ISI Admission Test 2017 Question Paper M.Tech(QROR) PQB

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ISI Admission Test
2017
Question Paper ·
M.Tech(QROR)
PQB
EXAM YEAR TYPE

ISI Admission Test 2017 Question Paper
SUBJECT

M.Tech(QROR) PQB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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PART I (STATISTICS / MATHEMATICS STREAM)

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ATTENTION : ANSWER A TOTAL OF SIX QUESTIONS TAKING

m . AT LEAST TWO FROM EACH GROUP - S1 AND S2.
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a 1. (a) Let 𝑋𝑋 , ⋯ , 𝑋𝑋 be a random sample from N(𝜇𝜇, 𝜎𝜎 ) with
GROUP S1: Statistics
2
1 𝑛𝑛
unknown 𝜇𝜇 and known 𝜎𝜎 2 . Find the UMVUE of 𝜇𝜇 3 . State
clearly any result that you may use.
(b) A random sample of size 𝑛𝑛 is drawn without replacement from
a population of unknown size 𝑁𝑁, having 𝑁𝑁1 (known) members
of a specified type. If 𝑥𝑥 is the number of individuals of this type

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𝑛𝑛𝑁𝑁1

.
in the sample, show that the MLE of 𝑁𝑁 is approximately .

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𝑥𝑥

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[10+10 = 20]

2. (a)
experiment, called? ag
What are the variables, influencing the response in an

(b) What is the name of the phenomenon where the difference in
response between the levels of one factor is not the same at all
levels of the other factors?
(c) What is the name of the design of experiments
(i) which systematically allows blocking in two directions?
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(ii) in which all possible combinations of the levels of the
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factors are investigated?

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(d) Which basic principle of experimental design makes the

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assumption that the errors are independently distributed
random variables valid?
(e) Consider the following fixed effects linear model under the
usual Gauss-Markoff setup for the data collected from an
experiment involving two treatments:

1

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𝜇𝜇 + 𝛽𝛽1 , 𝑗𝑗 = 1,3,4
𝐸𝐸(𝑦𝑦𝑖𝑖 ) = �
𝜇𝜇 + 𝛽𝛽2 , 𝑗𝑗 = 2,5
where 𝛽𝛽𝑖𝑖 is the effect of the ith treatment, 𝑖𝑖 = 1,2 and 𝜇𝜇 is the
overall mean.
(i) Name the design of experiments which might have been
used to collect the above data.
(ii) How will you compute the sum of squares due to
treatment? [No derivation is required, just write the
expression.] What is its degrees of freedom?
(iii) Derive the expected value of the above sum of squares.
[1 × 5 + 2 + (3 + 1) + 9 = 20]

3. It is believed that working in shipbuilding industry and smoking are
two important causes of occurrence of lung cancer. In order to verify
these beliefs several people are studied and the following are
observed:
• 11 people who did not smoke but worked in shipbuilding had
lung cancer
• 35 people who did not smoke but worked in shipbuilding did not
have lung cancer
• 50 people who did not smoke and did not work in shipbuilding
had lung cancer
• 203 people who did not smoke and did not work in shipbuilding
did not have lung cancer
• 84 people who smoked and worked in shipbuilding had lung
cancer
• 45 people who smoked and worked in shipbuilding did not have
lung cancer
• 313 people who smoked but did not work in shipbuilding had
lung cancer
• 270 people who smoked but did not work in shipbuilding did not
have lung cancer

(a) Present this data suitably in a tabular structure.

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(b) Do you think working in shipbuilding is likely to lead to lung
cancer irrespective of whether a person smokes or not?

o m m
c
Explain. (You may argue probabilistically without giving any

.
(c)m Do you see any interaction effect of habit of smoking / not
formal test procedure).
s e
s e smoking and working / not working in shipbuilding on the l a
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occurrence of lung cancer? Justify.
[10 + 5 + 5 = 20]

4. (a) Suppose we have 𝑛𝑛 observations 𝑥𝑥1 , 𝑥𝑥2 , ⋯ 𝑥𝑥𝑛𝑛 . Let
1
𝑥𝑥̅𝑖𝑖 = (∑𝑖𝑖𝑗𝑗=1 𝑥𝑥𝑗𝑗 )/𝑖𝑖, 𝑖𝑖 = 1,2, ⋯ 𝑛𝑛 and 𝑠𝑠 2 = 𝑛𝑛 ∑𝑛𝑛𝑖𝑖=1(𝑥𝑥𝑖𝑖 − 𝑥𝑥̅𝑛𝑛 )2.
𝑖𝑖
Show that 𝑛𝑛𝑠𝑠 2 = ∑𝑛𝑛𝑖𝑖=2 𝑖𝑖−1 (𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖 )2 .

m
(b) Consider the polynomial model:

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. identically distributed random
𝑌𝑌𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝑋𝑋𝑖𝑖 + 𝛽𝛽2 𝑋𝑋𝑖𝑖2 + 𝛽𝛽3 𝑋𝑋𝑖𝑖3 + 𝜖𝜖𝑖𝑖 ,
𝑖𝑖 = 1, ⋯ , 𝑛𝑛,

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e that 𝑛𝑛 = 12 and
variables with mean 0.sSuppose
where 𝜖𝜖 ′s are independent and
𝑖𝑖

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ag𝑋𝑋 = �−1,0, 𝑖𝑖𝑖𝑖 == 1,5, ⋯⋯ ,, 48
𝑖𝑖
1, 𝑖𝑖 = 9, ⋯ , 12.
(i) If the model is written as 𝑌𝑌 = 𝑋𝑋𝑋𝑋 + 𝜖𝜖, write down the 𝑋𝑋
matrix.
(ii) Are 𝛽𝛽0 − 𝛽𝛽1 and 𝛽𝛽0 + 𝛽𝛽1 + 𝛽𝛽2 + 𝛽𝛽3 estimable? Justify
your answers.

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[10 + (4 + (3+3)) = 20]

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Let 𝑝𝑝 be the probability that a coin will turn up head in a single
e
5. (a)

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1 3
toss. In order to test H0 : 𝑝𝑝 = 2 against H1 : 𝑝𝑝 = 4 , the coin

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is tossed 5 times and H0 is rejected if more than 3 heads are
obtained. Find the probability of type I error and power of the
test.
(b) The principle of ordinary least squares is used to estimate the
regression coefficients in simple linear regression studies of

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y on x, and x on y, based on a random paired sample
{(𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), 𝑖𝑖 = 1, ⋯ , 𝑛𝑛}. Let 𝑒𝑒𝑦𝑦∙𝑥𝑥 represent the residual of y
when y is regressed on x. Similarly, 𝑒𝑒𝑥𝑥∙𝑦𝑦 represents the residual
of x when x is regressed on y. If r (≠ 0, ±1) is the sample
correlation coefficient then
(i) find the correlation coefficient between x and 𝑒𝑒𝑦𝑦∙𝑥𝑥 , and
interpret 𝑒𝑒𝑦𝑦∙𝑥𝑥 based on the value so obtained, and
(ii) compute the correlation coefficient between 𝑒𝑒𝑥𝑥∙𝑦𝑦 and 𝑒𝑒𝑦𝑦∙𝑥𝑥 .
[(5+5) + ((3+1) + 6) = 20]

GROUP S2: Probability

6. (a) A and B throw alternatively a pair of dice in that order. The
score in a throw is the sum of the outcomes of the two dice. A
wins if he scores exactly 6 points before B gets exactly 7
points, and vice versa. If A starts the game what is the
probability that he wins?
(b) Let 𝑌𝑌1 , 𝑌𝑌2 and 𝑌𝑌3 be the order statistics of 𝑋𝑋1 , 𝑋𝑋2 and 𝑋𝑋3, where
𝑋𝑋𝑖𝑖 ’s are independent and uniformly distributed between 0 and
1. Find the distribution of 𝑍𝑍 = 𝑌𝑌3 − 𝑌𝑌1 .
[10 + 10 = 20]

7. (a) The market for laundry detergent in a certain place deals with
three brands, namely A, B and C. The customers show some
brand loyalty, but tend to switch brands at a predictable rate
from week to week. The proportion of buyers who switch
from one brand of this week to another brand in the next week
is given in the following table:
Next Week
Brand A B C
Current A 0.4 0.4 0.2
Week B 0.5 0.4 0.1
C 0.3 0.2 0.5

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Thus the transition probability that a person who buys B in the
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current week will buy A in the next week is 0.5, and similarly

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other transitions are interpreted. Assuming that the probability

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of transition from brand to brand is not affected by the past

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history of brand transition, find the steady state market share

a (b) The random variables 𝑋𝑋 and 𝑌𝑌 have joint probability density
of the brands.

function:
2 2
𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 𝑐𝑐𝑒𝑒 −8𝑥𝑥 −6𝑥𝑥𝑥𝑥−18𝑦𝑦 , − ∞ < 𝑥𝑥, 𝑦𝑦 < ∞.
Find the value of 𝑐𝑐 and the correlation coefficient between 𝑋𝑋
and 𝑌𝑌.

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[8 + 12 = 20]

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8. (a) A car-wash facility has two service stations, S1 and S2, with

s
service times distributed as exponential with rates 𝛿𝛿1 and 𝛿𝛿2

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respectively. An entering car has to visit both the stations in

a
that order. On completion of service at station S1, the car
proceeds to S2 provided that S2 is unoccupied. Otherwise, it
is made to wait at S1, blocking the entrance of any newly
arriving car. The car leaves the facility immediately after its
service at S2 is completed. When you arrive at the facility, you
find a car being served at S1, and S2 is busy servicing another
car. Compute your expected waiting time before you exit from

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the wash facility.

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(b) Let 𝑋𝑋 and 𝑌𝑌 be independent random variables each having

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the standard normal distribution. Consider the circle

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centered at the origin and passing through the point (𝑋𝑋, 𝑌𝑌) in

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the 𝑥𝑥𝑥𝑥-plane. What is the probability density function of the
area of the circle? What is the expected value of this area?
[10 + (8+2) = 20]

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9. (a) Let 𝑋𝑋 be the outcome when a die is rolled once. Suppose that
the probability of a particular outcome 𝑋𝑋 = 𝑥𝑥 is proportional
to 𝑥𝑥 (𝑥𝑥 = 1,2, ⋯ ,6). At first this die is rolled and its outcome
𝑋𝑋 is noted. Next a fair coin is tossed 𝑋𝑋 times and the number
of heads so obtained is denoted by 𝑌𝑌. Compute the followings:
(i) P[𝑌𝑌 = 4]
(ii) P[𝑋𝑋 = 5 |𝑌𝑌 = 4]
(iii) E[𝑌𝑌]
(iv) E[𝑋𝑋𝑋𝑋]
(b) Processing time of a job on the machine has Gamma
distribution with shape parameter 𝛼𝛼 = 4 and scale (rate)
parameter 𝜆𝜆 = 2 per hour. Use the central limit theorem to
find the approximate probability that the machine completes
100 such jobs in 200 hours.
[(4 + 2 + 3 + 3) + 8 = 20]

10. (a) Suppose that 𝑌𝑌 is a discrete random variable defined on the
sample space {0, 1, 2, ⋯ }. Let the conditional distribution of
𝑋𝑋 given 𝑌𝑌 = 𝑦𝑦 be binomial with size 𝑦𝑦 and probability 𝑝𝑝.
Show that if 𝑌𝑌 has Poisson distribution with mean 𝜆𝜆, then the
marginal distribution of 𝑋𝑋 is Poisson with mean 𝜆𝜆𝜆𝜆.
(b) In a coin tossing experiment, the probability of head is 𝑝𝑝 and
that of tail is 𝑞𝑞, where 0 < 𝑝𝑝 < 1 and 𝑝𝑝 + 𝑞𝑞 = 1. Suppose that
the coin is tossed three times. Let 𝑋𝑋 be the number of heads in
the first two tosses and 𝑌𝑌 be the number of heads in the last
two tosses. What is the joint probability distribution of (𝑋𝑋, 𝑌𝑌)?
Find the expectation of 𝑋𝑋𝑋𝑋.
(c) Let 𝑋𝑋 be an absolutely continuous random variable
symmetric about 0. Assume that 𝑋𝑋 has finite second moment.
Define 𝑌𝑌 = |𝑋𝑋|. Show that 𝑋𝑋 and 𝑌𝑌 are uncorrelated but are
not independent.
[7 + (5 + 2) + 6 = 20]

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PART II (ENGINEERING STREAM)

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ATTENTION : ANSWER A TOTAL OF SIX QUESTIONS TAKING

m . AT LEAST TWO FROM EACH GROUP - E1 AND E2.
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GROUP E1: Mathematics

ag 1
If 𝑧𝑧 + = 2 cos 𝜃𝜃, where z is a complex number, show that
𝑧𝑧
𝑧𝑧 2𝑛𝑛 − 1
� 2𝑛𝑛 � = |𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛𝑛𝑛|
𝑧𝑧 + 1

(b) If 𝑝𝑝 ≠ 0, 𝑞𝑞 ≠ 0 and
𝑝𝑝 𝑞𝑞 𝑝𝑝𝑝𝑝 + 𝑞𝑞
det � 𝑞𝑞 𝑟𝑟 𝑞𝑞𝑞𝑞 + 𝑟𝑟 � = 0,
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𝑝𝑝𝑝𝑝 + 𝑞𝑞 𝑞𝑞𝑞𝑞 + 𝑟𝑟 0

einmgeometric progression,
then prove that either of the following statements is true:

s
laof the equation 𝑝𝑝𝑥𝑥 + 2𝑞𝑞𝑞𝑞 + 𝑟𝑟 = 0.
(i) p, q and r are

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(ii) 𝛼𝛼 is a root 2

[8+12=20]

2. (a) Show that the sum of all positive integers up to 1000 which are
divisible by 5 and not divisible by 2 is 50000.

(b) Suppose that a geometric progression (GP) 1, 𝑞𝑞, 𝑞𝑞 2 , … (where
|𝑞𝑞| < 1) is to be constructed such that every term of this GP is
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a constant multiple (say k) of sum of all the subsequent terms.

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For what values of k is this problem solvable?

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(c) A rectangle ABCD is inscribed in a circle. Let PQ be the
diameter parallel to the side AB of the rectangle. Suppose that
the measure of the angle BPC is 300. Show that the ratio of the
area of the rectangle and the area of the circle is √3⁄𝜋𝜋.

[4+6+10=20]

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3. (a) Let 𝐼𝐼 = ∫𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠2 𝑥𝑥 𝑑𝑑𝑑𝑑, 𝑎𝑎 > 0. Find the value of I.
−𝜋𝜋 1+𝑎𝑎𝑥𝑥

(b) A circle, having center at (2, 3) and radius 6, crosses y-axis at
the points P and Q. The straight line with equation x = 1
intersects the radii CP and CQ at points R and S respectively.
Find the area of the trapezium PQSR.
[10+10=20]

4. (a) Consider a parallelogram ABCD. Suppose that P is a point on
the side AD so that AP : AD = x : y. Let Q be the intersection
point of AC and PB. Show that AQ : AC = x : x + y.

(b) Find the derivative of 𝑓𝑓(𝑥𝑥) at 𝑥𝑥 = 0, where
4 6 100
√1 + 2𝑥𝑥 ∙ √1 + 4𝑥𝑥 ∙ √1 + 6𝑥𝑥 ⋯ √1 + 100𝑥𝑥
𝑓𝑓(𝑥𝑥) = 3 5 7 101
√1 + 3𝑥𝑥 ∙ √1 + 5𝑥𝑥 ∙ √1 + 7𝑥𝑥 ⋯ √1 + 101𝑥𝑥

(c) A spherical balloon is being inflated so that its volume
increases uniformly at the rate of 40 cm3 ⁄minute. What is the
rate at which the surface area is increasing when the radius of
the balloon is 8 cm?
[5+7+8=20]

GROUP E2: Engineering & Technology

Engineering Mechanics and Thermodynamics

5. (a) A block ‘1’of weight W1 rests on a horizontal surface and
supports on the top of it a block ‘2’of weight W2. The block
‘2’is attached to a vertical wall by the inclined string AB as
shown in Fig. 1. Find the magnitude of the horizontal force P,
applied to the lower block ‘1’, which will be necessary to
cause slipping to impend. The coefficient of static friction for
all contiguous surfaces is µ = 0.3.

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Fig. 1

(b) Three identical cylinders, each weighing W, are stacked on
smooth inclined surfaces as shown in Fig. 2. Determine the
smallest angle θ to prevent the stack from collapsing.

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a
Fig. 2
[10+10=20]

6. (a) Using an engine of 40% thermal efficiency to drive a

m
refrigerator having COP of 5, what will be the heat input into

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the engine for removal of 2 MJ from the cold body by the

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refrigerator? If this system is used as a heat pump, how many

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MJ of heat will be available for heating for each MJ of heat

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input to the engine?
(b) An engine cylinder has a piston area of 0.12 m2 and contains a
gas at a pressure of 1.5 MPa. The gas expands according to a
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process which is represented by a straight line on the P-V
diagram. The final pressure of the gas is 0.15 MPa. Calculate
the work done by the gas on the piston if the stroke is 0.30 m.

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(c) The efficiency of a Carnot heat engine can be increased either
by increasing the source temperature while the sink
temperature is held constant or by decreasing the sink
temperature while the source temperature is held constant.
Which of the above two possibilities is more effective?
[(3+3)+4+10=20]

7. (a) An engine shaft running at 120 rpm is required to drive a
machine shaft by means of a belt. The pulley on the engine
shaft is of 2 m diameter and that of the machine shaft is of 1
m diameter. If the belt thickness is 5 mm, find the speed of the
machine shaft when (i) there is no slip, and (ii) there is a slip
of 3%.

(b) From the first law of thermodynamics, show that the change
of entropy of a perfect gas (∆𝑆𝑆) having mass m is
𝑇𝑇
∆𝑆𝑆 = 𝑚𝑚𝐶𝐶𝑝𝑝 ln 𝑇𝑇2 , where 𝐶𝐶𝑝𝑝 is the specific heat at constant
1
pressure, 𝑇𝑇1 is the initial temperature of the gas and 𝑇𝑇2 is the
final temperature of the gas in K.
[(5+5)+10=20]

Electrical and Electronics Engineering

8. (a) A three-phase star-connected synchronous generator driven at
750 rpm is required to generate a line-to-line voltage of 440 V
at 50 Hz on open circuit. The stator is wound with 2 slots per
pole per phase and each coil has 4 turns. Calculate the useful
flux per pole. [Assume a full-pitch coil. sin 150 = 0.26].

(b) A three-phase, 6-pole, 50 Hz induction motor develops the
maximum torque at 940 rpm. The rotor resistance per phase is
0.1 Ω. Calculate the value of the standstill rotor reactance.

(c) A dc shunt machine has an armature resistance of 0.5 Ω and
a field-circuit resistance of 750 Ω. When run as a motor on
no-load at 500 V applied terminal voltage, the line current

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was 3 A. Estimate the efficiency of the machine when it
operates as a generator with an output of 20 kW at 500 V, the
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field circuit resistance remaining unchanged.

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a
[8+5+7=20]

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g la 𝑖𝑖(𝑡𝑡) = 4 + 10 sin 𝜔𝜔𝜔𝜔 A.
(a) Calculate the average and rms values of the current
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a
(b) Show that a full adder can be constructed with two half adders
and an OR gate using necessary truth tables, equations and
circuit diagrams.

(c) A cube is formed by joining equal wires, each of resistance 1
Ω. The cube is shown in Fig. 3. Calculate the equivalent

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resistance between the points A and B.

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a

Fig. 3

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

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Engineering Drawing

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10. (a) Sketch the sectional front view, top view and right side view

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of the object shown in Fig. 4.

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Fig. 4

(b) Sketch the three views of the object shown in Fig. 5.

Fig. 5

[12+8=20]

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