ISI Admission Test 2024 Question Paper MS (QE) PEA

FOR ISI EXAM PREPARATION

ISI 2024
Question Paper · MS
(QE) PEA
EXAM YEAR TYPE SUBJECT

ISI 2024 Question Paper MS (QE) PEA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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m .co s e m
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1. If f : R → R is a differentiable function at a ∈ R such that
f � (a) = af (a), then what is limx→a xf (a)−af (x)
?
m
x−a

.co
(A) af (a) (B) f (a) (C) (1 − a2 )f (a) (D) None of the previous
m
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a
options
e m
em l as
s
la2. Suppose S is defined as follows for every positive integer n ≥ 2: ag
ag n

� 1 �� 1� � 1�
Sn = 1 − 1− ··· 1 −
22 32 n2

The value of limn→∞ Sn is then
(A) 0 (B) 12 (C) 1 (D) ∞

3. Suppose limx→0 a2ex12 +a
−1 a x

o m
= 1, where a1 , a2 and a3 are given real
c
numbers. Then it is necessarily .true that
3x

s em (B) a = a �= 0
la
(A) a = a = a = 1
1 2 3 1 3

(C) a = 0
2
a g (D) a + a �= 0 2 3

4. Let a function f : R → R be defined as:



 cx2 + ax + b if x < 0

f (x) = bx2 + cx + a if 0 ≤ x < 2


m
ax2 + bx + c if x ≥ 2


m . c o
.co m
where a, b, c are positive real numbers. Which of the following

m s e
a
statements is correct, under the assumption that f is continuous?
s e (A) f is continuous for all values of a, b and c gl
g la a
a (B) f is continuous if and only if a − b = b − c
(C) f is continuous if and only if a = b and c = 2a
(D) f is continuous if and only if a = b = c

1
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x
5. Consider f : [0, 1] → [0, 1] such that f (x) = 2−x . Which of the
following statements is incorrect?
(A) f (0) = 0 and f (1) = 1
(B) f (1 − f (x)) = 1 − x
(C) f is strictly concave in the interval (0, 1)
(D) f is strictly increasing in the interval (0, 1)

6. Let f : R → R be the function

f (x) = |x| + x2 ∀x∈R

Which of the following statements about f is correct?
(A) f is differentiable
(B) f is concave but not differentiable
(C) f is convex but not differentiable
(D) f is discontinuous

2
x3 ex dx equals
�
7.
2 2
(A) x(x−1) (B) (x 2−1) ex (C) x(x+1) (D) (x 2+1) ex
2 2 2 2
2
ex 2
ex

8. Let A be a 3 × 3 matrix having eigenvalues 2, 7, 5. What is the
determinant of A + 2I?
(A) 252 (B) 70 (C) 420 (D) 84

9. Let x and y be two column vectors of length 3 such that
�3 T T
i=1 xi yi �= 0. What is the rank of xy , where y is the trans-
pose of y?
(A) 0 (B) 1 (C) 2 (D) 3

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10. Let A be a 3 × 3 matrix such that Ax = x for all x where x is
a column vector of length 3. Which of the following statements
m
.co
is correct?
m
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(A) No such A exists
m s e m
e a
   
l
1 1 1 0 0 1

l as(B) A = 1 1 1
 
(C) A = 0 1 0
 
ag
ag 1 1 1 1 0 0
(D) A exists but is different from the options given in parts (A)
or (B) above

11. Let
S = {(x1 , 0) : x1 is any real number}

o m
c
and
T = {(0, x ) : x is .any real number}.
e m 2 2

la s
Which of the following statements is correct?
g
(A) S, T and S ∩ Taare vector spaces
(B) S, T and S ∪ T are vector spaces
(C) S ∪ T and S ∩ T are vector spaces
(D) Neither S ∪ T nor S ∩ T are vector spaces

12. In a chess tournament, there are both boys and girls. Each
o m
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.co m
player plays with another player exactly once in the tournament.
If there are 45 games in the tournament and exactly 15eof them
e m l asa boy and
las ag
feature only boys, then how many games will feature

ag a girl?
(A) 6 (B) 15 (C) 20 (D) 24

3
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13. What is the number of possible arrangements of the letters of the
word ‘madam’ such that the two ‘a’s never appear in consecutive
positions?
(A) 30 (B) 24 (C) 18 (D) 12

14. A monkey starts at (0, 0) on the xy-plane in period 1. From
any position (x, y) in a period, the monkey can only jump
to (a, b) in the next period, where a ∈ {x + 1, x, x − 1} and
b ∈ {y + 1, y, y − 1}. How many possible positions can the mon-
key be in period 2?
(A) 9 (B) 6 (C) 4 (D) 3

15. In the previous question, suppose in every period the monkey
can go to any of the possible positions in the next period with
equal probability. Then what is the probability that the monkey
is at a distance of more than 1 from (0, 0) in period 2?
16
(A) 0 (B) 81 (C) 31 (D) 49

16. For a given data set, let the least squares regression line be
y = 10 + 2x. It is given that variance of x is 9 and variance of y
is 81. What is the correlation coefficient between x and y?
(A) 13 (B) 12 (C) 23 (D) 43

17. An urn contains 4 white, 6 red, and 5 black balls. 5 balls are ran-
domly selected from the urn. Let X and Y denote respectively
the number of white and black balls selected. Suppose Y = 2,
i.e., 2 of the 5 balls selected are black. What is the probability
that X takes the value 2?
3 4 3 4
(A) 10 (B) 10 (C) 15 (D) 15

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18. In answering a question on a multiple-choice test with 4 choices
available as possible answers, a student either knows the answer,
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with probability 1/4, or guesses the answer, with probability 3/4.

o m m
c
Assume that a student who guesses the answer will be correct
with. probability 1/4. What is the probability that a student s e
m the answer given that he answered it correctly?
eknew l a
s
la (A) ag
ag 3
7
(B) 4
7
(C) (D) 1 3
4

19. Let X1 , X2 , · · · , X5 be independently and identically dis-
tributed random variables with mean 10 and variance 4. Let
X̄ = 51 5i=1 Xi . What is the value of Cov(X̄, X1 − X̄)?
�

(A) 0 (B) 1 (C) -1 (D) None of the previous options

o m
20. Consider a discrete random m
c
. which follows a binomial
s en and p, where n, a positive integer,
variable

l a
distribution with parameters
is the number of trialsgand p ∈ (0, 1) is the probability of success
a
in any trial. Which of the following statements is incorrect?
(A) If np is an integer, then the mean and mode of this distri-
bution coincide
(B) If n is even and p = 12 , then the median of this distribution
is n2

o m
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(C) If the probability of k successes out of n trials is equal to the
m probability of n − k successes out of n trials, ∀k ∈ {0, 1, ..,. n},
.co s em
em lais unimodal
1
then p =
s
2

l a a g
(D) If (n+1)p−1 is an integer, then this distribution
ag

5
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21. Consider an economy where all factors of production are fully
employed and which has an aggregate production function (in
per capita form) y = Ak, where k and y are the capital-labour
ratio and the output-labour ratio respectively, and A is a positive
constant. A constant proportion s ∈ (0, 1) of income is saved
and invested in this economy. Suppose labour force grows at the
rate n > 0 and the rate of depreciation of capital is given by
δ ∈ (0, 1). Assume the parameter values are such that positive
long run growth of y can be ensured. Then which of the following
statements regarding this model economy is incorrect?
(A) Output, capital and consumption grows at the same rate
(B) If the level of investment (I) is higher than the depreciation
of capital (K) (i.e. I > δK), then output grows at a positive
rate
(C) The economy is always on the steady state growth path (i.e.,
if you consider any variable you will find its growth rate to be a
constant at all times
(D) Increasing the savings rate s will not have any effect on the
long run growth rate of output per worker

The following data are for Question numbers 22 and
23: There is a Solow economy without population growth or
technological change which has a production function given by
1 2
Y = K 3 L 3 , and a depreciation rate δ = 0.05.

22. Suppose the savings rate is s = 0.2. What will be the capital-
labour ratio in steady state?
(A) 6 (B) 8 (C) 2 (D) 4

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23. Suppose that a social planner wishes to maximize steady state
per-capita consumption in this economy. What savings rate will
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be compatible with the level of per-capita consumption chosen

o m m
. c
by the planner?
s e
e m
(A) 2
(B) 1
(C) (D) 1 1

l a
l as 3 2 3 4

ag
ag 24. Consider a Solow economy with the aggregate production func-
1
tion in intensive form given by y = k 2 , where k and y are the
capital labour ratio and the per capita output respectively. Sup-
pose all factors are fully employed in this economy, there is no
depreciation of capital, and the labour force grows at rate n > 0.
If the steady state value of capital labour ratio for this economy

m
is 50 and the current value of capital labour ratio is 2, what is
the current growth rate of per capita output?.co
(B) 25 s em(D) None of the previous options
la
(A) 24n n
(C) 0.3

ag
The following data are for Question numbers 25 and
26: There are two countries, A and B, and two goods, wheat
and television. A produces only wheat, with YA = 200, and
B produces only television, with YB = 250. Residents of both
countries consume both wheat and television. Treating televi-
sion as the numeraire, and therefore fixing its price at 1, let the
relative price of wheat be p. Let A’s total expenditure mea-
o m
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sured in terms of wheat be denoted by E , and let B ’s total
eby E .
A

em as
expenditure measured in terms of television be denoted
lexpendtiure,
B

las Now, total world income must equal total world g
a trade balance
ag i.e., pY + Y = pE + E , so it follows that the
A B A B

of A, measured in wheat, is given by TA = YA − EA . Suppose
initially EA is 100, and in an attempt to improve trade balance,
A reduces expenditure EA to 70.

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25. Suppose preferences are such that 31 rd of the expenditure of res-
idents of any country is on wheat and the rest is on television.
What is the impact on equilibrium price (i.e., the price at which
all markets clear) p of A reducing its expenditure from 100 to
70?
(A) p will stay the same
(B) p will increase
(C) p will decrease
(D) The effect on p will be ambiguous

26. Suppose preferences are such that 32 rd of the expenditure of res-
idents of any country is on their own good, and the rest is on
the other good (i.e. 23 rd of the expenditure of residents of A is
on wheat and the rest is on television, while 23 rd of the expendi-
ture of residents of B is on television and the rest is on wheat).
What is the impact on equilibrium price (i.e., the price at which
all markets clear) p of A reducing its expenditure from 100 to
70?
(A) p will stay the same
(B) p will increase
(C) p will decrease
(D) The effect on p will be ambiguous

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27. A consumer consumes two goods, X and Y . It is observed that
the consumer’s consumption of good X always falls when the
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price of X falls, ceteris paribus. Suppose the consumer’s in-

o m m
c
come rises, given prices of X and Y . What will happen to the
. consumption of X? s e
m
consumer’s
e(A) Consumption of X falls l a
l as ag
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a (B) Consumption of X rises
(C) Consumption of X remains unchanged
(D) Indeterminate: consumption of X could rise, fall or remain
unchanged

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28. An individual A has a pond in which he can set up a net cost-

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lessly only at time t = 0. The net will yield an output of f
m
s e
fish at t = 1 and nothing at t = 2. A eats only fish and his
only source of fish is theapond. A can consume in periods t = 1
l
ag he ceases to exist. A has the option of
and t = 2, after which
costlessly storing fish at t = 1 for consumption at t = 2, without
depreciation, and does not discount the future. His utility from
consumption ct in any period t is u(ct ) = (ct )n , where 0 < n < 1.
How much fish will A optimally consume in the two periods?
(A) Any division of f fish across the two periods is optimal

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(B) c1 = c2 = f2
m
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s e m
s e (D) c1 = 0, c2 = f
g la
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a

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29. Country C, which has a closed economy, has 10, 000 iden-
tical farmers producing rice, with any farmer bearing cost
0.5qi2 + 4qi + 100 if he produces qi units of rice. C has many
consumers of rice who generate an aggregate demand function
Q = −10, 000p + 400, 000, where p is the price per unit of rice.
The price of rice in C used to be determined by competitive
market conditions, but is currently regulated, and set at p = 30.
Consumers pay this amount to buy a unit of rice, with the gov-
ernment of C buying all unsold units at 30 per unit. The gov-
ernment proposes to remove the price regulation and go back to
competitive markets, with each farmer getting an equal flat pay-
ment, total payment across all farmers being equal to the savings
of the government from not having to purchase unsold rice. In
order to implement this proposal, each farmer has to open a
bank account at own cost. By how much does a farmer gain or
lose from the policy shift, if it costs 300 to open an account?
(A) Policy shift causes income to rise by 8
(B) Policy shift causes income to rise by 4
(C) Policy shift causes income to fall by 4
(D) Policy shift causes income to fall by 8

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30. A consumer consumes three goods, X, Y and Z. She is observed
over three periods. In period 1, the unit prices of X, Y and Z
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are respectively 2, 3 and 3, in period 2, the respective prices are

o m m
c
3, 2 and 3, while in period 3, the respective prices are 3, 3 and
. following consumption pattern is observed: in period 1, s e
m
2. The
eshe consumes 3 units of X, 1 units of Y and 7 units of Z, in l a
as
l period 2, the consumption amounts are 7 units of X, 3 units of ag
g
a Y and 1 unit of Z, while in period 3, the consumption amounts
are 1 unit of X, 7 units of Y and 3 units of Z. Which of the
following statements regarding the preferences of the consumer
over bundles of X, Y and Z is correct?
(A) Preferences are intransitive though not necessarily incom-
plete
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(B) Preferences are transitive though not necessarily complete

s em
la and transitive
(C) Preferences are incomplete and intransitive

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(D) Preferences are complete

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