ISI Admission Test 2021 Question Paper M.Tech(QROR) PQB

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ISI Admission Test
2021
Question Paper ·
M.Tech(QROR) PQB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2021 Question Paper M.Tech(QROR) PQB

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PART I (STATISTICS / MATHEMATICS STREAM)

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

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GROUP S1: Statistics

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1. (a) Let X1, X2, …, Xn be a random sample from N(𝜃 + 𝜃 , 1), and Y1, Y2,
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…, Yn be a random sample from N(𝜃 − 𝜃 , 1) independent of the first

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one. Find the maximum likelihood estimators of 𝜃 and 𝜃 . Derive a
100(1 − 𝛼)% confidence interval for 𝜃 .
(b) Let X1, X2, …, Xn be a random sample from the distribution with
probability density function

𝑓(𝑥) = 𝜆 𝑥𝑒 , 𝑥>0
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
for 𝜆 > 0. Find a uniformly most powerful level 𝛼 test for testing H0:

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𝜆 = 𝜆 versus H1: 𝜆 > 𝜆 . Provide the power function of the test.
[(5+5) + 10 = 20]

s em from a random variable X having
probability density function gl
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2. A random sample of size n is generated

a 𝜃𝑒 , 𝑥 > 0
𝑓(𝑥) =
0, 𝑒𝑙𝑠𝑒
for unknown parameter 𝜃 > 0. However, for some unknown reason, only
observations having value less than A (a known positive constant) are
recorded. Let the recorded observations be denoted by x1, x2, …, xm for 0 ≤
𝑚 < 𝑛.

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(a) Write down the likelihood function. Justify.

c. o (b) Find a sufficient statistic for θ.
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estimator

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of θ will exist.
(d) Obtain mle of θ under the condition stated above in (c).
[6 + 6 + 3 + 5 = 20]

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3. (a) Let 𝑋 , 𝑋 , … , 𝑋 be a random sample from N(0, 𝜎 ). Find an unbiased
estimator of σ based on the statistic 𝑇 = ∑ |𝑋 |. Find its Cramer-
Rao lower bound.
(b) Suppose that 𝑋 , 𝑋 , … , 𝑋 constitutes a random sample from a
distribution having mean µ and variance 𝜎 . Let 𝑋 = ∑ 𝑋 and
𝑆 = ∑ (𝑋 − 𝑋) be sample mean and variance respectively.
Show that 𝐸(𝑆) ≤ 𝜎.
[(7+7) + 6 = 20]

4. (a) Consider a random sample 𝑋 , 𝑋 , … , 𝑋 from the discrete distribution:
( )
𝑃 ( 𝑋 = 1) = , 𝑃 ( 𝑋 = 2) =

for unknown 𝜃 ∈ (0,1).
(i) Find a moment estimator of 𝜃.
(ii) Is this estimator consistent? Justify your answer.
(b) Suppose that in a population of size 𝑁 = 800, P (unknown) is the
percentage of people having a particular blood group. In order to
estimate P, a random sample of size n is to be drawn with replacement.
(i) Determine the value of n so that P can be estimated within an
absolute error of 4%, with probability 0.95.
(ii) If you have prior knowledge that P is less than 30%, would you
consider changing the value of 𝑛 obtained in (i)? Justify your
answer.
[(5+5) + (6+4) = 20]

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5. (a) Consider the linear model: 𝑦 = 𝜇+𝛼 + 𝜖 for 𝑖 = 1, 2, 3,
𝑗 = 1, 2, 3 with 𝛼 + 𝛼 + 𝛼 = 0.
(i) Derive least-square estimators of the unknown parameters µ, 𝛼 , 𝛼
and 𝛼 . Are these estimators unique? Justify your answer.
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(ii) If the condition 𝛼 + 𝛼 + 𝛼 = 0 is dropped, what can you say
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regarding uniqueness of the above estimators?
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(b) Suppose

ag Explain with justification, whether variance of the remaining numbers
is larger than, less than or equal to the variance of the original set of
numbers.
[(5+5) + 10 = 20]

GROUP S2: Probability

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6. (a) Consider a square having each side as 5 cm. Pick a random point inside

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it. Denote the four perpendicular distances of the sides, from this point,
by X1, X2, X3 and X4.
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(i) Give a graphical representation of the above, and write down the
relationship among the four distances, if there is any.
(ii) Let Y be the distance of the furthest side from the point picked.
Write Y as a function of X1, X2, X3 and X4.
(iii) Hence or otherwise, determine the probability that Y is at least p
cm.

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(b) There are three chests, each having two drawers. Each drawer of Chest-
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1 contains a gold coin, and each of Chest-2 has a silver coin. In the

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Chest-3, there is a gold coin in one drawer and a silver coin in the other.

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A chest is selected at random. Then one of its drawers is selected

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randomly again, and is opened. It is found to contain a gold coin. What
a is the probability that the other drawer of the selected chest also
contains a gold coin?
[(3+2+5) + 10 = 20]

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7. (a) A parent particle gets split into 0, 1 or 2 child particles with probability
¼, ½ and ¼ respectively, and then disappears immediately. Beginning
with 1 particle as the progenitor, let us denote Xi = number of particles
in the ith generation, find
(i) P[X2 > 0],
(ii) P[X1=2 | X2=1].

(b) Two athletic teams A and B play a series of independent games
between them until one of them wins 4 games. The probability of every
team winning any single game is ½. Find the probability that the series
will end in at most 6 games.
[10 + 10 = 20]

8. (a) Suppose that 𝑋 , 𝑋 , … , 𝑋 are i.i.d. uniform distribution over {1, 2, …,
N}, where N is a positive integer. Let Mn = max {𝑋 , 𝑋 , … , 𝑋 }. Find
the probability distribution of Mn.
(b) Suppose the joint distribution of 𝑋 , 𝑋 , 𝑋 is N3(µ, Ʃ), where
3 4 0 2
µ= 1 and Ʃ = 0 1 −1 .
2 2 −1 3
(i) Find the conditional distribution of (𝑋 , 𝑋 ) given 𝑋 .
(ii) Find the partial correlation coefficient between 𝑋 and 𝑋 for fixed
𝑋 .
[10 + (7+3) = 20]

9. Suppose that (X, Y) is uniformly distributed over {(𝑥, 𝑦) ∈ 𝑅 : 𝑥 + 𝑦 <
1}. The probability density function of (X, Y) is given by
1
𝑓(𝑥, 𝑦) = 𝜋 if 𝑥 + 𝑦 < 1,
0 otherwise.
(a) Find the conditional distribution of X given 𝑌 = 𝑦, where |𝑦| < 1.
(b) Find E[X|Y].
(c) Find the correlation coefficient between X and Y.
(d) Are X and Y independent? Give reasons.
[8 + 3 + 4 + 5 = 20]

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10. (a) The failures of a certain machine occur according to a Poisson process
with rate 2 per week. Find the probability that the machine will have at
least one failure in each of the first three weeks.
(b) Consider the set of all 11-letter words that can be formed by arranging

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the letters in DISTURBANCE. Any two such words are called
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equivalent if they maintain the same relative order of the vowels and no

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two vowels appear adjacent to each other. For example,

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TURDISBANCE and TRUSIDBANCE are equivalent. Find the

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probability that a randomly picked word from the set will be equivalent
to DISTURBANCE.
[10 + 10 = 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) Find
∑ 𝑥 − 199
lim .
→ 𝑥−1
(b) Let
𝑎𝑥 − 𝑏, 𝑥≤1
𝑓(𝑥) = 3𝑥, 1<𝑥<2
𝑏𝑥 − 𝑎, 𝑥≥2
be continuous at 𝑥 = 1, 2. Find the values of a and b.
(c) Let 𝑝(𝑥) be a polynomial of degree 100, with all real coefficients,
satisfying 𝑝(𝑥) > 0 for every real 𝑥. Prove that 𝑝(𝑥) is a product of
quadratic polynomials having real coefficients.
[5 + 5 + 10 = 20]

2. (a) Give the solution set (in compact form) of the following inequality
|𝑥 + 3| + 𝑥
> 1.
𝑥+2
(b) Show that
𝑛 𝑛 𝑛
−2 + ⋯ + (−1) 𝑛 =0,
1 2 𝑛
where is the number of different groups of r objects that can be
chosen from a set of n objects.

(c) Show that the series + + + ⋯
converges for all 𝑥 ∈ ℝ.
[8 + 6 + 6 = 20]

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3. (a) Suppose α, β, γ, δ are the roots of the equation 𝑥 + 𝑥 + 1 = 0. Find
the value of
𝛼𝛽(𝛼 + 𝛽) + 𝛼𝛾(𝛼 + 𝛾) + 𝛼𝛿(𝛼 + 𝛿) + 𝛽𝛾(𝛽 + 𝛾) +
𝛽𝛿(𝛽 + 𝛿) + 𝛾𝛿(𝛾 + 𝛿) .

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(b) Consider the parabola 𝑦 = 4𝑎𝑥. Inscribe an equilateral triangle to it
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such that the parabola’s vertex coincides with one of the vertices of the

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triangle. Then a circle is inscribed to the above triangle. Work out areas

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of the
l (i) triangle, and a
ag (ii) circle.
[10 + (5+5) = 20]

4. (a) If a, b, c are all positive, and are the 𝑝th, 𝑞th and 𝑟th terms respectively
of a GP, then show that

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log 𝑎 𝑝 1

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log 𝑏 𝑞 1 = 0.

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log 𝑐 𝑟 1
(b) Consider the function
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𝑓(𝑥)

(i) Draw the graph of the function 𝑓(𝑥).
(ii) Now consider the equation
1
|𝑥| −= 𝑎𝑥 + 𝑏 .
|𝑥|

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Show that there exist real numbers 𝑎 and 𝑏 such that this equation

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has three distinct real roots, each with multiplicity one, and one of

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them being unity.

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[8 + (6+6) = 20]

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GROUP E2: Engineering & Technology

Engineering Mechanics and Thermodynamics

5. (a) A uniform wheel of 600 mm diameter, weighing 5 kN rests against
a rigid rectangular block of 150 mm height as shown in Figure 1.
Considering the entire surface to be smooth, find
(i) The least pull required through the centre of the wheel just to
turn the wheel over the corner A of the block.
(ii) The reaction of the block.

Figure 1
(b) An electric light fixture weighing 15 N hangs from a point by two
strings. One string makes an angle of 450 to the vertical plane and
the other string makes an angle of 600 to the horizontal plane.
Determine the forces acting on the two strings.
(c) An aeroplane requires a velocity of 80 km/hour and a run of 100 m
prior to take off. The mass of the aeroplane is 10,000 kg and the
frictional coefficient between the wheel and the ground surface is
0.2. Assuming that the aeroplane is accelerating uniformly, find the
minimum force required to take off.
[(7+3) + 5 + 5 = 20]

6. (a) A piston cylinder device is initially at 150 kPa pressure and 270 C
temperature. The piston rests on a stop and the volume of air
enclosed is 0.4 m3. To move the piston, a pressure of 350 kPa is
required. The system is heated until the volume is doubled.
Determine
(i) The final temperature of the air.
(ii) Work done.

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(iii) Total heat transfer.
(b) A fuel has the composition by mass like: C 90%, H 3%, O 2.5%, N
1%, S 0.5%, and ash 3%. Calculate the volumetric air supply rate
required to burn the fuel at the rate of 500 kg/hour considering 20%
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excess air.

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[(2+3+5) + 10 = 20]

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s e Determine if a tray of ice cubes could remain frozen when placed gla
la in a food freezer having a COP of 9, operating in a room at a a
7. (a)

ag temperature of 320 C.
(b) It is proposed that solar energy be used to warm a large collector
plate. This energy would, in turn, be transferred as heat to a fluid
within a heat engine, and the engine would reject energy as heat
to the atmosphere. Experiments indicate that 2000 kJm-2/hour of
energy can be collected when the plate is operating at 900 C.

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Estimate the minimum collector area that would be required for a

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plant producing 1 kW of useful shaft power. The atmospheric

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temperature may be assumed to be 250 C.
(c)
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Determine the moment of inertia of the object in Figure 2 about
both centroidal axes.
a

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a
Figure 2
[5 + 5 + 10 = 20]

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Electrical and Electronics Engineering

8. (a) For the given circuit in Figure 3,
(i) Determine the time constant.
(ii) Derive the expression of the current "𝑖 " as a function of time
after the switch closes at 𝑡 = 0 second.

Figure 3
(b) A 50 Hz single phase transformer has 80 turns on the Low Voltage
(LV) winding side and 400 turns on the High Voltage (HV)
winding side. The net cross sectional area of the core is 0.02 m2. If
the LV side is connected to 222 V, 50 Hz supply, determine
(i) The emf induced on the HV side.
(ii) The maximum flux density in the core when the HV side is
connected to an R-L load, which draws 555 W power at 0.5
lagging power factor.
(iii) The magnitude of the RMS current flowing through the HV
winding and the LV winding, neglecting the no-load current
and voltage drop in the winding.
[(6+8) + (1+2+3) = 20]

9. (a) Simplify the Boolean function 𝑌 = 𝐴̅𝐵𝐶 + 𝐴𝐵 𝐶 + 𝐴𝐵𝐶̅ + 𝐴𝐵𝐶
using a Karnaugh map and draw the simplified logic circuit.
(b) In the npn transistor, shown in Figure 4, 109 holes/µs move from
the base to the emitter region while 1010 electrons/µs move from the
emitter to the base region. An ammeter reads the base current as
𝑖 = 16𝜇A. Considering the charge of 1 electron = -1.602 × 10-19 C,
determine
(i) The emitter current 𝑖 .
(ii) The collector current 𝑖 .

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ag (c) Subtract 6702 from 8415 using 9’s complement method.
Figure 4

(d) For the circuit given in Figure 5, 𝑉 = 12V, 𝑉 = 6V, 𝑅 =
4 Ohms, and 𝑅 = 6 Ohms. Assuming that the internal resistances
of the batteries are zero, find the Thevenin equivalent for the
network to the left of the terminals a, b.

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a
Figure 5
[4 + (3.5+1.5) + 3 + 8 = 20]

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Engineering Drawing

10. (a) Draw the front view and top view of the object shown in Figure 6
using the third angle of projection.

Figure 6
(b) Draw the isometric view from the projection drawings (in the first
angle) as shown in Figure 7.

Figure 7
[12 + 8 = 20]

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