ISI Admission Test 2022 Question Paper M.Tech(QROR) PQB

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ISI Admission Test
2022
Question Paper ·
M.Tech(QROR) PQB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2022 Question Paper M.Tech(QROR) PQB

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PART I : Section M (Both Streams)1

(Answer any TWO questions)

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1. (a) A positive integer is called palindrome if it reads the same forward

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or backward. Find the number of 8-digit palindromes divisible by

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5.
(b) m a
s e non-empty
Consider the set
= 1,2, ⋯ , 100 of numbers. Find the number of
g l
l a subsets of
such that the product of all the numbers
a
ag in each such subset is even.
(c) If n is a positive integer, then prove that the integral part of

3 + √7 is an odd number.
[5 + 5 + 10 = 20]

8 −6 2
(a) Let  = −6 7 −4. Find a matrix  such that 

 =
,
m
2.
2 −4 3
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where
is a diagonal matrix.
(b) Show that the series cotem2 + cot 8 + cot 18 + cot 32 +
s








⋯ ∞ converges to la
#
.
a g $

(c) Let  and % be the two nonempty sets for which &'( = 7 and
&'%( = 10. Find the number of ways by which one-one function
can be formed from  to %.
[8 + 8 + 4 = 20]

3. (a) Find the altitude of a right circular cone, with semi-vertical angle ),

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of least volume that can be circumscribed around a sphere of
radius *. Consequently, find the value of sin ).
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c. o (b) Find the solution to the differential equation .
/01

41
3 = ./01023
se
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/01
2 4/

las given that 5 = 1, when 6 = 1.
a g
ag (c) Evaluate lim
/→:
;<'
02/(
2 ;<'
0/(
/=
without using L’Hospital’s Rule.
[(8+2) + 6 + 4 = 20]

1
Mathematics is now compulsory for Statistics Stream also.

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4. (a) A triangle ∆ABC, right-angled at C, moves such that the vertices A
and B always lie in the first quadrant of the X-Y plane. Prove that
the locus of C is a straight line.
/ 1
(b) A variable line drawn through the point of intersection + =1
> ?
/ 1
and + = 1 meets the axes at A and B. Show that the locus of the
? >
midpoint of AB is the curve 2xy(a+b) = ab(x+y).
(c) Suppose that @ is a sequence of numbers such that @ > 0 and
>
>
lim C >DEF C = G . Does the series ∑
0>D converge? Justify.
→B D D
[8 + 7 + 5 = 20]

PART II : Section S (Statistics Stream)

(Answer any FOUR questions)

1. Let I
, I2 , … , I be a random sample from a uniform distribution over
'0, K(.
(a) Find an unbiased estimator of K based on largest order statistic
I'( = max'I
, … , I (.
(b) Find an unbiased estimator of K based on sample mean IN.
(c) Compare the relative efficiency of the unbiased estimators obtained
in (a) and (b).
[7 + 3 + 10 = 20]

2. (a) Let I
, … , I be a random sample from the exponential distribution
with mean O and P
, … , P be a random sample from another
exponential distribution with mean Q. Assume that IR ’s and PR ’s are
independent.
(i) Derive the sampling distribution of IN/PN, where IN and PN are the
sample means based on IR ’s and PR ’s.
U
(ii) Find a 100'1 − )(% confidence interval for V.

(b) Let I
, I2 , … , I be a random sample from W'Q, 1(.

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(i) Show that the estimator IN 2 − is unbiased and consistent



estimator of Q 2 , where IN is the sample mean.
(ii) Find the Cramer-Rao lower bound of an unbiased estimator of
Q2.
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[(8+2) + (5+5) = 20]

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3. A study was conducted to assess whether job satisfaction is independent
of gender. Accordingly, 150 females and 400 males were interviewed
and the observations are given below
Fairly Completely
Gender Dissatisfied Neutral
Satisfied Satisfied
Female 20 40 40 50
Male 60 40 100 200

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(a) Estimate the proportion of males whose job satisfaction is higher

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than that of females and the proportion of females whose job

emof males whose job satisfaction is
satisfaction is higher than males.

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a and the proportion of females whose job
(b) Assume that the proportions

g
a than males may be construed as the
higher than that of females
satisfaction is higher
probabilities of these events.
(i) Write down the probability statements for the two events given
above.
(ii) Explain how you will use these probabilities to judge whether
gender and job satisfaction are independent.

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[10 + (5+5) = 20]

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a
4. (a) There are 25 similar balls in a box, numbered 01 to 25. Five balls
would be drawn randomly from the box. You have to guess the
number(s) on these 5 balls, before the draw starts.
(i) What is the probability that you get all of them right?
(ii) What is the probability that you get at least 3 of them right?

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(b) Suppose I ∼ W'Q, Y 2 (. Show that Z[Φ'@ + ]I(] = Φ . >0?V 3,
√
0?= = _
where Φ is the cumulative distribution function of standard normal
variable and, @ and ]'≠ 0( are constants.
[(5+5) + 10 = 20]


(a) Suppose I ∼ abcca&'1( and P ∼ %b&adb@e .2, 3 are two
2
5.
independent random variables. Find 'I ≥ P(.
(b) Suppose I = 'I
, I2 , Ig (h ∼ Wg 'Q, Σ(, where Q = '5, 4, 3(′ and
2 1 0
Σ = 1 2 1
0 1 2
(i) Find the conditional distribution of 'I2 − I
, Ig − I2 ( given
I
= 6
.
(ii) Are I2 − I
and Ig − I2 independently distributed? Justify
your answer.
[5 + (12+3) = 20]

6. (a) A certain drug is tested chemically for its toxic nature. Let the event
“the drug is toxic” be denoted by Z and the event “the test reveals
that the drug is toxic” be denoted by l.
Let 'Z( = m, 'l|Z( = 'l o |Z o ( = 1 − m. Show that the
probability that the drug is not toxic given that the test reveals that
it is toxic is free from m.
(b) An unbiased coin is tossed 'd + &( times (d > &(. Find the
probability of occurrence of at least d consecutive heads.
[10 + 10 = 20]

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PART II : Section E (Engineering Stream)

(Answer any FOUR questions)

1. (a) Two equal parallel forces, acting in opposite direction at fixed
points A and B, form a couple of moment p. If their lines of actions
m
m
are turned through a right angle, they form a couple of moment q.
co m .co
m . s e
a
Show that when they both act at right angles to AB, they form a

s e couple of moment √p 2 + q 2 .
g l
g la(b) There is a solid right circular cylinder whose radius is @ and altitude a
a is ℎ. From this cylinder, a right cone of the same radius and altitude
is cut. If s is the mass of the remaining solid, show that the moment
of inertia of the remaining solid about the axis of the cylinder is
g
t
s@2 .

(c) A heavy chain of length 100 m and weighing 10 kgm

hangs
vertically with one end attached to a peg and carries a weight of one

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tonne at the other end. Find the work done in winding 50 m of the

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chain round the peg.

s
(a) It is well known thatawhen NaCl is added to water then the
[4 + 8 + 8 = 20]

gl decreases but if we add CaCl instead
temperature of theasolution
2.
2
of NaCl, then the temperature of the solution increases. Explain the
reasons behind this contrasting behavior, indicating the changes
involved in ∆q, ∆
and ∆p in each of the two cases (notations have
their usual meaning).
(b) The polymerization of ethylene is a spontaneous process. Explain

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whether the sign of the enthalpy change will be positive or negative

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during polymerization.
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e at the
How long would it take to vaporize a sample of 100 gswater
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(c) Standard vaporization enthalpy of water is 40.657 kJmol at 100

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boiling point using a 100 W electric heater?

ag (d) A 1.25 g sample of octane is burned in excess oxygen in a bomb
calorimeter. The temperature of the calorimeter rises from 21oC to
31oC. If the heat capacity of calorimeter is 8.93 kJK-1, calculate the
enthalpy of combustion of octane. Given that combustion equation
is 2C8H18(l) + 25O2(g) = 16CO2(g) + 18H2O(l).

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(e) Explain the difficulty of eliminating environmental pollution using
the argument of entropy changes.
[3 + 2 + 5 + 8 + 2 = 20]

3. (a) From the first law of thermodynamics, show that the change in
entropy '∆
( of a perfect gas having mass d is
2 ~2
∆
= d xyz ln { | + 'y} − yz (ln { |

~

where
~
= initial volume of the gas

= initial temperature of the gas
~2 , 2 = corresponding values for the final condition
y} = specific heat at constant pressure
yz = specific heat at constant volume.
(b) An engineer claims to have designed a device which absorbs 2500
kJ of heat and produces 2000 kJ of work. If the heat sink for the
device is maintained at 270C, what would be the minimum source
temperature?
(c) In a workshop, an engine drives a shaft by a belt. The diameters of
the engine pulley and the shaft pulley are 500 mm and 250 mm
respectively. Another pulley of 700 mm diameter on the same shaft
drives a pulley of 280 mm diameter on the motor shaft. If the engine
runs at 180 rpm, find the speed of the motor shaft.
[8 + 7 + 5 = 20]

4. (a) Show that the maximum power obtainable from a source having an
open circuit voltage Z and an internal impedance '* + €I(Ω is
‚=
given by .$ƒ3 W.

(b) Calculate the average and rms values of current represented in
Figure 1 below.

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Figure 1
[12 + 8 = 20]
5. (a) In the circuit, represented in Figure 2 below, express : in terms of
†
, †2 and other depicted parameters.

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a

Figure 2

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(b) Subtract 10101 from 10011 using 2’s complement method.

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(c) (i) Apply Thevenin’s theorem to derive an expression for the
current flowing through IΩ in Figure 3 below.
em (ii) Determine the value of I in Figure 3 for whichla
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las a g the maximum

ag power is dissipated in the resistance.
(iii) Determine the value of the dissipated power.

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Figure 3
[5 + 3 + (6+4+2) = 20]
6. (a) A cylinder with a base of 40 mm diameter and 50 mm long rests on
a point of its base on HP such that the axis makes an angle of 30°
with HP. Draw the front view and top view of the cylinder. Follow
the rules of the orthographic projection.
(b) Draw the isometric view from the front view and top view of an
object provided below in Figure 4.

Figure 4
[12 + 8 = 20]

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