ISI Admission Test 2020 Question Paper MS (QE) PEA

FOR ISI EXAM PREPARATION

ISI 2020
Question Paper · MS
(QE) PEA
EXAM YEAR TYPE SUBJECT

ISI 2020 Question Paper MS (QE) PEA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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1. Consider the functions
1


x sin x if x 6= 0
f (x) =
0 if x = 0
and
1
x2 sin


x if x 6= 0
g(x) =
0 if x = 0
m
Then
m . co
c. o at zero but g is not differentiable at zero
(a) f is differentiable
s e m
e m at zero but f is not differentiable at zero l a
s
(b) g is differentiable

l a
(c) f and g are both differentiable at zero
ag
ag f nor g is differentiable at zero
(d) Neither

2. How many ordered pairs of numbers (x, y) are there, where x, y ∈
{1, 2, . . . , 100}, such that |x − y| ≤ 50?

(a) 2550
(b) 5050
(c) 7550
m
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3. Let ABC be a right angled isosceles triangle with angle ∠ABC being
m
e
right-angled. Let D be the mid-point of AB, E be the foot of the
s
a
perpendicular drawn from D to the side AC, and F be the foot of the
l
ag
FC
perpendicular drawn from E to the side BC. What is the value of BC ?

(a) √12
(b) 34
√
(c) 2 − 2

4. Suppose that there are 30 MCQ type questions where each question has
four options: A, B, C, D. For each question, a student gets 4 marks
m
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for a correct answer, 0 marks for a wrong answer, and 1 mark for not

m attempting the question. Suppose in each question, the probability that

m .co option A is correct is 0.5, option B is correct is 0.3, option C is correct
s e m
e la
is 0.2, and option D is correct is 0. Two students Gupi and Bagha have

las no clue about the right answers. Gupi answers each question randomly,
ag
g
that is, ticks any of the options with probability 0.25. Whereas Bagha
a attempts each question with probability 0.5, but whenever he attempts a
question, he randomly ticks an option. Which of the following is correct?

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(a) Both Gupi and Bagha have expected scores more than 30
(b) Gupi’s expected score is greater than or equal to 30 and Bagha’s
expected score is strictly less than 30
(c) Gupi’s expected score is less than or equal to 30 and Bagha’s ex-
pected score is strictly more than 30
1
5. Evaluate: lim e3x − 5x x

x→∞

(a) e3
(b) 3
(c) 1
( )
|x−3|
, for x =
6 3
6. Suppose f (x) = x−3 . Then, lim f (x) :
0, for x = 3 x→3

(a) is -1
(b) is 0
(c) does not exist
(d) is 1

7. Consider the following system of equations in x, y, z :

x + 2y − 3z = a
2x + 6y − 11z = b
x − 2y + 7z = c

For what values of a, b, c, does the above system have no solution?

(a) c + 2b − 5a 6= 0
(b) c + 2b − 5a = 0
(c) c + 2b − 4a = 0

8. The sequence xn is given by the formula: for every positive integer n

xn = n3 − 9n2 + 631.
The largest value of n such that xn > xn+1

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(a) is 4
(b) is 5
(c) is 6

m
9. Suppose an unbiased coin is tossed 10 times. Let D be the random

co
variable that denotes the number of heads minus the number of tails.

. c om
What is the variance of D?
e m .
(a) 10
e m l as
(b) 1
l as ag
(c) 0g
a

10. Suppose we are given a 4 × 4 square matrix A, which satisfies Aij = 0
if i < j. Suppose the each diagonal entry Aii is drawn uniformly at
random from {0, 1, . . . , 9}. What is the probability that A has full rank?

(a) 1014
(b) 35
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(c) 1 − 1014
9 4
e m


(d) 10

l as b 6= 0 and g : R → R is a
ag
11. Let a and b be two real numbers where
continuous function which satisfies

g(g(x)) = ag(x) + bx ∀ x ∈ R.

Which of the following must be true?

(a) g is strictly increasing
(b) g is strictly decreasing

m
(c) limx→∞ g(x) is finite

m (d) Either (a) or (b)

c. o12. For every positive integer n, let S(n) denote the sum of digits in n. Forsem .co
s em instance, S(387) = 3 + 8 + 7 = 18. The value of the sum g l a
la a
ag
S(1) + S(2) + . . . + S(99)

is

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(a) 450
(b) 495
(c) 900
(d) 990

13. Suppose five cards are randomly drawn without replacement from an
ordinary deck of 52 playing cards, with four suits of 13 cards each, which
has been well shuffled. Let a flush be the event that all five cards are of
the same suit. What is the probability of getting a flush?
4C 13 C
(a) 1
52 C
5
5
4 C 13 C
(b) 2
52 C
4
5
4 P 13 C
(c) 1
52 P
5
5
4 C 12 C
(d) 1
52 C
5
5

xn ln xdx, where n > 1
R
14. Evaluate:
n+1 1
(a) ln x xn+1 − (n+1) 2x
n+1 + c

x n+1 1
(b) ln x 2(n+1) − (n+1) n+1 + c
2x

n+1 1
(c) ln x xn+1 − (n+1) xn+1 + c

15. The area of the region bounded by the curve y = ln(x), the Y − axis,
and the lines y = 1 and y = −1

(a) is 2e
(b) is 2
(c) is e − 1e

16. A price discriminating monopolist finds that a person’s demand for its
product depends on the person’s age. The inverse demand function of
someone of age y, can be written p = A(y)−q where A(y) is an increasing
function of y. The product cannot be resold from one buyer to another
and the monopolist knows the ages of its consumers. (This is often the
case with online subscriptions.) If the monopolist maximizes its profits,
then

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(a) older people will pay higher prices and purchase more of this prod-
uct compared to younger people
(b) everyone pays the same price but older people consume more
(c) older people will pay higher prices compared to the younger people
but everyone will consume the same quantity of the product
m
17. Pam’s family consistsm . co
. cosector
of herself and her 3 sisters. They own a small farm

s e m
m
in the agricultural in Agri-land. The value of their total output

s e
is $4000 which is divided equally amongst the four. The urban sector
g l a
l a
has two kinds of jobs: informal sector (which anyone can get) pays $500
a
agincome and calculates
and formal
total
sector jobs give $1200. Pam would like to maximize her own
her own expected returns to migration. The
proportion of formal sector jobs to urban labor force that would deter
her from migrating is:

(a) Less than 2/3
(b) More than 5/6
(c) More than 21
(d) Less than 5/7
o m
c
18. Hu and Li are two dealers of used tractors in a. rural area of China. Hu
e mLi sells low quality ones.
l as tractor at $8000 while Lu
sells high quality second hand tractors while

g Consumers are willing to pay
Hu would be willing to sell his high quality
a
would sell his low quality one for $5000.
up to $10,000 for a high quality tractor and $7000 for a low quality
one. They expect a 50% chance of buying a high quality second-hand
tractor. In order to signal the quality of their tractors Hu and Li can
offer warranties. The cost of warranty for a high quality tractor is 500Y
and 1000Y for a low quality one (Y is the number of years of warranty).
What is the optimal number of years of warranty that Hu should offer
so that consumers know his tractors are of good quality?

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(a) Less than 2 years
m (b) 0.5 years
c. o (c) More than 1.5 years s e m
e m l a
las (d) 3 years
a g
ag 19. Suppose the capacity curve for each laborer is described as follows: for
all payments up to $100, capacity is zero and then begins to rise by 2

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units for each additional $ paid. This happens until the payment rises
to $500. Thereafter, an additional $ payment increases work capacity by
only 1.1 units, until total income paid is $1000. At this point additional
payments have no effect on work capacity. Assume all income is spent
on nutrition. Suppose you are an employer faced by the above capacity
curve of your workers. You need 8000 units of work or capacity units.
How many workers would you hire and how much would you pay each
worker so that you get 8000 units of work at minimum cost?

(a) 5 workers; $1000 per worker
(b) 10 workers; $700 per worker
(c) 10 workers; $500 per worker
(d) 15 workers; $400 per worker

20. Suppose you were to believe that “money illusion” exists, that is as
prices and income rise proportionally, then people buy more. Which of
the following statements about demand should not be true?

(a) Demand functions are downward sloping
(b) Demand functions are homogenous of degree zero
(c) Demand has a positive vertical intercept
(d) Demand functions are homogenous of degree one

21. Consider a Bertrand price competition model between two profit maxi-
mizing widget producers, say A and B. The marginal cost of producing
a widget is 4 for each producer. Each widget producer has a capacity
constraint to produce only 5 widgets. There are 8 identical individuals
who demand 1 widget only, and value each widget at 6. If the firms are
maximizing profits, then the following statement is true:

(a) Firm A and B will charge 4
(b) Firm A and B will charge 6
(c) Firm A and B will charge greater than or equal to 5

22. The government estimates the market demand (QD ) and market supply
(QS ) for turnips to be the following: QD = 30 − 2P, QS = 4; where P
is the per unit price and Q is the quantity measured in kilograms. The
government aims to increase the market price of turnips to $8 per unit

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to improve the welfare of domestic producers of turnips. Is is consider-
ing three possible choices: (i) a per unit subsidy; (ii) a price floor and
purchase of any surplus production, and (iii) a production quota. Which
of these policies should the government adopt if it wants to maximize
the producers’ welfare but minimize the loss of efficiency?

(a) A production quota
m
(b) A price subsidy
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c. osubsidy or a price floor
(c) Either a price
s e m
(d) Either a m a
s e production quota or a price floor g l
g la faces a demand curve: q = 5 . Her cost function is: C (q) = a
a
23. A monopolist
p
3q. Suppose, in the same market, there are some competitive suppliers
ready to sell the good at the price p = 5. The monopolist’s profit
maximizing price and output could be given by

(a) p = 3, q = 53 .
5
(b) p = 3.01, q = 3.01
5
(c) p = 2.99, q = 2.99
5
(d) p = 4.99, q = 4.99
o m
24. The consumption function is given by C m
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eis
β
= AY with β = 0.5 and

as
A = 0.3. The marginal propensity to save
l
(a) equal to 0.5 a g
(b) increasing in income, Y
(c) equal to 0.3
(d) equal to 0.7

25. The production function is given by Y = AL. The wage rigidity con-
straint is given by W ≥ B. The labour endowment is given by C. Here
A, B, and C are finite and positive constants. Assume that the entire
m
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labor endowment is supplied. If A > B, then in a labour market equi-
m
c. o (a) L = C
librium
e m
e m las
las (b) L = 0
ag
ag (c) 0 < L < C

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26. Consider the Mundell-Flemming model with perfect capital mobility and
a flexible exchange rate in the short run. A monetary expansion leads
to a/an in output; a fiscal expansion leads to in output.

(a) decrease; no change
(b) increase; decrease
(c) increase; no change
(d) increase; increase

27. Mr. X has an exogenous income W and his utility from consumption c
is u(c). Mr. X knows that an accident can occur with probability p and
if it occurs, the monetary equivalent to the damage is T . Mr. X can
however affect the accident probability p through the prevention effort
e. In particular, e can take two values - zero and a and an assumption
is that p(0) > p(a), that is by putting prevention effort, probability of
occurring an accident can be reduced. Let us also assume that if Mr.
X puts an effort e, the disutility from the effort is Ae2 where A is the
per unit effort cost. What is the critical value of A, A∗ , below which
the effort will be undertaken, and above which the effort will not be
undertaken, by Mr. X ?

(a) A∗ = [p(a)−p(0)][u(W
a2
−T )−u(W )]

[p(a)−p(0)]a 2
(b) A∗ = [u(W −T )−u(W )]
p(a)

(c) A∗ = u(Wp(0)
−T ) 2
u(W )
a
p(a)p(0)a 2
(d) A∗ = u(W −T )u(W )

28. Labor supply in macro models results from individual decision making.
Let c denote an individual’s consumption and L denote labor supply.
Assume that individuals solve the following optimization problem
11 2
M ax U (c, L) = log c − L
{c,L} 2b

subject to c + S̄ = wL where U (·) is the utility function, b > 0 is a
constant, S̄ is a constant exogenous level of savings, and w is the real
wage the person can earn in the labor market. Derive the optimal labor
supply. It is

(a) increasing in w; increasing in c
(b) decreasing in w; decreasing in c

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(c) increasing in w; decreasing in c
(d) decreasing in w; increasing in c

29. Consider a Solow economy that begins in steady state. Then a strong
earthquake destroys half the capital stock. The steady state level of
capital , the level of output on impact, and the growth rate of
m
co
the economy as the economy approaches its steady state.

o m decreases m .
c
.same; decreases; decreases
(a) decreases; decreases;
s e
m
ethe same; decreases; increases
(b) remains the
l a
a s
l remains the same, decreases
(c) remains
ag
ag
(d) decreases;

30. Suppose the economy is characterized by the following equations

C = c0 + c1 YD
YD = Y − T
I = b0 + b 1 Y

where C = Consumption, c0 = Autonomous Consumption, c1 ∈ [0, 1],
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YD = Disposable Income, Y = Aggregate GDP, T = Taxes, I = Investment, b0 =
Autonomous Investment, and b1 ∈ [0, 1]. For the multiplier to be posi-
tive, what condition needs to be satisfied?
e m
l as
ag
(a) b1 + c1 = 0
(b) b1 + c1 = 1
(c) b1 + c1 < 1
(d) b1 + c1 > 1

m
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g la a
a
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