ISI Admission Test 2017 Question Paper MS (QE) PEA

FOR ISI EXAM PREPARATION

ISI 2017
Question Paper · MS
(QE) PEA
EXAM YEAR TYPE SUBJECT

ISI 2017 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 dimension of the space spanned by the vectors (−1, 0, 1, 2),
(−2, −1, 0, 1), (−3, 2, 0, 1) and (0, 0, −1, 1) is

A. 1
B. 2

m
.co
C. 3

o m
D. 4.
m
c
2. How .many onto functions are there from a set A with m > 2 ele- s e
s em to a set B with 2 elements? g l a
l a ments
a
ag A. 2 m

B. 2m − 1
C. 2m−1 − 2
D. 2m − 2.
3. The function f : R2+ → R given by f (x, y) = xy is

o m
c
A. quasiconcave and concave

m .
B. concave but not quasiconcave
e concave
C. quasiconcave butsnot
D. none of the g l a
aabove.
4. The function f : R2+ → R given by f (x, y) = xy is

A. homogeneous of degree 0
B. homogeneous of degree 1
C. homogeneous of degree 2

o m
c
D. not homothetic.
m .
m .co 5. You have n observations on rainfall in centimeters (cm) at m
s edeviation,
a certain

s e a
variance, and coefficient of variation (CV). Now,glif instead, you
location, denoted by x, and you calculate the standard

g la were given the same observations measured in a millimeters (mm),
a then

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 8

A. the standard deviation and CV would increase by a factor
of 10, and the variance by a factor of 100
B. the standard deviation would increase by a factor of 10,
the variance by a factor of 100, and the CV would be
unchanged
C. the standard deviation would increase by a factor of 10,
and the variance and CV by a factor of 100

6. You have n observations on rainfall in centimeters (cm) at two lo-
cations, denoted by x and y respectively, and you calculate the
covariance, correlation coefficient r, and the slope coefficient b of
the regression of y on x. Now, if instead, you were given the same
observations measured in millimeters (mm), then

A. the covariance would increase by a factor of 10, b by a
factor of 100, and r would be unchanged
B. the covariance and b would increase by a factor of 100,
and r would be unchanged
C. the covariance would increase by a factor of 100, and b
and r would be unchanged

7. Let 0 < p < 100. Any solution (x∗ , y ∗ ) of the constrained maxi-
mization problem
( )
−1
max +y
x,y x
subject to

px + y ≤ 10,
x, y ≥ 0,

must satisfy
A. y ∗ = 10 − p
B. x∗ = 10/p
√
C. x∗ = 1/ p

2

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 8

m
m .co

m .co s e m
se g l a
a

8. Suppose the matrix equation Ax = b has no solution, where A is a
3 × 3 non-zero matrix of real numbers and b is an 3 × 1 vector of
real numbers. Then,
A. The set of vectors x for which Ax = 0 is a plane.
m
.co
B. The set of vectors x for which Ax = 0 is a line.
m rank of A is 3.
c. D.C.o The s e m
s em Ax = 0 has a non-zero solution.
l a
l 9. k people get off a plane and walk into a hall where they are as- ag
a
ag signed to at most n queues. The number of ways in which this can
be done is

A. Ckn
B. Pkn
C. nk k!
D. n(n + 1) . . . (n + k − 1).
m
.co
10. If P r(A) = P r(B) = p, then P r(A ∩ B) must be
A. greater than p2
s em
B. equal to p
g la
2

a equal to p
C. less than or 2

11. If P r(Ac ) = α and P r(B c ) = β, (where Ac denotes the event ‘not
A’), then P r(A ∩ B) must be
A. 1 − αβ,
B. (1 − α)(1 − β)
m
.co
C. greater than or equal to 1 − α − β
m
.co m
m s e
12. The density function of a normal distribution withamean µ and
s e g l
g la standard deviation σ has inflection points at
a
a A. µ
B. µ − σ, µ + σ

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 8

C. µ − 2σ, µ + 2σ
D. nowhere.

13. In how many ways can five objects be placed in a row if two of
them cannot be placed next to each other?
A. 36
B. 60
C. 72
D. 24.

14. Suppose x = 0 is the only solution to the matrix equation Ax = 0
where A is m × n, x is n × 1, and 0 is m × 1. Then, of the two
statements (i) The rank of A is n, and (ii) m ≥ n,
A. Only (i) must be true
B. Only (ii) must be true
C. Both (i) and (ii) must be true
D. Neither (i) nor (ii) has to be true.

15. Mr A is selling raffle tickets which cost 1 rupee per ticket. In the
queue for tickets, there are n people. One of them has only a 2-
rupee coin while all the rest have 1-rupee coins. Each person in the
queue wants to buy exactly one ticket and each arrangement in the
queue is equally likely to occur. Initially, Mr A has no coins and
enough tickets for everyone in the queue. He stops selling tickets as
soon as he is unable to give the required change. The probability
that he can sell tickets to all people in the queue is:
A. n−2
n
B. n1
C. n−1
n
.
n−1
D. n+1 .

16. Out of 800 families with five children each, how many families would
you expect to have either 2 or 3 boys? Assume equal probabilities
for boys and girls.
A. 400
B. 450

4

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 8

m
m .co

m .co s e m
se g l a
a
C. 500
D. 550

17. The function f : R → R given by
{
x
, if x ̸= 0,
f (x) = |x|
m
.co
1, if x = 0.
m
is
.co
m A. concave s e m
s e l a
g l a B. convex ag
a C. neither concave nor convex
D. both concave and convex
n +1 2
18. As n → ∞, the sequence { 2n 2 +3 }

A. diverges
B. converges to 1/3
C. converges to 1/2
m
D. neither converges nor diverges.
.co
19. The function x1/3 is
s em
a
A. differentiableglat x = 0
a
B. continuous at x = 0
C. concave

20. The function sin(log x), where x > 0
A. is increasing
B. is bounded and converges to a real number as x → ∞
m
m .co
.co
C. is bounded but does not converge as x → ∞

m D. none of the above.
s em
s e laR, define the
21. For any two functions f : [0, 1] → R and f : [0, 1]g→
g la 1
a for all x ∈ [0, 1].
2

a function g : [0, 1] → R as g(x) = max(f (x), f (x))
A. If f1 and f2 are linear, then g is linear
1 2

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 8

B. If f1 and f2 are differentiable, then g is differentiable
C. If f1 and f2 are convex, then g is convex
D. None of the above
22. Let f : R → R be the function
f (x) = x3 − 3x ∀ x ∈ R.
Find the maximum value of f (x) on the set of real numbers x
satisfying x4 + 36 ≤ 13x2 .
A. 18
B. −2
C. 2
D. 52
23. A monkey is sitting on 0 on the real line in period 0. In every
period t ∈ {0, 1, 2, . . .}, it moves 1 to the right with probability p
and 1 to the left with probability 1 − p, where p ∈ [ 21 , 1]. Let πk
denote the probability that the monkey will reach positive integer
k in some period t > 0. The value of πk for any positive integer k
is
A. pk
B. 1
pk
C. (1−p) k

D. kp .
24. Refer to the previous question. Suppose p = 21 and πk now denotes
the probability that the monkey will reach any integer k in some
period t > 0. The value of π0 is
A. 0
B. 21k
C. 12
D. 1
25. Suppose f : R → R is a differentiable function with f ′ (x) > 0 for
all x ∈ R and satisfying the property
lim f (x) ≥ 0.
x→−∞

Which of the following must be true?

6

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 6 of 8

m
m .co

m .co s e m
se g l a
a
A. f (1) < 0
B. f (1) > 0
C. f (1) = 0
D. None of the above
26. For what values of x is
m
m
c. A.o 4 < x < 9
x2 − 3x − 2 < 10 − 2x
m .co
m B. x < 0 s e
s e l a
g l a ag
a C. −3 < x < 4
D. None of the above
∫ e2 1
27. e x(log x)3
dx =
A. 3/8
B. 5/8
C. 6/5
D. −4/5
o m
. c
m
28. The solution of the system of equations
e− 2y + z = 7
l as
x

ag 2x − y + 4z = 17
3x − 2y + 2z = 14
is
A. x = 4, y = −1, z = 3
B. x = 2, y = 4, z = 3
C. x = 2, y = −1, z = 5
o m
m . c
.co m
2
29. Let f : R → R be a twice-differentiable function with non-zero

s e there is
em
second partial derivatives. Suppose that for every x ∈ R,

s
∗
la
a unique value of y, say y (x), that solves the problem
g
g la a
a
max f (x, y).
y∈R

Then y ∗ is increasing in x if

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 8

A. f is strictly concave
B. f is strictly convex
∂ f2
C. ∂x∂y >0
∂ f2
D. ∂x∂y < 0.

30.
√
∫
3 2x+1 dx =

A. √ √
3 2x+1 2x + 1
+ +c
ln 3 ln 3
B. √ √ √
3 2x+1 2x + 1 3 2x+1
− +c
ln 3 (ln 3)2
C. √ √ √
3 2x+1 2x + 1 3 2x+1
− +c
(ln 3)2 ln 3

8

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