As per your request below I am uploading the PDF file of M.A Economics Entrance Exam previous year question paper of Delhi School of Economics whose some questions are as follows

Delhi School of Economics M.A Economics Entrance Exam question paper
QUESTION 1. Two women and four men are to be seated randomly around a circular
table. Find the probability that the women are not seated next to each other.
(a) 1/2
(b) 1/3
(c) 2/5
(d) 3/5
QUESTION 2. A fair coin is tossed until a head comes up for the first time. The
probability of this happening on an odd-numbered toss is
(a) 1/2
(b) 1/3
(c) 2/3
(d) 3/4
QUESTION 3. Let f(x) = x + |x| + (x − 1) + |x − 1| for x ∈ ℜ.
(a) f differentiable everywhere except at 0.
(b) f is not continuous at 0.
(c) f is not differentiable at 1.
(d) f is not continuous at 1.
QUESTION 4. What is the total number of local maxima and local minima of the
function
f(x) =
{
(2 + x)3, if x ∈ (−3,−1]
x2/3,
if x ∈ (−1,2]
(a) 1
(b) 2
(c) 3
(d) 4
QUESTION 5. Let f : ℜ++ → ℜ is differentiable and f(1) = 1. Moreover, for every x
lim
t→x
t2f(x) − x2f(t)
t − x
= 1
Then f(x) is
(a) 1/3x + 2x2/3
(b) −1/5x + 4x2/5
(c) −1/x + 2/x2
(d) 1/x
QUESTION 6. An n-gon is a regular polygon with n equal sides. Find the number of
diagonals (edges of an n-gon are not considered as diagonals) of a 10-gon.
(a) 20 diagonals
(b) 25 diagonals
(c) 35 diagonals
(d) 45 diagonals
QUESTION 7. The equation x7 = x + 1
(a) has no real solution.
(b) has a real solution in the interval (0,2).
(c) has no positive real solution.
(d) has a real solution but not within (0,2).
QUESTION 8. limn→∞
(√
n − 1 −
√
n
)
(a) equals 1.
(b) equals 0.
(c) does not exist.
(d) depends on n.
EEE 2013 B
QUESTION 9. A rectangle has its lower left hand corner at the origin and its upper
right hand corner on the graph of f(x) = x2 +x
−2. For which x is the area of the rectangle
minimized?
(a) x = 0
(b) x = ∞
(c) x =
(1
3
)1/4
(d) x = 21/3
QUESTION 10. Consider the system of equations
αx + βy = 0
µx + νy = 0
α,β,µ and ν are i.i.d. random variables, each taking value 1 or 0 with equal probability.
Consider the following propositions. (A) The probability that the system of equations has
a unique solution is 3/8. (B) The probability that the system of equations has at least one
solution is 1.
(a) Proposition A is correct but B is false.
(b) Proposition B is correct but A is false.
(c) Both Propositions are correct.
(d) Both Propositions are false.
Part II
Instructions.
• Answer any four of the following five questions in the space following the relevant
question. No other paper will be provided for this purpose.
You may use the blank pages at the end of this booklet, marked Rough work, to do
calculations, drawings, etc. Your “Rough work” will not be read or checked.
• Each question is worth 20 marks.
QUESTION 11. Suppose ℜ is given the Euclidean metric. We say that f : ℜ→ℜ is
upper semicontinuous at x ∈ ℜ if, for every ϵ > 0, there exists δ > 0 such that y ∈ ℜ and
|x − y| < δ implies f(y) − f(x) < ϵ. We say that f is upper semicontinuous on ℜ if it is
upper semicontinuous at every x ∈ ℜ.
(A) Show that, f is upper semicontinuous on ℜ if and only if {x ∈ℜ| f(x) ≥ r} is a
closed subset of ℜ for every r ∈ ℜ.
(B) Consider a family of functions {fi | i ∈ I} such that fi : ℜ → ℜ is upper
semicontinuous on ℜ for every i ∈ I and inf{fi(x) | i ∈ I}∈ℜ for every x ∈ ℜ. Define
f : ℜ→ℜ by f(x) = inf{fi(x) | i ∈ I}.
Show that {x ∈ℜ| f(x) ≥ r} = ∩i∈I{x ∈ℜ| fi(x) ≥ r} for every r ∈ ℜ.
(C) In the light of (A) and (B), state and prove a theorem relating the upper semi-
continuity of f and the upper semicontinuity of all the functions in the family {fi | i ∈ I}.
ANSWER.
QUESTION 12. Let |.| be the Euclidean metric on ℜ. Consider the function f : ℜ→ℜ.
Suppose there exists β ∈ (0,1) such that |f(x) − f(y)| ≤ β|x − y| for all x, y ∈ ℜ. Let
x0 ∈ ℜ. Define the sequence (xn) inductively by the formula xn = f(xn−1) for n ∈ N.
Show the following facts.
(A) (xn) is a Cauchy sequence.
(B) (xn) is convergent.
(C) The limit point of x, say x
∗
, is a fixed point of f, i.e., x
∗
= f(x
∗
).
(D) There is no other fixed point of f.
ANSWER.
QUESTION 13. Let V be a vector space and P : V → V a linear mapping with range
space R(P) and null space N(P).
P is called a projector if
(a) V = R(P) ⊕ N(P), and
(b) for every u ∈ R(P) and w ∈ N(P), we have P(u + w) = u.
In this case, we say that P projects V on R(P) along N(P).
Show the following facts.
(A) P is a projector if and only if it is idempotent.
(B) If U is a vector space and X : U → V is a linear mapping with R(P) = R(X),
then P is a projector if and only if PX = X.
(C) P is a projector if and only if I − P is a projector.
Let W be a vector space and A : V → W a linear mapping. Let B : W → V be a
linear mapping such that ABA = A.
Show the following facts.
(D) ρ(A) = ρ(AB), where ρ(.) denotes the rank of the relevant linear mapping.
(E) AB projects W on R(A).
ANSWER.
QUESTION 14. Given x, y ∈ ℜn, define (x, y) = {tx + (1 − t)y | t ∈ (0,1)}. We say
that C ⊂ ℜn is a convex set if x, y ∈ C implies (x, y) ⊂ C. We say that f : ℜn → ℜ is a
concave function if x, y ∈ ℜn and t ∈ (0,1) implies f(tx + (1 − t)y) ≥ tf(x) + (1 − t)f(y).
(A) Show that f : ℜn → ℜ is a concave function if and only if H(f) = {(x, r) ∈
ℜn ×ℜ| f(x) ≥ r} is a convex set in ℜn × ℜ.
(B) Consider a family of functions {fi | i ∈ I} where fi : ℜn → ℜ is a concave function
for every i ∈ I. Suppose inf{fi(x) | i ∈ I}∈ℜ for every x ∈ ℜn. Show that f : ℜn → ℜ,
defined by f(x) = inf{fi(x) | i ∈ I} is a concave function.
(C) Consider concave functions f1 : ℜn → ℜ and f2 : ℜn → ℜ. Define f : ℜn → ℜ
by f(x) = max{f1(x),f2(x)}. Is f necessarily a concave function? Provide a proof or
counter-example.
(D) Show that, if f : ℜn → ℜ is a concave function, then {x ∈ ℜn | f(x) ≥ r} is a
convex set for every r ∈ ℜ.
(E) Is the converse of (D) true? Provide a proof or counter-example.
ANSWER.
QUESTION 15. (A) An urn contains N balls, of which Np are white. Let Sn be the
number of white balls in a sample of n balls drawn from the urn without replacement.
Calculate the mean and variance of Sn.
(B) Let X and Y be jointly continuous random variables with the probability density
function
f(x, y) =
1
2π
exp
[
−
1
2
(x2 + y2)
]
(a) Are X and Y independent?
(b) Are X and Y identically distributed?
(c) Are X and Y normally distributed?
(d) Calculate Prob [X2 + Y 2 ≤ 4].
(e) Are X2 and Y 2 independent random variables?
(f) Calculate Prob [X2 ≤ 2].
(g) Find the individual density function of X2