ISI Admission Test 2019 Sample Paper M.Tech(QROR) PQB

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ISI Admission Test
2019
Sample Paper ·
M.Tech(QROR)
PQB
EXAM YEAR TYPE

ISI Admission Test 2019 Sample Paper
SUBJECT

M.Tech(QROR) PQB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Sample PQB 2019
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. I (STATISTICS / MATHEMATICS STREAM) em
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PART

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

GROUP S1: Statistics

1. (a) Assume that the variables , ,⋯, all have same
variance , and correlation coefficient between and is
given by for ≠ and , = 1,2, ⋯ ,2 . Prove that the
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correlation coefficient between and is

e m ,

a s =∑
1 + ( − 1)

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where =∑ and .

(b) If the random variables , and are such that
= + ,
where , ≠ 0; then show that
(i) the partial correlation coefficients are numerically

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equal to unity;

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(ii) has the same sign as , ∙ has the same sign as

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∙
and ∙ has the opposite sign of / .

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[10 + (5+5)

2. (a) Let ̅ be the arithmetic mean of three variatesa , and .
Show that
1 1
( − ) + ( + −2 ) = ( − ̅) .
2 3∙2

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(b) Suppose, in a city 40% of the households have no motorized
vehicle, 30% have one, 15% have two, 10% have three and 5%
have four or more vehicles.

(i) Draw a graph to represent the distribution function of
the number of vehicles per household.

(ii) Find the population median.

(iii) You have been asked to report a measure of location
for the number of vehicles owned by households in the
city. Do you think population median would represent
the reality better than the population mean? Explain
briefly.

(iv) Suppose you have surveyed 500 households in the city.
Let be the number of vehicles owned by the -th
household, = 1, 2, ⋯ , 500. Which diagram may be
used to present this data? Explain briefly.

(v) A random sample of 10 households revealed the
following values of the number of motorized vehicles
owned by them: 4, 2, 1, 0, 0, 0, 0, 0, 0, 1. Draw a graph
of the empirical distribution function.
[8 + (3+1+3+3+2) = 20]

3. (a) Suppose there are 200 students in a particular class. To study
the age of the students, a random sample of size 20 with
replacement is drawn and their ages are noted as
, ,⋯, .
It is given that ∑ = 280 and ∑ = 4000. On the
basis of these information, get a 90% confidence interval for
the average age of the students in the class. [Given that
. , = 1.729, . , 20 = 1.725, and . = 1.645. ]

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(b) Consider a process producing balls. The chance of each ball
being defective is (with ≤ 0.02). These balls are packed
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in boxes of 100 each. The value of is known to you. Let

m . be the number of boxes that do not contain any defective
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balls. Get an unbiased estimate of on the basis of .

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Also, give the standard error of the estimator.

ag [10 + 10 = 20]

4. A car tyre manufacturing company intends to test four
different types of rubber improver for their effects on the wear
of car tyres. Four test cars (four-wheeler) are available for the
experiment. Each improver can be tested on one tyre only. The
position of a tyre in a car may influence the wear. No
interaction is expected between different sources of variation.

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The cars will be driven under normal conditions for six

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months, after which the wear of each tyre on each car will be
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measured. These values will then be suitably compared for

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selecting the best rubber improver amongst the four.

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(i) Identify, with proper justification, a suitable design for
the experiment and name it.

(ii) Give the layout of the design.

(iii) Identify the treatments.

(iv) How will you randomize your design?

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(v) Write down the skeleton analysis of variance table,

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showing different sources of variation, degrees of

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freedom and the computational formulae for respective

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sum of squares.

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[Note: If the identification of the design is not logical then no
credit will be given for rest of the answers.]
[4 + 2 + 1 + 4 + 9 = 20]

5. (a) Consider a box containing smart phones, where is
unknown to you. Each phone is labelled 1 through . In order
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to estimate , you are allowed to draw a random sample
(without replacement) of three phones, and the observed labels
are 18, 5 and 27. Write down the likelihood function and
provide maximum likelihood estimate of .

(b) Ajay and Bhaskar are playing a game to see whether they are
equal in ability. The game is such that it cannot result in a
draw. It was decided that the game will be played 7 times and
if either person wins 6 times or more, they would accept that
their abilities differ.

(i) Formulate the hypotheses and .

(ii) Are and simple or composite? Explain briefly.

(iii) Write down any one point in the sample space.

(iv) List the points in the sample space that represent the
critical region.

(v) Find the level of significance.

(vi) What is the equation of the power function?

(vii) Is the test unbiased? Explain briefly.

(viii) What assumptions did you make?
[5 + (1+2+1+1+2+3+2+3) = 20]

GROUP S2: Probability

6. (a) A five-digit number is formed using the digits 0, 1, 2, 3, 4
without repetition. Find the probability that it will be divisible
by 4.
(b) A multiple-choice-test has 30 questions. Each question has
four possible answers, exactly one of which is correct. A
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student knows the correct answer to 16 questions, and guesses
at random for the remaining ones.
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c. o (i) Obtain s em
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the probability distribution of the number of

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total questions this student gets right.

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a (c) In ten independent throws of a biased die, the probability that
correct answers this student gets.

an even number will appear five times is twice the probability
that an even number will appear four times. Find the
probability that an even number will not appear in ten
independent throws.
[7 + (5+3) + 5 = 20]

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7. (a) Choose a point at random in the unit disk around origin. What
is its expected distance from the origin?
(b) At a measuring station form
and carbon particles a
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s recorded at noon every day. Let be
air pollutants, the amount of ozone

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the concentrationg of carbon particles ( g⁄cc), and
are
a
concentration of ozone (ppm). Suppose that ( , ) has a
the

bivariate normal distribution, where has mean 10.7 and
variance 32.0, and has mean 0.1 and variance 0.02. The
correlation coefficient between and is 0.80. The ozone is
considered unhealthy if it exceeds 0.35.
Suppose that the instrument used to measure ozone fails, so

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that we can measure only the carbon level. If this turns out to

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be 20.7 g⁄cc, what is the

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(i) predicted ozone level, and

(ii) probability that the ozone level is unhealthy.
a g
[You may leave the answer in terms of standard normal
distribution function Φ(∙).]
[10 + (4+6) = 20]

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8. (a) There are k + 1 machines in a shop, all engaged in the mass
production of an item. The ith machine produces defectives
with a probability of ⁄ , i = 0, 1, 2, …, k. A machine is
randomly selected and then the items produced are repeatedly
sampled. If first n products are all defectives then show that
the conditional probability that (n + 1)st sampled product will
also be defective is approximately equal to ( + 1)⁄( + 2)
when k is large.
[Hint: Use the integral approximation
1 1
≈ =
+1
for large k.]

(b) Suppose that the number of telephone calls received by an
operator between 11 a.m. to 12 noon follows Poisson
distribution with rate λ=2 per hour. Find the probability that
he will receive a total of one call in next three days during that
time-period.
[15 + 5 = 20]

9. (a) Let , , ⋯ , , ≥ 3, be independent random variables
with continuous uniform distribution over [0, 1].

(i) Find ( < < ).

(ii) Define = ⋯ . Find V(Y).

(b) Suppose that ( , , ) has trinomial distribution with
probability vector ( , , ), + + = 1, > 0, =
1, 2, 3; and given that + + = 20. Find expectation
and variance of given = 8.
[(5+5) + 10 = 20]

10. (a) Trucks arrive at a particular toll booth on the highway
according to Poisson process with rate one per minute.

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Assume that midnight 12 o’clock as the start-time of the
process. Compute the
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(i) probability that the 10th truck arrives two or more

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minutes after the arrival of 9th one,

la se (ii) probability that the 10 truck arrives after 00: 20 a.m., ag
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th

a (iii) probability that there are two arrivals during
[01: 01, 01: 04] a.m. and three arrivals in the interval
[01: 03, 01: 05] a.m.

(b) Consider a system having two possible states: 0 or 1. Let
be the state of the system at time for = 0, 1, 2, ⋯ and
P[ = 1| = 0] = , P[ = 0| = 1] = .

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(i) Write down the transition probability matrix.

m. 0 < < 1. Find the value of
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(ii) Let P[ = 1] = for
P[ = 0].
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(iii) Assuming P[ = 1] = for 0 < < 1, find
lim P[ = 1].
→
[(3+3+6) + (1+2+5) =20]

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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) How many numbers from 1 to 1000 are not divisible by 2, 3
and 5?
(b) Let
4
( )= .
4 +2
Find

.
2002

[8 + 12 =20]

2. (a) Find the greatest value of subject to + + = 18,
and , , > 0.

(b) For what values of , the following series converges:

1
.
(log )
[10 + 10 = 20]

3. (a) Show that the area lying in the region ≥ 4 and between the
circle + = 32, the -axis and the tangent drawn at the
point (4,4) on the circle is 4(4 − ) square units.

(b) Consider the triangle , where ∠ = 30 and ∠ = 45 .
Prove that ( − )= .
[10 + 10 = 20]

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4. (a) Find the equation of the curve which is such that the portion of

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the -axis cut-off between origin and the tangent at any point
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is proportional to the ordinate of that point.

g la (b) For real , , , , and , show that
a
( − ) ( − ) ( − )
( − ) ( − ) ( − )
( − ) ( − ) ( − )

(1 + ) (1 + ) (1 + )

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= (1 + ) (1 + ) (1 + ) .

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(1 + ) (1 + ) (1 + )

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[8 + 12 = 20]

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GROUP E2: Engineering & Technology

Engineering Mechanics and Thermodynamics

5. (a) With reference to the coordinate axes x and y, locate the
centroid of the area of the cross-section shown in Figure 1.

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Figure 1
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(b) A horizontal beam AB is hinged to a vertical wall at A and
supported at its mid-point C by a tie rod CD as shown in

9

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Figure 2. Find the tension in the tie rod and the reaction at A
due to a vertical load applied at B.

Figure 2
[10+10=20]

6. (a) A certain quantity of energy Q is transferred from a body at
constant temperature T1 to another body at constant
temperature T2 (T1 > T2). Show that the loss in the available
energy is T0 SUni, where T0 is the ambient temperature and
SUni is the total change in the entropy of the universe.
(b) An engineer claims to have developed a refrigerator that has a
COP of 14 and that maintains the cold temperature at 10C while
operating in a kitchen where the temperature is 250C. Is the
claim true or false?
[10+10=20]

7. (a) The specific heat at constant volume for a particular
substance is given by the relation
= + +
where a, b and c are constants. If it is desired to change the
temperature of the substance from T1 to T2 at constant volume,
then show that the relation to calculate the heat interaction is
= ( − )+ ( − )+ ( − ).
2 3
(b) A hammer weighing m1 kg is used to drive horizontally a nail
weighing m2 kg into a timber post. The striking velocity of the

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hammer is 10 m/s and the nail is driven 25 mm at each blow.
Find the resistance offered by the timber to the nail and the
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energy lost in driving it 75 mm inside the post.

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[10+(5+5)=20]

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Electrical and Electronics Engineering

8. (a) The open-circuit (OC) and short-circuit (SC) test data for a
single phase 1 kVA, 100 V / 200 V, 50 Hz transformer are
given below:

OC test from the
100 V 0.5 A 30 W
low voltage side

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SC test from the

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10 V 5A 40 W

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high voltage side

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Calculate the following:

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(i) magnetizing current;

a
(ii) core-loss current;
(iii) magnetizing reactance;
(iv) short-circuit impedance.

(b) A four-pole, wave-wound separately excited DC generator is
being driven by a turbine at 1500 rpm speed. Compute
(i) the induced voltage in the armature, if the flux per pole
is 12 mWb and total number of armature conductor is

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500,

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(ii) the line current and the terminal voltage when an

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electric heater of rating 250 V, 1250 W is connected

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across the armature. Consider the armature circuit

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resistance as 2 , and

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(iii) the line current and the terminal voltage when a 12.5
 resistor is connected across the heater.
[10+(4+4+2)=20]

9. (a) An npn transistor shown in Figure 3 below, is used in
common-emitter amplifier mode with β = 49, VCC = 10 V, and
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RL = 2 kΩ. If a 100 kΩ resistor RB is connected between the
collector and the base of the transistor, then calculate
(i) the quiescent collector current, and
(ii) the collector to emitter voltage drop between points A
and B. Assume base to emitter voltage drop is 0.7 V.

Figure 3

(b) For the circuit, given below in Figure 4, find the output voltage
v0 for vS = 0.5 V and vS = -1.0 V. Assume that the diode D2 is
ideal, R1 = 2 k, R2 = 8 k, R = 2 k, R3 = 8 k and vS2 = 0.5
V.

Figure 4

[(4+4)+(6+6)=20]

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Engineering Drawing
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10. (a) Sketch three views of the object shown in Figure 5.

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Figure 5

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(b) Two views of an object m
s e are shown in Figure 6. Sketch the

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isometric view of the object.

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c. o Figure 6
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a

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