ISI Admission Test 2019 Question Paper M.Stat PSA

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ISI 2019
Question Paper ·
M.Stat PSA
EXAM YEAR TYPE SUBJECT

ISI 2019 Question Paper M.Stat PSA

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PSA a
2019

1. Let A = ((aij )) be an m×n matrix with all non-zero real entries. Let B
be obtained from A by replacing a11 by 0 and keeping all other entries
m
m
unchanged. If r is the rank of A, then what is the set of possible values
.co
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for the rank of B?
s e m
s e(A)m{r} l a
g la (B) {r − 1, r, r + 1} (C) {r, r + 1} (D) {r − 1, r}
ag
a

o m
c
2. What is the period of the function .g(x) = | cos x| + | sin x|?
s em
g la
a
(A) π (B) π/2 (C) 2π (D) π/4

m
m .co
m .co s e m
la
3. If 3(cos 100◦ + i sin 100◦ )(cos 110◦ + i sin 110◦ ) = x + iy, where x and y
s e g
la
are real numbers, then

g a
a
√
(A) x = − 3 2 3 , y = − 32 .
√
(B) x = 3 2 3 , y = 23 .
√
(C) x = 3 2 3 , y = − 32 .
√
(D) x = − 3 2 3 , y = 32 .

1
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4. What is the number of 6 digit positive integers in which the sum of
the digits is at least 52?
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(A) 66
m .co (B) 24 (C) 28 (D) 120
s e m
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a

o m
c
5. Let the sum
3 + 33 + 333 +.· · · + �33 ��
e m . . . 3�

l a s 200 times

g
be . . . zyx in the decimal system, i.e., x is the unit’s digit, y the ten’s
digit, and so on. What a is z?
(A) 0 (B) 9 (C) 7 (D) 3

m
m .co
m .co s e m
s e g la
g la a
a 6. How many times does the digit ‘2’ appear in the set of integers
{1, 2, ..., 1000}?

(A) 590 (B) 600 (C) 300 (D) 299

2
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a

 
0 1 t

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 
7. Let t be a real number. Then the rank of 2 t −1 equals

c o m 2 2 0
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(A) .2 if t = −1, and 3 if t �= −1. e
e(B)m 2 if t = 1, and 3 if t �= 1. l as
s
la (C) 2 if t = ±1, and 3 if |t| �= 1. ag
g
a (D) 3 for all t.

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s e
�200�
l a
ag
8. The number 100 /4100 lies in

(A) [ 43 , 1) (B) (0, 12 ) (C) [1, ∞) (D) [ 21 , 34 )

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m .co s em
s e 9. Let P (x) = x + 4x − 8x − 1. Which of the following isla
la
4 3 2

ag false?

ag (A) P (x) has a real root in (−4, 1)
(B) P (x) has a real root < −4
(C) P (x) has a real root > 1
(D) P (x) has at least two real roots.

3
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10. Two friends, one from Kolkata and one from Delhi, start driving to-
wards each other at the same time. It is given that the distance between
m
m
Kolkata and Delhi is 1455 km. One of them drives at a constant speed
.co
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of 80 kmph (km per hour), while the other drives at a speed of 50 kmph
m s e m
s e
during the first hour, 55 kmph during the second hour, 60 kmph dur-
l a
g l a ing the third hour, and so on (i.e., his speeds over successive hours are
ag
a in an arithmetic progression). How long will it take for them to meet
each other?

(A) 8 hrs 56 mins
(B) 8 hrs 46 mins
(C) 10 hrs 26 mins
(D) 9 hrs 36 mins

m
m .co
s e
l a
ag
11. How many positive divisors of 25 53 114 are perfect squares?

(A) 60 (B) 18 (C) 120 (D) 4

m
m .co
m .co s e m
s e g la
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a 12. What is the set of numbers x in (0, 2π) such that log log(sin x + cos x)
is well-defined?

(A) [ π8 , 3π
8 ] (B) (0, π2 ) (C) (0, π4 ] (D) (0, π) ∪ ( 3π
2 , 2π)

4
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13. Let C1 and C2 be concentric circles with centre at O and radii r1 and
r2 respectively. The line OA2 intersects C1 at A1 . The line A1 D is
m
m
tangent to C1 at A1 . What is the length of the line segment A2 D?
.co
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e a
� � �

s
(A) 2r1 (r2 − r1 ) (B) r22 − r12 (C) r2 (D) 2r2 (r2 − r1 )
l
g l a ag
a C2 D

C1

O A1 A2
f

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14. The reflection of the point (1, 2) with respect to the line x + 2y = 15 is

(A) (3, 6). (B) (6, 3). (C) (10, 5). (D) (5, 10).

m
m .co
m .co s e m
s e g la
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a 15. How many solutions does the equation cos2 x + 3 sin x cos x + 1 = 0
have for x ∈ [0, 2π)?

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

5
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16. The functions f, g : [0, 1] → [0, 1] are given by f (x) = 12 x(x + 1) and
g(x) = 21 x2 (x + 1). What is the area enclosed between the graphs of
m
c o m
f −1 and g −1 ?
m .co
. s e
s e(A)m1/8 (B) 1/4 (C) 5/12 (D) 7/24
l a
g la ag
a

o m
17. If f (a) = 2, f (a) = 1, g(a) = −1m
. c
e
� �
and g (a) = 2, then what is

as
lim l
g
g(x)f (a) − f (x)g(a)
a
?
x→a x−a

(A) 5 (B) 3 (C) −3 (D) −5

m
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m .co s e m
s e g la
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a 18. Draw one observation N at random from the set {1, 2, . . . , 100}. What
is the probability that the last digit of N 2 is 1?

(A) 1/20 (B) 1/50 (C) 1/10 (D) 1/5

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19. Let X be the number of tosses of a fair coin required to get the first
head. If Y | X = n is distributed as Binomial(n, 21 ), then what is
m
P(Y = 1)?
c o m m .co
. s e
s e(A)m4/9 (B) 1/4 (C) 1/3 (D) 5/9
l a
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a

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20. Suppose X is distributed uniformly on (−1, 1). For i = 0, 1, 2, 3, let

.co
� �
pi = P X 2 ∈ ( 4i , i+1
4 ) . For which value of i is pi the largest?

e m
(A) 3
l
(B) 1
as (C) 0 (D) 2

ag

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21. In a simulation experiment, two independent observations X1 and X2

.co e m
are generated from the Poisson distribution with mean 1. The experi-

e m ment is said to be successful if X1 + X2 is odd. What is the expected
las
las ag
value of the number of experiments required to obtain the first success?

ag (A) 2(1 + e−2 )
(B) 2/(1 − e−2 )
(C) 2/(1 − e−4 )
(D) 2(1 + e−4 )

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22. A shopkeeper has 12 bulbs of which 3 are defective. She sells the bulbs
by selecting them at random one at a time. What is the probability
m
c o m
that the seventh bulb sold is the last defective one?
m .co
. s e
s e(A)m3/44 (B) 9/44 (C) 13/44 (D) 7/44
l a
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a

23. The chances of Smitha getting admitted to colleges A and B are 60%
and 40% respectively. Assume that colleges admit students indepen-
m
.co
dently of each other. If Smitha is told that she has been admitted to

m
at least one college, what is the probability that she got admitted to
college A?
s e
gl a
(A) 3/5 (B)a15/19 (C) 10/13 (D) 5/7

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24. Suppose X1 , X2 , . . . , Xn is a random sample from an exponential dis-

.co e m
tribution with mean λ. If λ̂1 and λ̂2 are, respectively, the maximum

e m likelihood estimators of the mean and the median of the underlying
las
las distribution, then
ag
ag (A) λ̂1 < λ̂2 .
(B) λ̂1 = λ̂2 .
(C) λ̂1 < λ̂2 and λ̂1 > λ̂2 are both possible.
(D) λ̂1 > λ̂2 .

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25. Suppose Y, X1 , X2 , . . . , Xn are i.i.d. N (µ, 1) random variables, and
In = (X̄n − an , X̄n + an ) is a 95% confidence interval for µ. Then
m
P(Y ∈ In )
c o m m .co
(A) .converges to 1 as n → ∞. s e
s e(B)m is greater than 0.95 for all n ≥ 1. l a
g la (C) is less than 0.95 for all n ≥ 1. ag
a
(D) equals 0.95 for all n ≥ 1.

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26. Suppose (X1 , Y1 ), (X2 , Y2 ), . . . , (Xn , Yn ) is an i.i.d. sample from
m
s e
N2 (0, 0, 1, 1, ρ) where |ρ| ≤ 1. Let (i1 , . . . , in ) be a random permu-

l a
tation of {1, 2, . . . , n}. Define Tn = n1
n
�

ag
Xj Yij . What is E(Tn )?
j=1

(A) 1/n (B) 0 (C) ρ/n (D) ρ

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m . cΦo
.co m
2

erandom
27. Suppose X is a N (µ, σ ) random variable, and Y = Φ(X), where

e m las
is the cumulative distribution function of a standard normal

las variable. What is E(Y )?
a g
ag √
(A) Φ(µ/ 2 + σ 2 )
√
(B) Φ(µ/ 1 + σ 2 )
(C) Φ(µ/σ)
√
(D) Φ(µ/ 4 + σ 2 )

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28. Let X1 , X2 , . . . , Xn be a random sample from a distribution with prob-
ability density function
m
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.co


m
(λ + 1) xλ 0 ≤ x ≤ 1
fλ (x) =
s e
em l a
0 otherwise,
s
la where λ > −1. What is the maximum likelihood estimator of λ? ag
g
a (A) 1 + n/( � log X )n
i
i=1
�n
(B) −1 − n/( log Xi )
i=1
n
�
(C) −1 − n1 log Xi
i=1
�n
(D) 1 − n/( log Xi )
m
.co
i=1

s em
g la
a of√(X , X ) is N (0, 0, 2, 2, −0.5). What
29. Suppose the joint distribution 1 2 2
is the value of P(2X1 + X2 ≤ 2 2)? Here Φ denotes the cumulative
distribution function of a standard normal random variable.

√ √ √
(A) Φ(2/ 7) (B) Φ(2/ 3) (C) Φ(2/ 5) (D) Φ(1)

m
m .co
m .co 30. Suppose the joint probability density function of (X, Y ) is
s e m
s e g la
la


a
e−x 0 ≤ y ≤ x < ∞

ag f (x) =
0 otherwise.

What is E(X)?

(A) 2 (B) 1 (C) 6 (D) 1/2

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