ISI Admission Test 2026 Question Paper M.Tech(QROR) PQB

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ISI Admission Test
2026
Question Paper ·
M.Tech(QROR) PQB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2026 Question Paper M.Tech(QROR) PQB

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Part I: Section M (Common for Both Streams)

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(Answer any TWO questions.)

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m a
1. (a) Find the minimum value of f (x, y), using only the fact that

laseAM → GM for two positive numbers, where ag l
ag f (x, y) = 4x2 +
x
y 2
4y
+ , x, y > 0.
x
Note: No credit will be given if results from calculus are
used.
(b) Without expanding show that the value of the matrix D is
real, where

m 
 

c. o↑1 + i .
2 3+i ↑1


↑1 ↑1m
D = 3 ↑ i 0

e
as
↑i 1

(c) If a , a , a and a glare the coe!cients of any four
1 2 3
a the expansion of (1 + x) , then show
consecutive terms in
4
n

that
a1 a3 2a2
+ = .
a1 + a2 a3 + a4 a2 + a3
[8+5+7 = 20]

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2. (a) Suppose A is a 3 ↓ 3 matrix, where

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 

se
↑2 0 1

em la
 

s
A =  1 1 0 .

la
0 ↑2
a g
g
0

a i. Obtain the eigen values and eigen vectors of A.
ii. Is it possible to obtain three linearly independent eigen
vectors for A? Justify your answer.
iii. Check whether A can be diagonalized as P →1 AP , where
P is the modal matrix of A.

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(b) If y1 and y2 be the solutions of the equation

dy
+ P y = Q,
dx
where P and Q are functions of only x and y2 = y1 z, then
show that ! Q
→ dx
z = 1 + ae y1
.

[(4+3+4)+9 = 20]

3. (a) Consider the following two functions:

f (x) = x3 + Ax2 + x + B
g(x) = x3 + x2 + Bx + A

where, A and B are integers and B is not even.
Suppose the following two conditions hold:

i. f (1) = 0.
ii. The roots of g(x) = 0 are squares of the roots of
f (x) = 0.

Find the value of A2025 ↑ B 2025 .

(b) Let f, g : R ↔ R be two functions defined as f (x) = |x| + x
and g(x) = |x|–x, ↗x ↘ R. Then find i) f og and ii) gof .

(c) Two parallel chords of a circle of radius 2 are at a distance
≃
(1 + 3) apart. If the two chords subtend angles ωk and
ω
2k
, k > 0, at the centre, then find the value of [k].

[7+(2 12 + 2 12 )+8 = 20]

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4. (a) Suppose AX, BY and CZ are three concurrent cevians

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other than altitudes, medians or angle bisectors of the

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triangle ABC given in Figure 1. (Cevian: a line segment

c o
. side)
that connects a vertex of a triangle to a point on the
e m
m a s
se l
opposite

la ag
ag

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Figure 1

↓ s= em1.
la
BX CY AZ
↓
g
Show that

a to express sin ω cos ω in terms of
XC YA ZB

4 4
(b) Use de Moivre’s thorem
sines and cosines of multiples of ω.
[12+8 = 20]

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Part II: Section S (Statistics Stream)

(Answer any FOUR questions.)

5. (a) A game host puts three balls randomly into three identical
boxes. Each box may contain 0, 1, 2 or 3 balls. The player
chooses any two boxes to open and if all the three balls are
found in these two chosen boxes, then the player wins the
game. Find the probability that the player wins the game.
(b) A pair of unbiased dice is thrown until the sum of two faces
is either six or seven. Find the probability that the sum
seven occurs first.
(c) If a fair coin is tossed an even number 2n times, show that
the probability of obtaining more heads than tail is
' ( ) ( )2n *
1 2n 1
1↑ .
2 n 2

[7+7+6 = 20]

6. (a) Suppose that the number of customers visiting a
restaurant on a given day is N , with N ⊋ Poisson(ε).
Assume that each customer purchases a drink with
probability p. Furthermore, purchase of one customer is
independent of the purchase of other customers and
independent of N .
Let X be the number of customers who purchase the drink
and Y be the number of customers who do not purchase
the drink.
i. Find the marginal probability mass function of X and
Y.
ii. Find the joint distribution of X and Y .
iii. Are they independent?

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(b) A population consists of three villages V1 , V2 and V3 with

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average farm size x1 , x2 and x3 respectively. A probability

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sample consisting of two villages is selected to estimate the

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two .estimators T and T as depicted below:
population average µ (µ = x1 +x32 +x3 ). A consultant suggests
em
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se g l
1 2

g la Sample S Prob (S) Estimator T Estimator T a
a 1 2

1 x1 +x2 x1 +x2
V 1 , V2 3 2 2

x2
V 2 , V3 1
3 2
+ x33 x2 +x3
2

x1
V 1 , V3 1
3 2
+ 2x33 x1 +x3
2

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Find the condition(s) under which the estimator T1 would

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be preferable.

l a s [(5+5+2)+8 = 20]

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7. (a) A company manufactures a certain type of rechargeable
battery and the lifetime of each battery under stress (in
days) is assumed to follow an exponential distribution
with mean ω. Suppose there are 3 types of batches of
rechargeable batteries, viz., “good ”, “medium”and “bad ”,

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having average lifetime under stress as 4 days, 1 day and

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1/4 day respectively. Five rechargeable batteries are

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selected from a batch at random and their lifespan under

s g a
lcan
la
stress turns out to be 4 days, 2 days, 1 day, 2 days

g a
a
1 day (in coded units). Based on the above data, you
find an MLE of batch-type (i.e., whether it is “good ”,
“medium”or “bad ”)?
[Assume log 3 ⇐ 1.1 and log 2 ⇐ 0.7]

(b) A marathon run is conducted where the total distance is

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ω km. Five runners participated and the lengths (in km)
covered by them are as follows:

8, 5, 9, 6, 10.

i. Assuming the length covered by the runners as iid
realizations from a uniform distribution, find an
unbiased estimate of ω based on a minimal su!cient
statistic.
ii. Find the BLUE of ω.
iii. Compare the estimators found in (i) and (ii) above with
proper justification.
[7+(5+6+2) = 20]

8. (a) The following model is assumed for weather prediction:
(1) If it is rainy today then the probability that it would
be rainy, cloudy (but not rainy) or sunny tomorrow
are 0.5, 0.3, 0.2 respectively.
(2) If it is cloudy today then the probability that it would
be rainy, cloudy or sunny tomorrow are 0.2, 0.4, 0.4
respectively.
(3) If it is sunny today then the probability that it would
be rainy, cloudy or sunny tomorrow are 0.1, 0.1, 0.8
respectively.
i. Find the transition matrix.
ii. If today is cloudy, then find the probability that it will
rain the day after tomorrow.
iii. Suppose it is known that tomorrow can either be
rainy or cloudy with equal probability. Can you
obtain the distribution of the weather status for the
day after tomorrow?

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(b) Obtain the regression equation of Y on X for the

m
distribution given by

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y

co
ye→ (1+x)

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f (x, y) = , x, y → 0.

s
(1 + x)4
m e l a
l a s [(2+4+4)+10 = 20]
ag
g
a9. (a) Given a single observation from the population with
probability density function
2
f (x) = (ω ↑ x),
ω2
find 100(1 ↑ ϑ)% confidence interval for ω.

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c. o positive quantities.
(b) Suppose Xi , i = 1, 2, · · · , n follows independent
2 2

empowerful test of level ϑ for
N (ϖ ω, ϖ ϱ ), where ϖ , ϱ are known
i i i

la
i. Find the uniformly most
s
testing
a g
H0 : ω = ω 0 against H1 : ω = ω1 > ω0 .

ii. Examine whether uniformly most powerful test of level
ϑ exists for H1 : ω ⇒= ω0 .
(c) Let X be a random variable with probability density

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function

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

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se
1 for 0 ⇑ x ⇑ ω

em
ε

la
f (x) =
0
s g
otherwise.

gla a
a
Consider the following two critical regions based on a single
observation x : C1 = {x → 0.5} and C2 = {1 ⇑ x ⇑ 1.5},
for the problem of testing H0 : ω = 1 against H1 : ω = 2.
Find out the probability of type I error of the two tests and
their powers.
[5+(6+3)+6 = 20]

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10. (a) Suppose a fair coin is tossed 144 times. Find an
approximate value of the probability that the number of
heads is at most 66.
[Assume ”(0.5) = 0.84, ”(1) = 0.69, ”(1.5) = 0.93,
”(2) = 0.98]
(b) Let X1 , X2 , . . . be iid samples from exponential distribution
with mean ε. Define Yi = Xi 2 , i = 1, 2, · · · . Find an and
bn such that

Ȳn ↑ an d
↔
↑ Z, as n ↔ ⇓,
bn

where Ȳn = (Y1 + · · · + Yn )/n, Z is distributed as N(0,1),
d
and ↔
↑ indicates convergence in distribution.
(c) Let W1 < W2 < · · · < Wn be the order statistics of n
independent observations from a U(0,1) distribution.
i. Show that the probability density function of Wr
follows Beta distribution.
ii. Show that
r(r + 1)
E(Wr 2 ) = .
(n + 1)(n + 2)

[4+6+(5+5) = 20]

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Part II: Section E (Engineering Stream)

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(Answer any FOUR questions.)

o m m .
c
. a constant velocity of 54 km/h crossed the car. The
5. (a) A car stood in front of a tea shop. A passenger bus traveling
s e
m a
secar started after 5 seconds and accelerated by 25 m/s until g l
with

g la it reached the velocity of 72 km/h. It continued with the a
2

a same velocity and eventually crossed the bus after some
time. What distance does the car travel before it overtakes
the passenger bus?
(b) A thin cylinder of diameter d and thickness t is
pressurized with an internal pressure of p. Assume that E

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is the modulus of elasticity and ς is the Poisson’s ratio. If

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em
ς = 0.2, find the ratio of longitudinal strain to

a s
circumferential strain.

g l
a
(c) Find the moment of inertia of the shaded region, as shown
in Figure 2, about X and Y axes.

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a
Figure 2

[6+6+8 = 20]

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6. (a) An induction motor drives the main shaft of a workshop
by means of a flat belt. The diameter of the pulleys on
the motor shaft and the main shaft are 0.4 m and 0.8 m
respectively. Another pulley of diameter 0.7 m, which is
keyed to the main shaft, drives a countershaft having a
pulley of diameter 1.4 m keyed to it. If there is a 2% slip
on each drive, determine the speed of the countershaft if
the motor runs at 1400 rpm. Neglect thickness of the belt.
(b) A person does 30 kJ work on 2 kg of water by stirring
with the help of a paddle wheel. While stirring, around
20 kJ of heat is released from water through its container
to the surface and surroundings by thermal conduction
and radiation. What is the change in internal energy of
the system?
(c) An inventor claims that his new engine will develop
30 kW for a heat addition of 2400 kJ/min. The highest
and lowest temperatures of the cycle are 2200 ↑ C and
327 ↑ C respectively. Would you agree to his claim?
[10+5+5 = 20]

7. (a) A system undergoes a cycle involving three processes, with
details given in the table below, with the values of heat (Q),
work (W ) and change in internal energy (#U ).

Process Q(kJ) W (kJ) #U (kJ)
1↑2 a 150 150
2↑3 b ↑75 c
3↑1 150 d ↑300

Find out the values of a, b, c and d.
(b) Butane is burned in dry air. The volumetric analysis of
the products is found as 11% CO2 , 1% CO, 3.5% O2 and

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84.5% N2 . Determine the percent of theoretical air used in

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combustion.

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(c) For heating water, an electric coil with a resistance of
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c
. the process, the temperature of the heater remains e m
m s
40 $ is connected to a power source of 240 V for one hour.

seconstant at 90 C. Determine
During
g l a
g la a
↑

a i. the work done by the heater,
ii. the heat transferred to the heater,
iii. change in entropy of the heater and the water,
iv. entropy change of the universe.
[4+10+(2+1+2+1) = 20]

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8. (a) For the circuit given in Figure 3, assume that the diode D

emthe output voltage v for
is ideal, R1 = 2 k$, R2 = 8 k$, R = 2 k$, R3 = 8 k$

i) v = 1.5 V and ii) v la
and v = 0.75 V.
s2
s
Find 0

s
a g = ↑1.5 V.
s

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a Figure 3

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(b) i. Explain the operation of a half adder.
ii. Show that a full adder can be constructed with two half
adders and an OR gate using the necessary equations,
truth tables and circuit diagrams.
[(6+4)+(4+6) = 20]

9. (a) A 4-pole 404 V wave winded DC shunt motor is running at
1200 rpm. It has 400 copper conductors and draws 12 A
from the supply. The armature resistance is 0.4 $ and the
shunt field resistance is 202 $.
i. Find the values of armature current, back emf and flux
per pole.
ii. Calculate the speed at which the motor will run if 3.6 $
resistor is introduced in the series with the armature.
(b) For the circuit shown in Figure 4,
i. Obtain the value of R such that maximum power will
be transferred to R.
ii. Also prove that, the value of R thus obtained,
corresponds to the maximum power.

Figure 4

[(8+4)+(4+4) = 20]

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10. Draw the top and the front views of a regular hexagonal prism of

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base side 25 mm and axis length 50 mm when it is lying on the

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ground on one of its rectangular faces and the axis is inclined at

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↑
45 to the vertical plane.

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