ISI Admission Test 2016 Syllabus and Sample Paper MS (QE) PEA

FOR ISI EXAM PREPARATION

ISI 2016
Syllabus and Sample
Paper · MS (QE) PEA
EXAM YEAR TYPE SUBJECT

ISI 2016 Syllabus and Sample Paper MS (QE) PEA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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SYLLABUS FOR MSQE
(Program Code: MQEK and MQED) 2016

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Syllabus for PEA (Mathematics), 2016

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Algebra: Binomial Theorem, AP, GP, HP, Exponential, Logarithmic Series,
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Sequence, Permutations and Combinations, Theory of Polynomial Equations;
(up to third degree).

Matrix Algebra: Vectors and Matrices, Matrix Operations, Determinants.

Calculus: Functions, Limits, Continuity, Differentiation of functions of one
or more variables. Unconstrained Optimization, Definite and Indefinite
Integrals: Integration by parts and integration by substitution, Constrained
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optimization of functions of not more than two variables.

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Elementary Statistics: Elementary probability theory, measures of central

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tendency, dispersion, correlation and regression, probability distributions,

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standard distributions-Binomial and Normal.

Syllabus for PEB (Economics), 2016

Microeconomics: Theory of consumer behaviour, theory of production,
market structure under perfect competition, monopoly, price discrimination,
duopoly with Cournot and Bertrand competition (elementary problems) and
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Macroeconomics: National income accounting, simple Keynesian Model of
s e income determination and the multiplier, IS-LM Model, models of aggregate
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a money, banking and inflation.

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Page 1 of 11

PEA 2016 (Mathematics)

Answer all questions

1. Consider the polynomial P (x) = ax3 + bx2 + cx + d, where a, b, c, d ∈ {1, 2, . . . , 9}. If
P (10) = 5861, then the value of c is

(a) 1.

(b) 2.

(c) 6.

(d) 5.

2. Let A ⊂ R, f : A → R be a twice continuously differentiable function, and x∗ ∈ A be
∂f ∗
such that (x ) = 0.
∂x
∂2f ∗
(a) (x ) ≤ 0 is a sufficient condition for x∗ to be a point of local maximum of f
∂x2
on A;
∂ 2f ∗
(b) (x ) ≤ 0 is a necessary condition for x∗ to be a point of local maximum of f
∂x2
on A;
∂ 2f ∗
(c) (x ) ≤ 0 is necessary and sufficient for x∗ to be a point of local maximum of
∂x2
f on A;
∂ 2f ∗
(d) (x ) ≤ 0 is neither necessary nor sufficient for x∗ to be a point of local
∂x2
maximum of f on A.

3. You are given five observations x1 , x2 , x3 , x4 , x5 on a variable x, ordered from lowest to
highest. Suppose x5 is increased. Then,

(a) The mean, median, and variance, all increase.

(b) The median and the variance increase but the mean is unchanged.

(c) The variance increases but the mean and the median are unchanged.

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(d) None of the above. a
4. Suppose the sum of coefficients in the expansion (x+y)n is 4096. The largest coefficient
in the expansion is:

(a) 924.

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(b) 1024.
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(c) 824. .co s e m
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(d) 724.
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5. There are three cards. The first is green on both sides, the second is red on both sides
and the third is green on one side and red on the other. I choose a card with equal
probability, then a side of that card with equal probability. If the side I choose of the
card is green, what is the probability that the other side is green?

(a) 13 .
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(b) 1
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2

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(c) 2
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3

(d) 43 .

6. The value of  π
2
x sin xdx
0

is:

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(a) 0.
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e m (b) −1.
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ag (c) 21 .

(d) 1

2

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7. Let f : R → R be defined as follows:

⎧
⎨ ax + b if x ≥ 0
f (x) =
⎩ sin 2x if x < 0

For what values of a and b is f continuous but not differentiable?

(a) a = 2, b = 0.

(b) a = 2, b = 1.

(c) a = 1, b = 1.

(d) a = 1, b = 0.

8. A student wished to regress household food consumption on household income. By
mistake the student regressed household income on household food consumption and
found R2 to be 0.35. The R2 in the correct regression of household food consumption
on household income is

(a) 0.65.

(b) 0.35.

c) 1 − (.35)2 .

(d) None of the above.

9. Let f : R2 → R be defined by

f (x, y) = 3xey − x3 − e3y

.

Which of the following statements is true?

(a) (x = 1, y = 0) is a local maximum of f .

(b) (x = 1, y = 0) is a local minimum of f .

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(c) (x = 1, y = 0) is neither a local maximum noraa local minimum of f .

(d) (x = 0, y = 0) is a global maximum of f .

10. Let √
x+ 3
f (x) = √
1 − 3x
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for all x = √13 . What is the value of f (f (x))?
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√

s
x−√ 3

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(a) 1+ .

(b) as
e
3x

l a
ag
√

l
x2 +2
√ 3x+3 .

g
1−2 3x+3x

a
(c)
√
x+√ 3
1− 3x
.
√
x+√ 3
(d) 1− 3x
.

11. The continuous random variable X has probability density f (x) where
⎧
⎪
⎪ a if 0 ≤ x < k

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⎪
⎨

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f (x) = b if k ≤ x ≤ 1
⎪

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⎪
⎩ 0 otherwise
⎪

s
a E(X) is given by:
where a > b > 0 and 0 < k < 1.glThen
a
b(1−a) 2
(a) 2a(a−b) .

(b) 21 .
a−b
(c) (a+b) .

(d) 1−2b+ab .
2(a−b)
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c. o12. The set of values of x for which x − 3|x| + 2 < 0 is given by: sem
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(a) {x : x < −2} ∪ {x : x > 1}.

ag (b) {x : −2 < x < −1} ∪ {x : 1 < x < 2}.

(c) {x : x < −1} ∪ {x : x > 2}.

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(d) None of the above.

13. The system of linear equations

(4d − 1)x + y + z = 0

−y + z = 0

(4d − 1)z = 0

has a non-zero solution if:

(a) d = 14 .

(b) d = 0.

(c) d = 14 .

(d) d = 1.

14. Suppose F is a cumulative distribution function of a random variable x distributed in
[0, 1] defined as follows:
⎧
⎨ ax + b if x ≥ a
F (x) =
⎩ x2 − x + 1 otherwise

where a ∈ (0, 1) and b is a real number. Which of the following is true?

(a) F is continuous in (0, 1).

(b) F is differentiable in (0, 1).

(c) F is not continuous at x = a.

(d) None of the above.

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15. The solution of the optimization problem a
max 3xy − y 3
x,y

subject to

2x + 5y ≥ 20

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x − 2y = 5
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x, y ≥ 0.

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is given by:
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(a) x = 19, y = 7.

(b) x = 45, y = 20.

(c) x = 15, y = 5.

(d) None of the above.

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16. Let f : R → R be a strictly increasing function. Let g be the inverse of the function
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f . If f  (1) = g(1) = 1, then g  (1) equals to

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(a) 0. a
(b) 12 .

(c) −1.

(d) 1.

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17. Consider a quadratic polynomial P (x). Suppose P (1) = −3, P (−1) = −9, P (−2) = 0.
m Then, which of the following is true.
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(a) P ( 12 ) = 0.
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a (b) P ( 52 ) = 0.

(c) P ( 54 ) = 0.

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(d) P ( 34 ) = 0.

18. For any positive integers k,  with k ≥ , let C(k, ) denote the number of ways in
which  distinct objects can be chosen from k objects. Consider n ≥ 3 distinct points
on a circle and join every pair of points by a line segment. If we pick three of these
line segments uniformly at random, what is the probability that we choose a triangle?

C(n,2)
(a) C(C(n,2),3) .
C(n,3)
(b) C(C(n,2),3)
2
(c) n−1 .
C(n,3)
(d) C(C(n,2),2) .

19. Let X = {(x, y) ∈ R2 : x + y ≤ 1, 2x + y2 ≤ 1, x ≥ 0, y ≥ 0}. Consider the optimization
problem of maximizing a function f (x) = ax + by, where a, b are real numbers, subject
to the constraint that (x, y) ∈ X. Which of the following is not an optimal value of f
for any value of a and b?

(a) x = 0, y = 1.

(b) x = 13 , y = 23 .

(c) x = 41 , y = 14 .

(d) x = 21 , y = 0.

20. Let F : [0, 1] → R be a differentiable function such that its derivative F  (x) is increasing
in x. Which of the following is true for every x, y ∈ [0, 1] with x > y?

(a) F (x) − F (y) = (x − y)F  (x).

(b) F (x) − F (y) ≥ (x − y)F  (x).

(c) F (x) − F (y) ≤ (x − y)F  (x).

(d) F (x) − F (y) = F  (x) − F  (y).

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21. A bag contains N balls of which a (a < N ) are red.aTwo balls are drawn from the bag
without replacement. Let p1 denote the probability that the first ball is red and p2 the
probability that the second ball is red. Which of the following statements is true?

(a) p1 > p2 .

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(b) p1 < p2 .

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(c) p2 = Na−1
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−1

(d)
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p2 = Na .
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22. Letgt = x + x + 2bx + c where b > c. Which of the following statements is true?
√
a
2 2

(a) dx
dt
= t−x
t+b
.

(b) dx
dt
= t+2x
2t+b
.
1
(c) dx
dt
= 2x+b .

(d) None of the above.
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c
. the i-th row and j-th column is min(i, j).
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23. Let A be an n × n matrix whose entry on
e
The determinant of A is:
la s
ag
(a) n.

(b) 1.

(c) n!

(d) 0.

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24. What is the number of non-negative integer solutions of the equation x1 +x2 +x3 = 10?
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(a) 66.
g l a
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a (c) 100.

(d) None of the above.

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 9 of 11

25. The value of  2b
xdx
,
b x 2 + b2
b > 0 is:

(a) 1b .

(b) ln 4b2 .

(c) 21 ln( 52 ).

(d) None of the above.

26. Let f and g be functions on R2 defined respectively by

f (x, y) = 31 x3 − 23 y 2 + 2x,

and
g (x, y) = x − y.

Consider the problems of maximizing and minimizing f on the constraint set C =
{(x, y) ∈ R2 : g (x, y) = 0} .

(a) f has a maximum at (x = 1, y = 1) , and a minimum at (x = 2, y = 2).

(b) f has a maximum at (x = 1, y = 1) , but does not have a minimum.

(c) f has a minimum at (x = 2, y = 2) , but does not have a maximum.

(d) f has neither a maximum nor a minimum.

27. A particular men’s competition has an unlimited number of rounds. In each round,
every participant has to complete a task. The probability of a participant completing
the task in a round is p. If a participant fails to complete the task in a round, he is
eliminated from the competition. He participates in every round before being elimi-
nated. The competition begins with three participants. The probability that all three
participants are eliminated in the same round is:

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(a) (1−p)
1−p3
.
3
a
(b) 31 (1 − p).

(c) p13 .

(d) None of the above.

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28. Three married couples sit down at a round table at which there are six chairs. All of

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the possible seating arrangements of the six people are equally likely. The probability
. sits next to his wife is: s e
e m
that each husband
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(a)gl .
a
2
15

(b) 31 .
4
(c) 15 .

(d) None of the above.

29. Let f : R2 → R be a function. For every x, y, z ∈ R, we know that f (x, y) + f (y, z) +
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f (z, x) = 0. Then, for every x, y ∈ R2 , f (x, y) − f (x, 0) + f (y, 0) =

e m
(a) 0.
las
(b) 1.
ag
(c) −1.

(d) None of the above.

30. The minimum value of the expression below for x > 0 is:
6 
x + x1 − x6 + x16 − 2
 

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3 
x + x1 + x3 + x13
 

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(b) 3.
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a (c) 6

(d) 12.

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