ISI Admission Test 2022 Question Paper MS (QE) PEA

FOR ISI EXAM PREPARATION

ISI 2022
Question Paper · MS
(QE) PEA
EXAM YEAR TYPE SUBJECT

ISI 2022 Question Paper MS (QE) PEA

Notes · Sample Papers · Previous Year Papers · Mock Tests

m
m .co

m .co s e m
se g l a
a

1. The number of ways in which the word PANDEMIC can be
arranged such that the vowels appear together is
m
(A) 6 × (3!)(5!)
m (B) 5 × (3!)(5!)
.co
.co
(C) 4 × (3!)(5!) (D) 1 × (3!)(5!)
s e m
s em l a
g la2. Consider the functions f (x) = x − x − 1 and g(x) = x + 1,
2 ag
a both defined for all real values of x. Let α > 0 be the positive 1

real root and α2 < 0 be the negative real root of the equation
f (x) = 0. Let β1 > 0 be the positive real root and β2 < 0 be the
negative real root of the equation f (g(x)) = 0. After identifying
the exact values of α1 , α2 , β1 and β2 , identify which one of the
following four statements is incorrect.
m
.co
(A) α1 − β1 = α2 − β2 = 1

e m√
(B) α1 + β2 = α2 + β1 = 0
s
(C) α + β = −(α + βa) = 5
l
(D) α + α = −(βag
1 1 2 2

1 2+ β ) = −11 2

√
3. Let the function f (x) = 1 − 1 − x2 be defined only over all x
belonging in [0, 1]. Then f (1 − f (x)) equals

(A) x (B) 1 − x
(C) x2 (D) 1 − x2
m
m .co
m .co s em
4. Suppose f (x) is increasing, concave and twice differentiable and

s e la Then the
g(x) is decreasing, convex and twice differentiable.
g
g la function G(x) = g(f (x)) is a
a (A) increasing and convex
(B) decreasing and convex
(C) increasing and concave
(D) decreasing and concave

1
m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 9

5. Let A and B be two non-singular matrices of the same order
and let C be a matrix such that C = BAB −1 . Then for any
scalar λ, the value of det(C + λI) (where I is the identity
matrix) is

(A) detA (B) detB
(C) det(A + λI) (D) det(B + λI)

6. Suppose that f : R → R is a twice differentiable function
satisfying f ′′ (x) > 0 for all x ∈ R. Furthermore, assume that
f (1) = 1 and f (2) = 2. Then,

(A) 0 < f ′ (2) < 1 (B) f ′ (2) > 1
(C) f ′ (2) = 1 (D) f ′ (2) = 0

7. The value of limx→e logx−e
e x−1
is

(A) 0 (B) e
(C) 1
e
(D) None of these

8. Let f : [0, ∞) → R be a function such that f (0) = 0 and
f ′′ (x) > 0 for all x > 0. Then the function g : (0, ∞) → R,
defined by g(x) = f (x)
x
, is
(A) increasing in (0, ∞)
(B) decreasing in (0, ∞)
(C) increasing in (0, 1] and decreasing in (1, ∞)
(D) decreasing in (0, 1] and increasing in (1, ∞)

9. Let f : A → B be a function where A = {1, 2, 3, 4, 5} and
B = {1, 2}. How many onto functions can one generate?

(A) 52 − 1 (B) 52 − 2
(C) 25 − 1 (D) 25 − 2

2

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 9

m
m .co

m .co s e m
se g l a
a

x
10. Let f : R → R be a function defined by f (x) = for all
1 + x2
x ∈ R. Then,
m
m
(A) −1 ≤ f (x) ≤ 1 (B) −1 ≤ f (x) ≤ 1/2
.co
.co
(C) −1/2 ≤ f (x) ≤ 1 (D) −1/2 ≤ f (x) ≤ 1/2
s e m
s em l a
g a If a 3 × 3 matrix A has rank 3 and a 3 × 4 matrix B has rank
l11. ag
a 3, then the rank of AB is
(A) 3 (B) 4 (C) 6 (D) 7

 
2 0 3 1 −1
 
2 3 −1
1 0
 
12. Let A =  m
 . Which one is an eigenvalue of
0 −1
.co
3 1 2
 
m
1 2 3 −1 0 
2 1 −1 0
s e
a
3
A?
agl
(A) 2 (B) 1 (C) 3 (D) 5

13. Let A be a 5 × 5 non-null singular matrix. Then which of the
following statement is true?
(A) Ax = 0 has only a trivial solution
m
.co
(B) Ax = 0 has 5 solutions
m
m .co (C) Ax = 0 has no solution
s e m
s e (D) Ax = 0 has infinitely many solutions
g la
g la a
a 14. A family has two children. What is the probability that both
are boys given that at least one is a boy?

(A) 1
2
(B) 2
3
(C) 1
3
(D) 1
4

3
m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 9

15. Consider two boxes, one containing one black ball and one
white ball, the other containing two white balls and one black
ball. A box is selected at random, and a ball is selected at
random from the selected box. What is the probability that the
ball is black?

(A) 5
12
(B) 2
5
(C) 1
6
(D) 5
11

16. The function f : R → R, defined by f (x) = (x2 + 1)2022 , is

(A) one-one but not onto
(B) onto but not one-one
(C) both one-one and onto
(D) neither one-one nor onto

17. Consider an economy with two goods X and Y . Let the utility
√
function be given by u(x, y) = A xy where A > 0, x ≥ 0 is
the amount of good X consumed and y ≥ 0 is the amount of
good Y consumed. Suppose that the budget constraint is given
by PX x + PY y ≤ M where M > 0 is the money income of
the consumer and PX and PY are the prices of the goods X
and Y , respectively. Let PX = PY > 1 and let (x∗ , y ∗ ) be the
equilibrium quantities of this consumer who maximizes utility
subject to the budget constraint. Then,

(A) it must always be that x∗ > y ∗
(B) it must always be that x∗ = y ∗
(C) it must always be that x∗ < y ∗
(D) it must always be that x∗ + y ∗ = M

18. Consider the utility function u(x1 , x2 ) = 3x1 + 2x2 of a
consumer defined for all x1 ≥ 0 and x2 ≥ 0. Let the price

4

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 9

m
m .co

m .co s e m
se g l a
a

of good 1 be p1 > 0 and that of good 2 be p2 > 0. Let M > 0 be
the money income of the consumer. Consider the optimization
m
.co
problem maxx1 ≥0,x2 ≥0 3x1 + 2x2 subject to 2x1 + 3x2 ≤ M . The

o m
associated Lagrangian function for this maximization problem
m
c
. , x ; λ) = 3x + 2x + λ[M − 2x − 3x ]; where λ denotes s e
m
is L(x
ethe non-negative Lagrangian multiplier. Then the equilibrium
1 2 1 2 1 2
l a
as
l solution (x , x , λ ) to this Lagrangian function maximization ag
g
a problem is
∗
1
∗
2
∗

(A) (x∗1 = M2 , x∗2 = 0, λ∗ = 32 )
(B) (x∗1 = M2 , x∗2 = 0, λ∗ = 23 )
(C) (x∗1 = 0, x∗2 = M3 , λ∗ = 23 )
(D) (x∗1 = 0, x∗2 = M3 , λ∗ = 32 )

o m
c
. the two goods are X and
19. Consider a two good economy where
m
e A and B. In a month when the
a s
Y and consider two consumers
l 2 and that of good Y was Rs. 3,
price of good X wasgRs.
a
consumer A consumed 3 units of good X and 8 units of good Y
and consumer B consumed 6 units of both goods. In the next
month, when the price of good X was Rs. 3 and that of good
Y was Rs. 2, consumer A consumed 8 units of good X and 3
units of good Y and consumer B consumed 4 units of good X
and 9 units of good Y . Given this information which one of the
following statements is correct?
o m
m . c
m .co (A) Both consumers satisfy the weak axiom of revealed
s e m
laof revealed
preference
s e g
g la a
(B) Neither consumer satisfies the weak axiom
a preference
(C) Consumer A satisfies the weak axiom of revealed preference
but not consumer B
(D) Consumer B satisfies the weak axiom of revealed preference
but not consumer A

5
m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 5 of 9

20. Let the production function be Y (L, K) = min{2L, K}, where
L and K are the amounts of labor and capital, respectively.
Consider the cost function C(L, K) = wL + rK, where w > 0
denotes the price of labor and r > 0 denotes the price of capital.
Suppose that (L∗ , K ∗ ) is the combination of labor and capital at
which cost is minimized subject to the constraint Y (L, K) ≥ Ȳ .
Then,

(A) L∗ = Ȳ and K ∗ = Ȳ /2
(B) L∗ = Ȳ and K ∗ = Ȳ
(C) L∗ = Ȳ /2 and K ∗ = Ȳ
(D) None of the other options is correct

21. Suppose that there are two firms 1 and 2 that produce the same
good. Let the inverted demand function be P (q1 , q2 ) = 1−q1 −q2 ,
where firm 1 produces q1 ≥ 0 and firm 2 produces q2 ≥ 0.
Suppose that the cost function of firm i ∈ {1, 2} is given by
ci (qi ) = κi qi , where κi ∈ 0, 12 . Note that there is no fixed cost
( )

for either firm. Then, the Cournot equilibrium profit of firm 2
is

(A) (1−κ1 +κ2 )2
9
(B) (1−κ2 +κ1 )2
9
(C) (1−2κ1 +κ2 )2
9
(D) (1−2κ2 +κ1 )2
9

22. A non-transitive preference relation can be represented by a
utility function

(A) Always
(B) Only if preferences are complete
(C) Only if preferences are complete and convex
(D) Never

6

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 6 of 9

m
m .co

m .co s e m
se g l a
a

23. Which of the following statements is correct in a two-good
world?
m
om
c. diminishing
(A) Diminishing marginal utility of both goods is sufficient for
m .co
s e
s em marginal rate of substitution
l a
g la (B) Diminishing marginal utility of both goods is necessary for ag
a diminishing marginal rate of substitution

(C) Diminishing marginal utility of at least one good is
necessary for diminishing marginal rate of substitution

(D) Diminishing marginal utility of at least one good is neither
necessary nor sufficient for diminishing marginal rate of
substitution m
.co
s em
g la
a
24. Rahul consumes two goods, X and Y , in amounts x and y,
respectively. Rahul’s utility function is U (x, y) = min{x, y}.
Rahul makes Rs 200; the price of X and price of Y are both
Rs 2. Rahul’s boss is thinking of sending him to another town
where the price of X is Rs 2 and the price of Y is Rs 3. The boss
offers no raise in pay. Rahul, who understands compensating
and equivalent variations perfectly, complains bitterly. He says
o m
m that although he doesn’t mind moving for its own sake and. the c
m .co m
town is just as pleasant as the old, having to move is sase bad as
s e la says that
a cut in pay of Rs A in his current location. Hegalso
g la he would not mind moving if, when he moved, a he got a raise of
a Rs B. What are A and B equal to?

(A) A = 30, B = 70 (B) A = 40, B = 50
(C) A = 50, B = 75 (D) A = 60, B = 60

7
m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 7 of 9

25. Let U (x, y) = −[(10 − x)2 + (10 − y)2 ] be a utility function of
some consumer. All prices are equal to 1, and income is 40.
Then the optimal values of x and y will be

(A) 10, 10 (B) 0, 0
(C) 5, 5 (D) None of these

26. Consider a production function be F (K, L) = min{ Ka , Lb }; a, b >
0 and a ̸= b. For any given K = K > 0, the marginal
productivity of labor is

(A) 0
(B) 1
a
if L < ( ab )K and 0 otherwise
(C) 1
b
if L < ( ab )K and 0 if L > ( ab )K
(D) None of the above

27. Let ei (p0 ) be the price elasticity of demand for a good X
of consumer i (i = 2, · · · , N ) at price p0 , given its demand
function. Consumers do not consume identical amounts of X
at p0 . Then the price elasticity of demand at price p0 for the
aggregate demand function for X is
∑ ∑
(A) ∑ i (ei (p0 ))2 (B) i ei (p0 )
(C) i ei (p0 )
N
(D) None of these

28. There are m identical competitive firms in an industry. Every
firm has the (total) cost function C(q) = q 2 + 1, where q is the
level of its output, q ≥ 0. Industry demand for the product is
given by D(P ) = a − bP , where P is price, and a, b > 0. Then
the short-run equilibrium output of each firm is

(A) 0 (B) a
m+2b
(C) m
a
+b
(D) a
m+ 2b
2

8

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 8 of 9

m
m .co

m .co s e m
se g l a
a

29. Suppose the (total) cost function for a monopolist is
C = 3q 2 + 800, where q is its output. The inverse
m
.co
market demand function is p = 280 − 4q. What is the
m
.co
price elasticity of demand at the profit maximizing price?
e m
m −4.5 (B) −3.5 (C) −2.5 (D) −1.5
e(A) l as
las ag
g
a 30. Consider the Solow growth model with constant average saving
propensity s, rate of depreciation δ, and labor supply growth
rate n. There is no technological progress. Then, at steady
state, the capital-output ratio is

(A) s
n+δ
(B) n
δ+n
(C) δ
(D) 1

m
.co
s+n s+n+δ

e m
l as
ag

m
m .co
m .co s e m
s e g la
g la a
a

9
m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 9 of 9