2's endowment is (11,0). Agent…
Q7Consider an exchange economy with two agents, 1 and 2, and two goods X and Y. Agent 1's endowment is (0,10) and Agent 2's endowment is (11,0). Agent…
Q8Consider an exchange economy with two agents, 1 and 2, and two goods X and Y. Agent 1's endowment is (0,10) and Agent 2's endowment is (11,0). Agent…
Q9Consider an exchange economy with two agents, 1 and 2, and two goods X and Y. Agent 1's endowment is (0,10) and Agent 2's endowment is (11,0). Agent…
Q10Consider an exchange economy with two agents, 1 and 2, and two goods X and Y. Agent 1's endowment is (0,10) and Agent 2's endowment is (11,0). Agent…
Q11Households are classified as rich or poor; a generation's state depends only on its parents' state. If a parent is poor, the child is poor with…
Q12Toss two fair coins. Let A = head on the first coin, C = head on the second coin, D = both coins match, G = two heads. Which of the following is…
Q13Let Y denote the number of heads when 3 fair coins are tossed. Then the expectation of Z=4+5Y² is
Q14Let Y denote the number of heads when 3 fair coins are tossed. Then the variance of Z=4+5Y² is
Q15Let events E,F,G be pairwise independent with Pr (G)>0 and Pr (E∩ F∩ G)=0. Let X^C denote the complement of X. Then Pr (E^C∩ F^C| G)=
Q165 men and 5 women are seated randomly in a single row of chairs. The expected number of women sitting next to at least 1 man equals
Q17Let M be a 3×3 matrix such that M²=M. Which of the following is necessarily true?
Q18Suppose a straight line in ℝ³ passes through the point (-1,3,3) in the direction of the vector (1,2,3). The line will hit the xy -plane at the point
Q19X is a random variable. Which of the following statements is always true?
Q20Consider two disjoint events A and B in a sample space S. Which of the following is correct?
Q21lim_(n→∞)ⁿ√n=
Q22lim_(x→0)x²cos (1/x)=
Q23Suppose A₁,A₂,… is a countably infinite family of subsets of a vector space, all linearly independent, with A₁⊆ A₂⊆⋯. Then ⋃ᵢ₌₁^∞ Aᵢ is
Q24If u and v are distinct vectors and k and t are distinct scalars, then the vectors u+k(u-v) and u+t(u-v)
Q25Let d((x₁,x₂),(y₁,y₂))=max {|x₁-y₁|,|x₂-y₂|} be the distance between two points on the plane. Then the circle centred at the origin with radius 1 is
Q26The set of all pairs of positive integers a,b with a<b such that a^b=b^a
Q27Suppose c is a given positive real number. The equation ln x=cx² must have a solution if
Q28Sania's boat is at point A; the closest point on land, B, is 2 km away, and C on land is 6 km from B with triangle ABC right-angled at B. She rows…
Q29Suppose Aⱼ, j=1,2,… are non-empty sets of real numbers. Define Cₙ=⋂ₖ₌ₙ^∞⋃ⱼ₌ₖ^∞Aⱼ, n=1,2,…. Which of the following must hold for a given n?
Q30Suppose x and y are integers. Consider: A. If 2x+3y is divisible by 17, then 9x+5y is divisible by 17. B. If 9x+5y is divisible by 17, then 2x+3y is…
Q31Consider an exchange economy with two goods where agents i and j have the same preferences, with the property: if (a,b) and (c,d) are distinct…
Q32Consider an exchange economy with two goods where agents i and j have the same preferences, with the property: if (a,b) and (c,d) are distinct…
Q33Two firms produce the same commodity; quantities x₁,x₂, total X=x₁+x₂. Inverse demand is P=a-bX. Firms have constant average costs c₁,c₂. Assume…
Q34Two firms produce the same commodity; quantities x₁,x₂, total X=x₁+x₂. Inverse demand is P=a-bX. Firms have constant average costs c₁,c₂. Assume…
Q35Two firms produce the same commodity; quantities x₁,x₂, total X=x₁+x₂. Inverse demand is P=a-bX. Firms have constant average costs c₁,c₂. Assume…
Q36An economy has endowments of K units of capital and L units of labour. Two goods can be produced by X₁=√kl and X₂=√l, where k is capital and l is…
Q37A two-person two-commodity economy has social endowment x=1 (food), y=1 (wine). Preferences are increasing in own consumption but decreasing in the…
Q38Consider a two-person two-good pure exchange economy with endowments e¹=(1,0), e²=(0,1). Both have identical preferences u¹(x,y)=u²(x,y)=1 if x+y<1…
Q39A monopolist produces at constant marginal cost c≥0 and sells the entire output to a consumer whose utility from x units is θ√x-t, where t is the…
Q40An economy has n≥2 individuals with preferences over X. Let S={a,b,c,d,e} and T={a,b,c,d}⊂ S. Consider: A. If a is Pareto optimal (PO) in S, then a…
Q415 men and 5 women are seated randomly in a single circle of chairs. The expected number of women sitting next to at least 1 man equals
Q42Ms A selects X from Uniform [0,1]. Mr B repeatedly and independently draws Y₁,Y₂,… from Uniform [0,1] until he gets a number larger than X/2, then…
Q43Ms A selects X from Uniform [0,1]. Mr B repeatedly and independently draws Y₁,Y₂,… from Uniform [0,1] until he gets a number larger than X/2, then…
Q44There are two fair coins. Coin 1 is tossed 3 times; let X be the number of heads. Then Coin 2 is tossed X times; let Y be the number of heads. The…
Q45Two independent random variables X and Y have the same density f(x)=c(1+x) for x∈[0,1] and 0 otherwise. Then Var(X+Y) equals
Q46Two restaurants are located at random on a street 10 km long. What is the probability that they are located less than five kilometres apart?
Q47Consider the linear regression yᵢ=β₁ D1ᵢ+β₂ D2ᵢ+εᵢ, where D1ᵢ=1 for one group of observations and 0 for the other, and D2=1-D1. Can this model be…
Q48Consider the least squares regression of y on a single variable x. Which statement is true?
Q49An analyst runs two least squares regressions: first of y on a single variable x, and second of x on y, both with an intercept. Which of the…
Q50Consider two models: (i) y=β₀+β₁X₁+β₂X₂+u and (ii) y=γ₀+γ₁Z₁+γ₂Z₂+v, where Z₁,Z₂ are distinct from X₁,X₂. Assume u~ N(0,σᵤ²), v~ N(0,σᵥ²), estimated…
Q51Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q52Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q53Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q54Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q55Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q56Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q57Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q58Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q59Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…
Q60Consider an economy with N identical competitive firms, each producing Yⁱ=(Kⁱ)^α (Lⁱ)^(1-α), 0<α<1, with aggregate output Yₜ=(Kₜ)^α (Lₜ)^(1-α)…