ISI Admission Test 2024 Question Paper M.Tech(QROR) PQB

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ISI Admission Test
2024
Question Paper ·
M.Tech(QROR) PQB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2024 Question Paper M.Tech(QROR) PQB

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Part I: Section M (Common for Both Streams)

m
co
(Answer any TWO questions)

m .
.co s e m
m
e divisible by 3. Show that p − 1 is divisible by 24. l a
1. (a) Suppose that p is an odd integer greater than 3, but not

a s
l (b) Sketch the graph of the function f (x)= , x ∈ (−∞, ∞).
2

ag
ag x
1+x2

(c) Show that for positive integer n,
           
n n n n n n
+ + ··· +
0 1 1 2 n−1 n
    
(n + 1)n n n n
= ··· .

m
n! 0 1 n−1

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em
[8 + 7 + 5 = 20]

l a s
a g
2. (a) From a point h meters above the surface of a lake, the angle
of elevation of a cloud is θ and the angle of depression of
its reflection in the lake is ϕ. Find the height of the cloud
above the lake surface.
(b) Consider the pair of straight lines given by

x2 (tan2 θ + cos2 θ) − 2xy tan θ + y 2 sin2 θ = 0.

m
m c. o
Show that the difference between the gradients of the lines

c. o e10m= 20]
is 2.

em la s
s
[10 +

la a g
ag 3. (a) If f (x) = sin 2x+A sin
x3
x+B cos x
is continuous at x = 0, find the
values of A, B and f (0).
(b) Find the volume of the largest cylinder that can be placed
inside a hollow right circular cone with base radius R cm
and height H cm.

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(c) Find the area of the region bounded by the curves y = |x|
and y = 2x + x2 − 2x3 .
[6 + 8 + 6 = 20]

4. (a) Let p(x) = x4 − a1 x3 + a2 x2 − a3 x + 1470 be a polynomial
such that all roots of the equation p(x) = 0 are positive
integers greater than unity. Further suppose that

′ d
|p(7)| + |p (7)| = 0, where p′ (x) = p(x).
dx
(i) Find the least and greatest possible values of a1 .
(ii) Find all the possible polynomials.
(iii) From the set of polynomials found in (ii) above, write
down the polynomial which has the least value for any
x ∈ R.
(b) Consider the following matrix for a, b, c ∈ R,
 
a+b b+c c+a
 
A =  b + c c + a a + b .
c+a a+b b+c

(i) Find det(A).
(ii) If a2 + b2 + c2 = 4, find the maximum and minimum
numerical values of the above determinant.

[(4 + 3 + 5) + (3 + 5) = 20]

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Part II: Section S (Statistics Stream)

m
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(Answer any FOUR questions)

m .
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m
e randomly with replacement from the set A. Find the l a
5. (a) Let A = {1, 2, . . . , 49, 50}. Suppose two numbers are chosen

l a s ag
ag probability that their difference is divisible by 3.
(b) A slip of paper is given to A, who marks it with either
a plus sign (+) or a minus sign (−). The probability of
marking plus sign is 13 . Thereafter, the slip is passed on to
B, C and D sequentially. Each of them either alters the
sign with probability 23 or leaves it unchanged. Finally D

m
hands over the slip to the referee, who finds plus sign on

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it. What is the probability that A has marked plus sign on

s em
the slip?

g l a [10 + 10 = 20]

a
6. (a) The joint probability density function of (X, Y ) is given by
1 + xy
f (x, y) = , −1 < x, y < 1.
4
Find the joint probability density function of (X 2 , Y 2 ).
(b) Let X0 , X1 , . . . be a Markov chain with state-space {1, 2, 3}

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and transition probability matrix

c. o m
1 4 

se
0

em
5 5

la
 1 

s
P =  25 1
10 
.

la g
2

a
1 9

g
0

a
10 10
(i) Find the probability distribution of X1 , if the


probability distribution of X0 is 15 , 25 , 25 .
(ii) Classify the states.

[10 + (6 + 4) = 20]

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7. (a) Suppose the number (N ) of customers that arrive at an
ATM to withdraw money, during a given period of one hour,
has a Poisson distribution with mean 4. The amount (X)
of money withdrawn by any customer is a discrete random
variable with probability mass function
1
pX (x) = , if x = 2000, 4000, 6000, 8000, 10000.
5
Assume that N and X are independent.
(i) Find the expected total amount of money drawn during
the period given that at least one customer withdrew
money.
(ii) What is the probability that 3 customers have
withdrawn money given that the total withdrawal is
Rs. 8000 during the period?
(b) Let X be a random variable with probability mass function


 e−1 if x = 0

 e−1
P (X = x) = if x ∈ {±1, ±2, . . .}

 2(|X|)!

 0 otherwise.
Find the distribution of Y = |X|.
[(6 + 6) + 8 = 20]

8. (a) Let X1 , X2 , . . . , Xn be a random sample from N (µ, σ 2 ).
Find the UMVUE of µ3 , if σ 2 is known.
(b) A salesman of used cars assumes that the number of sales he
makes per day is a Poisson random variable with parameter
µ. Over the past 30 days, he made no sales on 20 days and
one or more sales on each of the remaining 10 days. Find
the maximum likelihood estimate of µ.
[12 + 8 = 20]

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9. Let X(1) , X(2) , . . . , X(n) be the order statistics corresponding to

m
a random sample from the distribution with probability density

m .co
function

c. o f (x) = λe , 0 < x < ∞, λ > 0. e m
−λx

m the transformations a s
lase g l
Consider

g a
a Z1 = nX(1) , Zi = (n − i + 1)(X(i) − X(i−1) ), i = 2, 3, . . . , n.

(a) Show that Z1 , Z2 , . . . , Zn are iid with pdf f .
(b) Hence or otherwise find mean and variance of sample range.

[10 + (4 + 6) = 20]

o m
c
. density function
10. (a) Let X1 , X2 , . . . , Xn be a random sample from the

e m
distribution with probability

a s
l x) , 0 < x < ∞, θ > 0.
a g
θ−1
f (x) = θ(1 −

(i) Derive an α-level most powerful test for H : θ = 1 0

versus H1 : θ > 1. Find the power function of the test.
(ii) Derive a 100(1 − α)% confidence interval for θ.

(b) Consider the following linear model under the
Gauss-Markov set-up:

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c. o
E(Y1 ) = β2
m
em la se
E(Y2 ) = 2β1

las E(Y3 ) = β1 + β2 .
a g
ag Derive least-square estimators of (β1 , β2 ). Verify whether
they are unbiased.
[(8 + 3) + (6 + 3) = 20]

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Part II: Section E (Engineering Stream)

(Answer any FOUR questions)

5. (a) Three solid spheres, each of mass m and diameter d, are
glued together in a manner that the lines connecting the
centers form an equilateral triangle of side of length d as
shown in Figure 1. The centroid of the system is located
at point O. Now, IO and IA are the moments of inertia of
the system about the axes (perpendicular to the plane of
the aforementioned triangle) passing through the centroid
O and point A, respectively. Find out the value for the
ratio: IO /IA .

Figure 1

(b) A particle of mass m is projected vertically upwards with
a velocity v0 in a medium whose resistance is kv where v is
the instantaneous velocity and k is a constant. Determine
the expression of time for the particle to come to rest.

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(c) Consider the stress elements as shown in Figure 2. The

m
values of σx , σy and τxy are −10 MPa, 20 MPa and −20 MPa

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respectively. Find out the values of the principal stresses.

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Figure 2
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[8 + 6 + 6 = 20]

l a
ag between two identical bodies which
6. (a) A heat pump operates
are initially at temperature T1 and cools one of the bodies to
a temperature T2 (T2 < T1 ). Prove that for this operation
the minimum work required by the heat pump is given by
 2 
T1
W = Cp + T2 − 2T1
T2

o m
m c
. a
where Cp is the specific heat for both the bodies.

c. o (b) Two flat blocks are resting on the inclined plane m
s e to the
em
and

s g la3. The 5 kg
la
force P is applied to the 10 kg lower block parallel

g a
inclined plane in the direction shown in Figure
a upper block is prevented from moving by a cable, attached
to a fixed support. The corresponding coefficients of static
friction for the two mating surfaces are shown in Figure 3.
Considering g = 10 m/s2 , determine
(i) the tension in the cable,

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(ii) the force that makes the 10 kg lower block start sliding
downward,
(iii) the feasibility of moving the blocks together upward by
applying a commensurate force parallel to the inclined
plane.

Figure 3

[10 + (4 + 4 + 2) = 20]

7. (a) Determine the entropy change of 4 kg of a perfect gas whose
temperature varies from 127 ◦ C to 227 ◦ C during a constant
volume process. The specific heat varies linearly with
absolute temperature and is represented by the relation:

Cv = (0.48 + 0.0096T ) kJ/kgK.

[ln 1.25 = 0.2 (approx.)]

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(b) Source 1 can supply energy at the rate of 12,000 kJ/min at

m
343 ◦ C. A second source 2 can supply energy at the rate of

mchoose to supply energy to an ideal reversible heat .co
1,20,000 kJ/min at 69.3 ◦ C. Which source (1 or 2) would

o
c.you e m
m engine to produce large amount of power if the temperature
e of the surroundings is 35 C? l a s
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l (c) 281.5 kW of heat is supplied at a temperature of 290 C to
◦

ag
ag ◦

◦
a heat engine. The heat rejection takes place at 7 C. The
following results are obtained:
(i) 280 kW is rejected,
(ii) 140 kW is rejected,
(iii) 70 kW is rejected.

m
c. o
Classify the results as reversible cycle or irreversible cycle

em [5 + 6 + (3 + 3 + 3) = 20]
or impossible cycle.

l a s
g in Figure 4, find the voltage across
8. (a) For the circuitashown
and current through the load resistor 3 Ω using Norton’s
theorem.

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a
Figure 4

(b) A rotor of 440 V, 50 Hz, 4-pole, 3-phase induction motor is
observed to make 120 complete alterations per minute.

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(i) What will be the rotor speed?
(ii) If the power input to the rotor is 90 kW, what will be
the rotor copper loss per phase?
√
(c) In a 300 kVA, 1000 3 V, 50 Hz, 3-phase star connected
alternator, the effective armature resistance is 3 Ω per
phase. A field current of 40 A produces a short circuit
√
current of 100 A and an open circuit emf of 500 3 V. Find
the percentage voltage regulation using emf method at full
load with 0.6 lagging power factor.
[8 + (4 + 2) + 6 = 20]

9. (a) For the circuit given in Figure 5, find
(i) the resistance between the terminals A and B using
star delta conversion,
(ii) the current I.

Figure 5

(b) (i) Draw the circuit diagram of a 4-bit shift register using
D-type flip-flops.
(ii) Explain how to store a binary serial data 1011 in the
shift register mentioned in (i).
[(7 + 2) + (4 + 7) = 20]

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10. Draw the top view and front view of the object shown in Figure 6

m
using third angle projection.

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m
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g l a
a Figure 6

[10 + 10 = 20]

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c. o se
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a

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