ISI Admission Test 2020 Sample Paper M.Tech(QROR) PQB

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ISI Admission Test
2020
Sample Paper ·
M.Tech(QROR)
PQB
EXAM YEAR TYPE

ISI Admission Test 2020 Sample Paper
SUBJECT

M.Tech(QROR) PQB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Sample PQB 2020

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PART I (STATISTICS / MATHEMATICS STREAM)

em each Group - S1 & S2. l a s
s g
ATTENTION : Answer a total of SIX questions taking at least TWO from

g la a
a GROUP S1: Statistics

1. (a) The registrar of a university keeps an alphabetical list of all
students and labels them as 1, 2, ⋯. The registrar wants to
conduct a survey to get some idea about the socio-economic
background of the students. Suppose there are 10,000 students
in the current year. It was suggested that a random number

m
from 1 to 100 be chosen and the corresponding student be

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included in the sample. Subsequently every 100th student,

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starting the count at the selected student’s label, is chosen for

s
the sample. Is this a simple random sample? Justify your
answer.
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a
[6]
(b) A measuring instrument is being used to obtain 𝑛𝑛 independent
measurements 𝑋𝑋𝑖𝑖 , 𝑖𝑖 = 1, 2, ⋯ , 𝑛𝑛 of a physical, unknown
constant 𝜇𝜇. Suppose that the measuring instrument is known to
be biased to the positive side by 0.1 units.

(i) Write a model to express the random variable 𝑋𝑋𝑖𝑖 in terms
of the bias, the unknown constant 𝜇𝜇 and the

m
measurement error.

m c. o
c. o m
(ii) Obtain the MLE of the variance of error assuming that

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the observed measurements follow normal distribution.

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[2+4]
(c) Let (𝑋𝑋, 𝑌𝑌) ∼ 𝑁𝑁2 (𝜇𝜇1 , 𝜇𝜇2 , 𝜎𝜎12 , 𝜎𝜎22 , 𝜌𝜌). The parameters are
unknown. Derive a procedure for testing the following
hypothesis:
𝜎𝜎12 𝜎𝜎12
𝐻𝐻0 : 2 = 𝜃𝜃0 vs 𝐻𝐻1 : 2 ≠ 𝜃𝜃02 .
2
𝜎𝜎2 𝜎𝜎2
[8]

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2. Consider the following linear model under the usual Gauss-
Markoff set-up.
𝐸𝐸(𝑦𝑦1 ) = 𝐸𝐸(𝑦𝑦3 ) = 𝛽𝛽0 + 𝛽𝛽1 , 𝐸𝐸(𝑦𝑦2 ) = 𝛽𝛽0 + 𝛽𝛽2.
(a) Obtain least square estimators of the parameters 𝛽𝛽0, 𝛽𝛽1 and 𝛽𝛽2.
Are these unique? Are these unbiased? [Justify the preceding
two answers.]
[6+2+4]
(b) Determine the condition of estimability of the parametric
function ∑2𝑖𝑖=0 𝑙𝑙𝑖𝑖 𝛽𝛽𝑖𝑖 , where 𝑙𝑙0 , 𝑙𝑙1 and 𝑙𝑙2 are known constants.
[5]
(c) Obtain the BLUE of (3𝛽𝛽0 + 𝛽𝛽1 + 2𝛽𝛽2 ), if it exists, and also
find its variance.
[2+1]

3. Some legal experts believe that the verdict of death penalty in
cases of homicide (murder) in a country depends on the race
(white / black) of the victim and the defendant (the person
accused of the murder). A detailed study covering 326 cases
of homicides was carried out and the following were observed:

Out of 151 cases where both defendant and victim
were white, 19 were awarded death penalty. Out
of 9 cases where defendant was white and victim
black, none was awarded death penalty. Out of 63
cases where defendant was black and the victim
white, 11 was awarded death penalty. Finally, out
of 103 cases where both defendant and victim
were black, 6 were awarded death penalty.

(a) What random variable was being studied? Write the probable
distribution of the random variable and list its parameters.
[2 + 2 + 2]

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(b) Suppose you are checking the belief regarding discrimination
towards awarding death penalty with respect to the race of the
m
c. o Formulate m
defendants (accused) and the victims.

m hypotheses, s e
se a
(i) this problem as testing of statistical

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stating the null and the alternative

g
hypotheses.

a (ii) Are the hypotheses simple or composite? Justify.
[4+2]

(c) Present the entire data on awarding death penalty in tabular
format showing race of both the defendant as well as the
victim. Use this table to estimate the parameters in the
proposed hypotheses. Suggest a way to present this data
graphically.

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[3 + 3 + 2]

.
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4. Let 𝑋𝑋1 , ⋯ , 𝑋𝑋𝑛𝑛 be a random sample from a distribution with

a s
probability density function

g l 𝑥𝑥 −𝑥𝑥
a
𝑒𝑒 𝜃𝜃 , 𝑥𝑥 > 0
𝑓𝑓(𝑥𝑥) = �𝜃𝜃 2
0, otherwise,
with 𝜃𝜃 > 0.

(a) Find the minimal sufficient statistic of 𝜃𝜃. [5]
(b) Find the MVUE of 𝜃𝜃. [5]
Derive a test for 𝐻𝐻0 : 𝜃𝜃 = 𝜃𝜃0 vs 𝐻𝐻1 : 𝜃𝜃 > 𝜃𝜃0 . Find the power of
m
(c)

c. o
the test.
m
c. o m
[8+2]

s e
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Let 𝑋𝑋1 , ⋯ , 𝑋𝑋𝑛𝑛 be a random sample from a population with

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5. (a)
mean 𝜇𝜇 and variance 𝜎𝜎 2 . Show that
1 2
𝑇𝑇𝑛𝑛 = � ��𝑋𝑋𝑖𝑖 − 𝑋𝑋𝑗𝑗 �
2𝑛𝑛(𝑛𝑛 − 1)
𝑖𝑖 𝑗𝑗
2
is an unbiased estimator of 𝜎𝜎 .
[7]

2

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(b) Consider a population having probability density function
(1 − 𝜃𝜃)𝑥𝑥 −𝜃𝜃 , 0 < 𝑥𝑥 < 1
𝑓𝑓(𝑥𝑥) = �
0, otherwise,
with 0 < 𝜃𝜃 < 1.
(i) Based on a random sample 𝑋𝑋1 , ⋯ , 𝑋𝑋𝑛𝑛 , find the MLE of
𝜃𝜃. Find the Fisher information about 𝜃𝜃.

(ii) Let 𝑥𝑥1 = 0.4, 𝑥𝑥2 = 0.1, 𝑥𝑥3 = 0.6 and 𝑥𝑥4 = 0.5 be a
random sample of size 4. Derive an estimate of 𝜃𝜃 by
the method of moments.
[(4+4)+5]

GROUP S2: Probability

6. (a) A student takes a multiple-choice test consisting of two
questions. The first one has three options and the second one
has five. Exactly one option is correct for each of the two
questions. The student chooses at random, one option from
each of the two questions. Let 𝑋𝑋 be the number of right
answers. Find 𝐸𝐸(𝑋𝑋) and 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋).
[4+6]
(b) Consider the schooling system in a locality, where 15% of the
children come from low-income families. Children from low-
income families in the locality graduate from college only
20% of the time. Children not from low-income families have
40% chance of graduating from college. One person, picked
randomly from the locality, is found to have graduated from a
college. What is the probability that this person comes from a
low-income family?

7. Suppose {𝑁𝑁(𝑡𝑡), 𝑡𝑡 ≥ 0} is a Possion process with rate 𝜆𝜆 > 0,
where 𝑁𝑁(𝑡𝑡) denotes the number of occurrences of an event ℰ
in [0, 𝑡𝑡).

(a) Let 𝑇𝑇 denote the time until the first occurrence of the event ℰ.
Find the distribution of 𝑇𝑇.
[5]
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(b) Find 𝑃𝑃(𝑁𝑁(𝑠𝑠) = 𝑘𝑘|𝑁𝑁(𝑡𝑡) = 𝑛𝑛) for 𝑘𝑘𝑘𝑘{0, 1, ⋯ , 𝑛𝑛} and
𝑛𝑛𝑛𝑛{0, 1, 2, ⋯ } for 0 ≤ 𝑠𝑠 < 𝑡𝑡.

o m m
c
[8]

(c) . Suppose that a fair coin is tossed until a head appears. Given s e
m a
la se that the coin is tossed four times and no head is obtained in ag
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ag these four tosses, what is the probability that the coin must be
tossed at least two more times to obtain the first head?
[7]

8. (a) Suppose 𝑋𝑋 and 𝑌𝑌 are jointly distributed with probability
density function
12 2
(𝑥𝑥 + 𝑥𝑥𝑥𝑥), 0 ≤ 𝑥𝑥 ≤ 1, 0 ≤ 𝑦𝑦 ≤ 1
𝑓𝑓(𝑥𝑥, 𝑦𝑦) = � 7
0, otherwise.
m
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Find the distribution of 𝑈𝑈 = max {𝑋𝑋, 𝑌𝑌}. [10]

(b) Suppose (𝑋𝑋, 𝑌𝑌) have aem
l a s bivariate distribution. Given that

g
𝐸𝐸(𝑋𝑋) = 0, 𝑉𝑉𝑉𝑉𝑉𝑉 (𝑋𝑋) = 1, 𝐸𝐸(𝑌𝑌|𝑋𝑋) = 2 + 0.5𝑋𝑋
a find the correlation coefficient between 𝑋𝑋
and
𝑉𝑉𝑉𝑉𝑉𝑉(𝑌𝑌|𝑋𝑋) = 0.75,
and 𝑌𝑌.

9. Consider the function defined as follows:
0, 𝑥𝑥 < 0
0.3, 0 ≤ 𝑥𝑥 < 2
𝐹𝐹(𝑥𝑥) = �
0.3 + 0.2𝑥𝑥, 2 ≤ 𝑥𝑥 < 3
1, 𝑥𝑥 ≥ 3.
o m
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c. o m
e 𝑋𝑋.
(a) Draw a rough sketch of 𝐹𝐹(𝑥𝑥) on your answer-sheet and argue

em s
la [6]
that it is the distribution function of some random variable

(b) Find 𝐸𝐸(𝑋𝑋). a g [10]

(c) What is the third quartile of 𝑋𝑋? [4]

4

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10. (a) Let 𝑋𝑋 be a random variable with the following probability
density function:
𝑥𝑥, 0 ≤ 𝑥𝑥 ≤ 1
𝑓𝑓(𝑥𝑥) = �1, 1 ≤ 𝑥𝑥 ≤ 1.5
0, otherwise.
Find 𝐸𝐸(𝑋𝑋|𝑋𝑋 ≥ 0.5).

(b) Let 𝑋𝑋 be a gamma random variable with shape parameter 3
and scale parameter 1. Let 𝑌𝑌 be exponentially distributed with
mean 1. Suppose that 𝑋𝑋 and 𝑌𝑌 are independently distributed.
A new random variable 𝑍𝑍 is defined as follows:

1, if 𝑌𝑌 ≥ 𝑋𝑋
𝑍𝑍 = �
0, otherwise.

Find the distribution of 𝑍𝑍.

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PART II (ENGINEERING STREAM)

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ATTENTION : Answer a total of SIX questions taking at least TWO from

m . each Group - E1 & E2.
s e
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g la ag
a 1. (a) You are given an acute angled triangle ABC. Draw the
GROUP E1: Mathematics

perpendiculars AD and BE on the sides BC and CA
respectively. Show that CD.BC = CE.AC.
[6]

(b) Find the number of divisors and the sum of the divisors of
5400.

o m
c
[8]

m . coefficient of 𝑥𝑥 is largest in
e ?
(c) What is the value of 𝑘𝑘 such that 𝑘𝑘

the expansion of �1 + s𝑥𝑥�
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3 15

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4
[6]

2. (a) Consider the curve 𝑦𝑦 = 𝑥𝑥 3 + 2. Find the equation of its tangent
passing through the origin. Also, evaluate the area of the region
enclosed by this tangent and the curve.
[6+8]

(b) Find the number of ways of choosing three numbers 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐

m
c. o
from the integers 1, 2, …, 201 such that these three numbers

m
c. o
form an A.P.

e m
[6]

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3. (a) Determine the radius of convergence of the series
∞
26 𝑛𝑛 𝑛𝑛3 𝑥𝑥 3𝑛𝑛
�� � 4 .
11 𝑛𝑛 + 1
𝑛𝑛=0

6

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(b) Suppose that
𝑥𝑥(1 + 𝑎𝑎 cos 𝑥𝑥) − 𝑏𝑏 sin 𝑥𝑥
lim =1
𝑥𝑥→0 𝑥𝑥 3
5 3
for some constants 𝑎𝑎 and 𝑏𝑏. Show that 𝑎𝑎 = − 2 and 𝑏𝑏 = − 2 .

4. (a) For 𝑛𝑛 ≥ 2, consider the following square matrix of order (n-1):
3 1 1 1 1
1 4 1 1 1
⎛1 ⋯
1 5 1 1 ⎞
⎜1
⎜ 1 1 6 1 ⎟⎟
⋮ ⋱ ⋮
⎝1 1 1 1 ⋯ 𝑛𝑛 + 1⎠
Find its determinant using only elementary row operations and
denote it by 𝐴𝐴𝑛𝑛 . Hence or otherwise, check whether the
𝐴𝐴
sequence { 𝑛𝑛�𝑛𝑛!}n≥2 is bounded.
[12+3]
𝑥𝑥+3
(b) Solve the inequality ≤ 0 for 𝑥𝑥.
𝑥𝑥 2 −5𝑥𝑥+6
[5]

GROUP E2: Engineering & Technology

Engineering Mechanics and Thermodynamics

5. (a) A particle, starting from rest, moves in a straight line, whose
acceleration is given by the equation 𝑎𝑎 = 10 − 0.006𝑆𝑆 2
where 𝑎𝑎 = acceleration in m-s-2, 𝑆𝑆 = distance traversed in m.
Determine
(i) the velocity of the particle, when it has travelled 50 m,
(ii) the distance travelled by the particle when it comes to rest.
[3+3]

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(b) In a certain weight lifting machine, a weight of 16000 N is
lifted by an effort of 400 N while the weight moves up by 150
m
c.(i)o the mechanical advantage m
mm. The point of application of effort moves up by 9000 mm.

m s e
Find

se (ii) the velocity ratio l a
g la ag
a
(iii) the efficiency of the machine
(iv) whether the machine is reversible or not with proper
explanation.
[2+2+2+2]

(c) A man wishes to move a wooden block a distance of 2 m to
the right with the least amount of work. If the block weighs
1000 N and the coefficient of friction is 0.3, should he slide it

m
as indicated in Figure 1a or should he tip it as indicated in

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Figure 1b?

s em
g l a
a

Figure 1a

m
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c. o s e m
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Figure 1b
[6]
8

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6. (a) The food compartment of a refrigerator is maintained at 40C
by removing heat from it at a rate of 360 kJ per minute. If the
required power input to the refrigerator is 2 kW, determine
(i) the coefficient of performance of the refrigerator,
(ii) the rate of heat rejection to the room that houses the
refrigerator.
[5+5]
(b) A heat engine is used to drive a heat pump. The heat transferred
from the heat engine and the heat pump are rejected to the same
sink. The efficiency of the heat engine is 27% and the COP of
the heat pump is 4. Determine the ratio of the total heat
rejection from the heat engine and the heat pump to the heat
transferred to the heat engine.

7. (a) A crate of weight 𝑊𝑊 rests with one end on an inclined plane,
the opposite edge being supported by means of a rope as
shown in Figure 2. The edge at A is on the verge of slipping
down. The angle of friction is 300. Show that the tension 𝑇𝑇 on
the rope is 𝑊𝑊/√3.

Figure 2

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(b) A stationary mass of gas is compressed without friction
from an initial state of 0.3 m3 and 0.105 MPa to a final state
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of 0.15 m3 and 0.105 MPa. There is a transfer of 37.6 kJ of

m .heat from the gas during the process. What is the change in
s e
se l a
internal energy of the gas during the process?

la ag

agElectrical and Electronics Engineering

8. (a) An inductive load of 0.8 lagging power factor is fed by a
250 V / 125 V, 1 kVA single phase transformer. The core
loss obtained from the no-load test and the copper loss
obtained from the short-circuit test of the transformer are
80 W and 120 W respectively.
(i) Determine the full load efficiency of the transformer.

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(ii) Write down the condition of maximum efficiency and

.
estimate the ratio of the load currents under maximum

s em
efficiency to the full load efficiency. Neglect the no-load

l a
current.

a g
[5+(2+3)]

(b) A six-pole, wave-wound DC shunt motor is drawing 11 A line
current when connected across a 220 V DC supply. The
armature resistance and field resistance of the motor are 4 Ω
and 220 Ω respectively.
(i) Determine the back emf developed in the armature.
(ii) Determine the speed of rotation in RPM, if the armature

m
has 400 conductors and the flux per pole is 0.01 Wb.

m c. o
[5+5]

c. o s e m
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(a) Design the circuit for the Boolean function 𝑌𝑌 = 𝐴𝐴𝐴𝐴 + 𝐴𝐴𝐶𝐶

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using NAND gates only.
[4]

(b) Subtract 11110 from 10011 using 1’s complement method.
[4]

10

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(c) Consider the circuit shown in Figure 3.
(i) Find the Thevenin equivalent for the network to the left of
terminal 𝐴𝐴𝐴𝐴 by applying Thevenin’s theorem.
(ii) Find the output voltage 𝑣𝑣0 using the Thevenin voltage
across the terminal 𝐴𝐴𝐴𝐴.

Figure 3
[8+4]

Engineering Drawing

10. (a) Sketch the isometric view of the object whose two views are
shown in Figure 4.

Figure 4
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(b) Sketch the front view and top view of the object shown in

m .Figure 5. Use third angle projection.
s e
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a

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em
Figure 5

a s

g l
a

m
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c. o s e m
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12

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