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

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

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

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PART I (STATISTICS / MATHEMATICS STREAM)

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

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each Group - S1 & S2.

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a1.g (a)
GROUP S1: Statistics

At a toll plaza, it takes at least α minutes for a car to pay the
toll and move on. The actual time (𝑋𝑋) varies from car to car.
It is further assumed that this time is well represented by an
exponential random variable with the following density
function.

1 (𝛼𝛼−𝑥𝑥)/𝜃𝜃
𝑓𝑓𝑋𝑋 (𝑥𝑥) = � 𝜃𝜃 𝑒𝑒 , 𝑥𝑥 ≥ 𝛼𝛼, 0 < 𝜃𝜃 < ∞

o m
c
0 Otherwise.

The following time values (in.minutes) are recorded from 10
cars, selected randomly.se
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g la5.3, 3.7, 6.9, 4.3, 3.5, 3.9, 4.2
a likelihood estimators of α and θ, and
4.1, 3.2, 4.6,
Derive the maximum
give the respective estimates.
[10]

(b) Consider the model 𝑦𝑦𝑖𝑖 = 𝛽𝛽𝑥𝑥𝑖𝑖 + 𝜖𝜖𝑖𝑖 for 𝑖𝑖 = 1(1)𝑛𝑛 with 𝑥𝑥𝑖𝑖 > 0,
where 𝜖𝜖𝑖𝑖 ’s are independent 𝑁𝑁(0, 𝑥𝑥𝑖𝑖 𝜎𝜎 2 ) and, β and 𝜎𝜎 2 are

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unknown constants. Suppose that 𝛽𝛽̂1 and 𝛽𝛽̂2 are the least-

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squares estimators of 𝛽𝛽 obtained by minimizing
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𝑛𝑛 𝑛𝑛

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1

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𝑆𝑆1 = �(𝑦𝑦𝑖𝑖 − 𝛽𝛽𝑥𝑥𝑖𝑖 ) and 𝑆𝑆2 = � (𝑦𝑦𝑖𝑖 − 𝛽𝛽𝑥𝑥𝑖𝑖 )2 ,
2

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𝑥𝑥𝑖𝑖

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𝑖𝑖=1 𝑖𝑖=1

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respectively.

a Show that Var �𝛽𝛽̂1 � ≥ Var �𝛽𝛽̂2 �.
[10]

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2. The effect of five different additives on the yield of a chemical
process is to be studied. Each run of the experiment requires a
raw material which is supplied in batches. Each batch of
material is only large enough to permit five runs to be made.
Also due to time and infrastructure constraints, only five runs
can be made in a day. There is no interaction between the
factors.

(a) Plan the experiment using any of the standard complete
block designs known to you, in not more than twenty five
runs, to compare the effects of the additives, and give the
layout of the experiment. Also, write down the associated
linear model and state the assumptions, if any. How do you
honour the three basic principles of experimentation?
[5+3+5 = 13]
(b) If the experiment is to be conducted at two different
locations, adopting the same design proposed by you in
(a), then outline a method of combined analysis of the data
collected from the two places, assuming error variances
are the same at both the places. [No derivation is
necessary]
[7]

3. (a) A batch of chemicals may pass or fail a particular test carried
out after production. A failed batch is corrected and it
undergoes the same test for the second time, where the batch
may pass or fail. Notice that there are three possible outcomes
for a batch – pass, fail-pass and fail-fail. Suppose we have
collected test data for 𝑛𝑛 batches.
(i) Find the maximum likelihood estimators of the
probabilities of the three possible outcomes.

(ii) Test the hypothesis that the probability of failing in the
first test is same as the conditional probability of failing in
the second test against the alternative that these are not
same.
[5+8 = 13]

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(b) Consider the random variable 𝑋𝑋 having probability mass
function

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𝜃𝜃 + (1 − 𝜃𝜃)𝑒𝑒 −𝜆𝜆 , 𝑥𝑥 = 0

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𝑃𝑃[𝑋𝑋 = 𝑥𝑥] = � (1 − 𝜃𝜃)𝑒𝑒 𝜆𝜆
−𝜆𝜆 𝑥𝑥

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, 𝑥𝑥 = 1,2, ⋯ ,
. e
𝑥𝑥!

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e In a random sample of size 𝑛𝑛 from the above distribution,
𝜆𝜆 > 0 and 0 < 𝜃𝜃 < 1.
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ag suppose that 𝑛𝑛1 of the observations are 0 and the rest are non-
zero. For a known 𝜆𝜆, obtain an unbiased estimator of 𝜃𝜃 based
on the given information.
[7]

4. (a) Suppose we want to estimate the average number of school
going children per household in a city. We have the complete
list of schools. We choose a school at random and talk to the

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children in the school to find the number of school going

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children in their respective households. Assume that the

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children have given the number correctly and households are
well defined.
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(i) What are the study and target populations?
(ii) Do you think the sample average of the number of school
going children per household would be an unbiased
estimator of the average of the target population? Justify.
[2+5 = 7]

(b) A trial involves randomly choosing a number from the interval

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(0, 1) following some fixed and unknown scheme. Based on

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𝑛𝑛 = 80 independent trials, the frequencies observed in
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different subintervals are

s em g la(3/4, 1)
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Subinterval (0, 1/4] (1/4, 1/2] (1/2, 3/4]

ag Frequency 6 18 20 36
Consider the null hypothesis (𝐻𝐻0 ) that the underlying scheme
of choosing each number is governed by the pdf 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥

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for 0 < 𝑥𝑥 < 1. Carry out a test to check whether 𝐻𝐻0 can be
accepted at 5% (approximate) level of significance.
2 2 2 2
[𝜒𝜒0.025,4 = 11.143, 𝜒𝜒0.05,4 = 9.488, 𝜒𝜒0.025,3 = 9.348, 𝜒𝜒0.05,3 =
2 2
7.815, where 𝜒𝜒𝛼𝛼,𝑚𝑚 is the upper 𝛼𝛼-point of 𝜒𝜒 distribution with
𝑚𝑚 degrees of freedom.]
[13]

5. Let 𝑋𝑋1 , 𝑋𝑋2 , … , 𝑋𝑋𝑛𝑛 be a random sample from 𝑁𝑁(0, 𝜃𝜃), 𝜃𝜃 > 0.
(a) Derive a sufficient statistic for 𝜃𝜃.
(b) Find the Cramer-Rao lower bound for the variance of an
unbiased estimator of 𝜃𝜃. Is the unbiased estimator
efficient? – Justify your answer.
(c) Derive a 95% confidence interval for 𝜃𝜃.
(d) Derive an α-level most powerful test for 𝐻𝐻0 : 𝜃𝜃 = 𝜃𝜃0 versus
𝐻𝐻1 : 𝜃𝜃 > 𝜃𝜃0 .
[3+(4+4)+5+4 = 20]

GROUP S2: Probability

6. Suppose the bivariate random variable (𝑋𝑋, 𝑌𝑌) is uniformly
distributed over the region bounded below by 𝑦𝑦 = 𝑥𝑥 − 1 for
1 ≤ 𝑥𝑥 ≤ 2 and 𝑦𝑦 = 3 − 𝑥𝑥 for 2 ≤ 𝑥𝑥 ≤ 3 and, bounded above
by 𝑦𝑦 = 𝑥𝑥 for 1 ≤ 𝑥𝑥 ≤ 2 and 𝑦𝑦 = 4 − 𝑥𝑥 for 2 ≤ 𝑥𝑥 ≤ 3.
(a) Find the joint probability density function 𝑓𝑓𝑋𝑋,𝑌𝑌 (𝑥𝑥, 𝑦𝑦).
(b) Find the marginal probability density functions 𝑓𝑓𝑋𝑋 (𝑥𝑥) and
𝑓𝑓𝑌𝑌 (𝑦𝑦) of X and Y respectively.
(c) Are X and Y independent?
(d) Find 𝐸𝐸(𝑌𝑌|𝑋𝑋 = 𝑥𝑥).
[7+(3+5)+1+4 = 20]
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Let 𝑋𝑋1 , 𝑋𝑋2 , 𝑋𝑋3 be independent and identically distributed

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7. (a)

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continuous random variables. Compute 𝑃𝑃(𝑋𝑋1 > 𝑋𝑋3 |𝑋𝑋1 > 𝑋𝑋2 ).

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[10]

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(b) If 𝑋𝑋 is a non-negative integer-valued random variable, then

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show that 𝐸𝐸(𝑋𝑋) = ∑∞𝑥𝑥=1 𝑃𝑃(𝑋𝑋 ≥ 𝑥𝑥).
[10]

8. (a) Suppose (𝑋𝑋1 , 𝑋𝑋2 , 𝑋𝑋3 )𝑇𝑇 ~𝑁𝑁3 (𝜇𝜇, Σ), where

−3 1 −2 0
𝜇𝜇 = � 1 � and Σ = �−2 5 0�.
4 0 0 2

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(i) Find the conditional variance-covariance matrix of (𝑋𝑋1 , 𝑋𝑋2 )

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given 𝑋𝑋3 = 𝑥𝑥3 .

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(ii) Find the partial correlation coefficient between 𝑋𝑋1 and 𝑋𝑋2
for fixed 𝑋𝑋3 .
a [8+2 = 10]

(b) Let 𝑋𝑋 be a random variable with probability density function
1/3 if − 1 ≤ 𝑥𝑥 ≤ 2
𝑓𝑓(𝑥𝑥) = �
0 otherwise.

(i) Find the probability density function of 𝑌𝑌 = 𝑋𝑋 2 .

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(ii) Find the 95th percentile point of 𝑌𝑌.
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[8+2 = 10]

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(a) Consider a Markov chain {𝑋𝑋 , 𝑛𝑛 ≥ 0} with state space 𝑆𝑆 =
a
9. 𝑛𝑛

g
{1, 2} and the transition probability matrix

a 1/2 1/2
𝑃𝑃 = �
1/3 2/3
�.

(i) Find the distribution of 𝑋𝑋2 , given the probability
distribution of 𝑋𝑋0 as (1⁄2 , 1⁄2)𝑇𝑇 .
5

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(ii) Find 𝑃𝑃(𝑋𝑋8 = 2, 𝑋𝑋7 = 1, 𝑋𝑋5 = 2 | 𝑋𝑋3 = 2, 𝑋𝑋2 = 1).

(iii) What are the steady-state probabilities?
[5+5+2 = 12]
(b) Two dice are being rolled together repeatedly, where the
outcome of interest is the sum of their faces as shown up. What
is the probability that Seven occurs before Eight?
[8]

10. (a) Suppose that 𝑋𝑋1 , 𝑋𝑋2 , … , 𝑋𝑋𝑛𝑛 are independent and identically
distributed 𝑈𝑈(0, 𝜃𝜃) variables. Let 𝑌𝑌𝑛𝑛 = max{𝑋𝑋1 , … , 𝑋𝑋𝑛𝑛 }. Find
the limiting distribution of 𝑍𝑍𝑛𝑛 = 𝑛𝑛(𝜃𝜃 − 𝑌𝑌𝑛𝑛 ).
[10]

(b) Customers arrive at a service station according to a Poisson
process with rate 𝜆𝜆 = 10 per hour.
(i) Suppose that eight customers arrive during a given hour,
what is the probability that at most two customers will
arrive over the following hour?

(ii) Suppose that a customer arrived during a 30-minute period,
what is the probability that she arrived during the first 15
minutes of the period considered?
[5+5 = 10]

PART II (ENGINEERING STREAM)

ATTENTION : Answer a total of SIX questions taking at least TWO from
each Group - E1 & E2.

GROUP E1: Mathematics

1. (a) Find the area of the closed figure bounded by 𝑥𝑥 = −1, 𝑦𝑦 = 0,
𝑦𝑦 = 𝑥𝑥 2 + 𝑥𝑥 + 1 and the tangent to the curve 𝑦𝑦 = 𝑥𝑥 2 + 𝑥𝑥 + 1 at
(1, 3).
[8]

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(b) A conical tent of a given capacity has to be constructed. Find
the ratio of the height to the radius of the base for the minimum

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amount of the canvas required for the tent.

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[12]

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Two complex numbers 𝑍𝑍1 and 𝑍𝑍2 lie on a circle with center at

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2. (a)

s e the origin. Further, the tangents to the circle at 𝑍𝑍1 and 𝑍𝑍2
g
l a intersect at 𝑍𝑍3 . Show that 𝑍𝑍3 can be expressed as
a
ag 𝑍𝑍3 =
2 𝑍𝑍1 𝑍𝑍2
.
𝑍𝑍1 + 𝑍𝑍2
[10]

(b) A periodic function 𝑓𝑓(𝑥𝑥) with period 2𝜋𝜋 is defined as

𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 , −𝜋𝜋 ≤ 𝑥𝑥 < 𝜋𝜋.

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(i) Sketch the graph of the function between −5𝜋𝜋 and 5𝜋𝜋.

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(ii) Determine its Fourier coefficients and Fourier series.

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[2+8 = 10]

3. (a) A total of N badminton players participated in a recently
concluded round robin singles tournament where each player
played every other player exactly once. Each match produced
either a winner or a loser. Let 𝑊𝑊𝑖𝑖 be the number of matches
won by the i-th player and 𝐿𝐿𝑖𝑖 be the number of matches lost by
the i-th player in the tournament. Show that ∑𝑁𝑁 2 𝑁𝑁 2
𝑖𝑖=1 𝑊𝑊𝑖𝑖 = ∑𝑖𝑖=1 𝐿𝐿𝑖𝑖 .

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[6]

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(b) Let 𝑢𝑢 = 𝒙𝒙𝑻𝑻 𝑨𝑨𝑨𝑨 where 𝒙𝒙 = (𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 )𝑇𝑇 is a 3 × 1 vector and

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1 3 −2
la se
s g
𝑨𝑨 = �−3 0 �. Find the vector x that minimizes u

la
2

g
2 0 3
a
a subject to the constraints 𝑥𝑥1 + 𝑥𝑥2 = 2 and 𝑥𝑥2 + 𝑥𝑥3 = 3.
[7]

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(c) The four points A(0,0), B(1,0), C(3,0) and D(5,0) are given.
Find the locus of the point P satisfying ∠𝐴𝐴𝐴𝐴𝐴𝐴 = ∠𝐶𝐶𝐶𝐶𝐶𝐶.
[7]

4. (a) Two concentric circles of radii R and r, R > r, are shown in
Figure 1. From a point P, on the smaller circle, a straight line
is drawn that intersects the larger circle at B and C. The
perpendicular to BC at P intersects the smaller circle at A.
Show that 𝑃𝑃𝑃𝑃2 + 𝑃𝑃𝑃𝑃2 + 𝑃𝑃𝑃𝑃 2 = 2(𝑅𝑅2 + 𝑟𝑟 2 ) .

Figure 1
[10]

(b) A sequence of numbers 𝑎𝑎𝑛𝑛 , 𝑛𝑛 = 1, 2, ⋯ is defined as follows
1 2𝑛𝑛−3
𝑎𝑎1 = and 𝑎𝑎𝑛𝑛 = � � 𝑎𝑎𝑛𝑛−1 for 𝑛𝑛 ≥ 2.
2 2𝑛𝑛

Prove that ∑𝑛𝑛𝑘𝑘=1 𝑎𝑎𝑘𝑘 < 1 for all 𝑛𝑛 ≥ 1.
[10]

GROUP E2: Engineering & Technology

Engineering Mechanics and Thermodynamics

5. (a) A car, starting from rest, is uniformly accelerated. The
5
acceleration at any instant is (𝑉𝑉+2) m-s-2, where 𝑉𝑉 is the
velocity of the car in m-s-1 at the instant. Find the distance in
which the car will attain a velocity of 36 km-h-1.
[6]

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(b) An axial pool of 50 kN is suddenly applied to a steel rod of
length 2 m and cross sectional area of 1000 mm2. Calculate

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the strain energy that can be absorbed if Young’s modulus of

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steel is 20×104 N-mm-2.

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[6]

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e Athebarlength la s
la s
(c) is hanging freely. The cross section of the bar is 𝐴𝐴 and
a g
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is 𝑙𝑙. It is rigidly fixed at the upper end. The Young’s
modulus of the bar is 𝐸𝐸. Derive an expression of total
elongation of the bar due to its own weight.
[8]

6. (a) A reversible heat engine receives 8 kJ of heat from one
thermal reservoir at a temperature of 800 K and 7 kJ of heat
from another thermal reservoir at a temperature of 700 K. If it
rejects heat to a third thermal reservoir at a temperature of 200

m
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K, find the thermal efficiency of the engine.
[10]

e m a temperature of -15 C. The
s
ambient air is at 30 C.aIf the heat leaks into the freezer at a
0

l kW, what is the least power necessary
(b) A domestic food freezer maintains

g
0

a
continuous rate of 1.75
to pump the heat out continuously?
[10]

7. (a) Find the amount of heat required to convert 4 kg of water at
200 C to ice at -20 C (specific heat of ice = 2.09 kJ kg-1K-1).
[5]

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(b) An ideal gas expands from a volume of 6×10 -3 m 3 to

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16 × 10−3 m3 against constant external pressure of

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m
2.026×105 N-m-2. Find the change in enthalpy if the change

em la
in internal energy, ∆𝑈𝑈 = 418 J.

las g
[5]

g a blocks A, B
a
(c) For the system shown in Figure 2, the masses of the
and C are 𝑚𝑚1 , 𝑚𝑚2 and 𝑚𝑚3 respectively, where (𝑚𝑚1 + 𝑚𝑚2 ) < 𝑚𝑚3 .
Find the tension 𝑇𝑇1 , 𝑇𝑇2 and the common acceleration of the
blocks. Assume the pulley to be light and the connecting
strings of negligible masses.
9

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Figure 2
[3+3+4 = 10]

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Electrical and Electronics Engineering

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8. (a) The number of turns in the low voltage (LV) side and the high

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voltage (HV) side of a 2 kVA, 50 Hz single phase transformer

c o m
m(i). Under rated supply conditions, if the ‘voltage per turn’ of e
are 100 and 500 respectively.

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the LV side is 2.0 V, determine the induced emf in the HV
side.
(ii) Estimate the rated currents in the LV side and the HV side
of the above-mentioned transformer.
(iii) Calculate the percentage voltage regulation of the
transformer under a certain loaded condition when the
terminal voltage in the HV side is measured to be 940 V.
[3+(2+2)+3 = 10]

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(b) A 2-pole, separately excited DC generator is required to

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produce 600 V for an electroplating factory. The total number

a s
of conductors in the armature of the generator is 600 and the

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g in rpm at which the generator should be
magnetic flux per pole is 0.02 Wb.

a
(i) Estimate the speed
driven by a prime-mover.
(ii) If the resistance of each armature conductor is 0.012 Ω,
determine the equivalent resistance of the armature circuit.
Neglect the brush contact resistances.
(iii) Calculate the net armature copper loss of the generator if
the load current is 10 A. Neglect the brush contact drop.

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[4+4+2 = 10]

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9. (a) An amplifier with negative feedback has an overall gain of

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100. Open-loop gain increase of 10% is expected owing

gla and feedback fraction 𝛽𝛽 for which closed-loop a
production limitations. Determine the value of open-loop gain

a
gain will only
increase by 1%.
[9+3 = 12]

11

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(b) Determine the voltage across the 10 Ω resistor in Figure 3
using the source transformation method.
[5]

Figure 3

(c) Find the octal equivalent of 879.
[3]

Engineering Drawing

10. (a) The front view and right side view of an object are shown in
Figure 4. Draw/sketch the top view and isometric view of the
object.
[8]

Figure 4

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(b) Draw/ sketch the three views of the object shown in Figure 5.
Put dimensions on the views.

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Figure 5

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[12]

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