ISI Admission Test 2025 Sample Paper M.Tech(CS), M.Tech(CrS) PCB

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2025
Sample Paper ·
M.Tech(CS),
M.Tech(CrS) PCB
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2025 Sample Paper M.Tech(CS), M.Tech(CrS) PCB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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PCB Model Question Paper

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Disclaimer: The following model question paper is pro-

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vided solely for illustrative purposes to demonstrate the

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structure and format of the actual examination. Please note
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that the difficulty level and topic-wise distribution of ques-

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tions in the final exam may vary.

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The question paper is divided into two groups: CS Group

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and Non-CS Group. Please choose ANY ONE group and

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answer the questions from that group only, according to the

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instructions.

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CS Group

Answer Question C1 (20 marks) and any 5 from the other questions (5 × 16 = 80 marks).

C1. Consider a reservoir, fitted with a set of taps and drains. The partial pseudocode
given below takes as input:
L capacity of the reservoir in litres
T number of taps attached to the reservoir
D number of drains attached to the reservoir
ti time in minutes that it takes to fill an initially empty reservoir if all drains
are stopped, and only the i-th tap is opened (1 ≤ i ≤ T )
dj time in minutes that it takes to drain an initially full
reservoir if all taps are shut off, and only the j-th drain is opened
(1 ≤ j ≤ D)
Fill in the blanks below, so that the statements printed by the pseudocode are
correct. Do not copy the pseudocode; simply write the serial number of each blank
(I, II, . . ., VI), and your corresponding answer.
Pseudocode:
Initially, set variables X and Y to 0.
for i in {1, 2, 3, . . . , T } X ←− X+ (I) .
for j in {1, 2, 3, . . . , D} Y ←− Y + (II) .
If X (III) Y
then print “A partially empty reservoir never fills up if all taps and drains are
kept open.”
else print “An initially empty reservoir fills up in (IV) minutes if all taps
and drains are kept open.”
If X (V) Y
then print “A partially full reservoir never becomes empty if all taps and drains
are kept open.”
else print “An initially full reservoir becomes empty in (VI) minutes if all
taps and drains are kept open.”

C2. You are given 68 identical looking balls, each with a distinct weight. You are given
a common balance using which you can compare weights of any pair of balls with
a single measurement. Describe how you would identify both the heaviest and the
lightest balls by using at most 100 measurements in total.

C3. Let A = {a1 , a2 , . . . , an } and B = {b1 , b2 , . . . bm } be two sorted arrays of n and m
numbers, respectively. Devise an algorithm to count the number of pairs (i, j), 1 ≤
i ≤ n, 1 ≤ j ≤ m, so that ai > bj . The efficiency of your algorithm will be counted
in terms of the number of comparisons you perform between the elements of A and
B. For full credit, your algorithm should perform O(n + m) comparisons.

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C4. Let k be a positive integer, and n = 2k − 1. Let X = {1, 2, . . . , n} be arranged as
a min-heap stored in a complete binary tree, T .

(a) What can be the smallest possible value that can be stored at a leaf node of
T ? Prove your result.

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(b) Define the level of the root of the min-heap T to be 1. What can be the

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maximum possible value that can be stored at level 2 of T ? Prove your

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. connected graph G, let χ(G) be the minimum number of colours se m
result.

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C5. For any undirected

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all the vertices of G in such a way that no two adjacent vertices

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avertices v , v , . . . , v such that (v , v ) is an edge for 1 ≤ i ≤ k.
have the same colour. Prove that for every vertex v of G, there exists a path of
length χ(G) − 1 starting from v. A path of length k is a sequence of k + 1 distinct
1 2 k+1 i i+1

C6. A 32-bit pattern (p31 , . . . , p0 ) is interpreted as a floating point number as follows.

• The sign bit is s = p31 . If s = 0, the number is considered positive, otherwise
it is considered negative.
• The 8-bit part (p30 , . . . , p23 ) is the exponent biased by 127. That is, if these

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bits represent an integer e, then the

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corresponding term would be considered as 2e−127 . The all-zero and the

• The mantissa part m = (p , . . . ,m
.
all-one bit patterns are not considered valid for these eight bits.

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p ) would be interpreted as 1.m, where the
22 0

glthe decimal value of the 32-bit pattern becomes
leftmost 1 is implicitly understood.

(−1) × 2 e−127
a
With the above understanding,
s × 1.m. For example, consider the 32-bit pattern (1 1011 0110 011
0000 0000 0000 0000 0000), which is actually (−1)1 × 210110110−01111111 × 1.011
as represented with bits, the same as −1.375 × 255 in decimal.

(a) Suggest how the decimal zero can be represented in the above format.
(b) Describe the largest positive value in this format.
(c) Describe the smallest non-zero positive value in this format.

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(d) Given two such bit patterns (x31 , . . . , x0 ), (y31 , . . . , y0 ) in the above format,

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construct the digital circuit for their

em C7. Consider the alphabet Σ = {0, 1}. se
addition.

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(a) Design a finite automaton accepting language
k k
A = {0 w0 | k ≥ 1 and w ∈ Σ }. ∗

(b) Show that no finite automaton accepts the language
B = {0k 1w0k | k ≥ 1 and w ∈ Σ∗ }.

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C8. Let P1 , P2 , . . . , P20 be a set of 20 cooperating processes, all of which have access
to a shared integer X. For i ∈ {1, 2, . . . , 20}, the process Pi calls do something(i),
the code for which is given below.
shared int X = 0;
void do_something(int i)
{
printf("%d %d", i, X);
X = X + i;
}
(a) What is the minimum possible final value of X? Justify your answer.
(b) Your task is to synchronise the execution of the above
processes using semaphores so that the combined output of all 20 processes
is as follows.
1 1 2 3 3 6 ... 20 210
i. Show how you would declare and initialise the
necessary semaphore(s). Follow the syntax used in the examples given
below.
shared semaphore S = 1;
shared semaphore mutex[m] = {0, 0, ..., 0};
ii. Modify the body of the above function by inserting appropriate wait()
and signal() system calls so that the messages are printed in the desired
order.

C9. Design a synchronous counter that has an Enable input. When Enable = 1, it
generates a binary sequence equivalent to 0, 3, 1, 2, 0, 3, 1, 2, . . . in decimal. When
Enable = 0, the counter remains in its current state. Use J-K flip flops only and a
minimal number of AND-OR-NOT gates in your design. Argue why your gate
count is minimal. Modify your circuit without changing the input-output be-
haviour, so that it uses universal gates only.

C10. Consider sending the 6-bit message 101000 using Hamming codes for single-bit
error-correction.

(a) Calculate the codeword for this message. Use even parity.
(b) Invert the last bit of the codeword and then show how the recipient can correct
this inverted bit.
(c) Invert the first bit of the codeword and then show how the recipient can
correct this inverted bit.

C11. Suppose the relationship between customers and products are stored in a table
BOUGHT(CUST,PROD,QT) such that there is a tuple (c, p, n) ∈ BOUGHT if and only if

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the customer c has bought the product p and n is the total quantity of p bought
by c.
Fill the blanks in the given SQL statement so that it outputs a table COMMON PRODS,
which consists of the number of common products bought by every pair of cus-
tomers who have bought some common product.

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Do not copy the statements; simply write the serial number of each blank (I, II,

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. . ., VII), and your corresponding answer.

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select CUST1, CUST2, (I) as NUM

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from (select (II) as CUST1,
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a
aExample: For the table BOUGHT as shown in Table 1, the output table COMMON PRODS
as B2 where (VI) ) as T group by (VII) ;

should be as shown in Table 2.

BOUGHT
CUST PROD QT
1 A 6 CUST1 CUST2 NUM
2 A 8 1 2 2

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1 B 2 1 3 2

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1 C 2 1 4 1

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2 C 1 2 3 1

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3 B 9 3 4 1

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3 C 5

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Table 2: COMMON PRODS

a
4 B 2

Table 1: BOUGHT

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Non-CS Group

Answer Question N1 (20 marks) and any 5 from the other questions (5 × 16 = 80 marks).

N1. Consider a reservoir, fitted with a set of taps and drains. The partial pseudocode
given below takes as input:
L capacity of the reservoir in litres
T number of taps attached to the reservoir
D number of drains attached to the reservoir
ti time in minutes that it takes to fill an initially empty reservoir if all drains
are stopped, and only the i-th tap is opened (1 ≤ i ≤ T )
dj time in minutes that it takes to drain an initially full
reservoir if all taps are shut off, and only the j-th drain is opened
(1 ≤ j ≤ D)
Fill in the blanks below, so that the statements printed by the pseudocode are
correct. Do not copy the pseudocode; simply write the serial number of each blank
(I, II, . . ., VI), and your corresponding answer.
Pseudocode:
Initially, set variables X and Y to 0.
for i in {1, 2, 3, . . . , T } X ←− X+ (I) .
for j in {1, 2, 3, . . . , D} Y ←− Y + (II) .
If X (III) Y
then print “A partially empty reservoir never fills up if all taps and drains are
kept open.”
else print “An initially empty reservoir fills up in (IV) minutes if all taps
and drains are kept open.”
If X (V) Y
then print “A partially full reservoir never becomes empty if all taps and drains
are kept open.”
else print “An initially full reservoir becomes empty in (VI) minutes if all
taps and drains are kept open.”

N2. You are given 68 identical looking balls, each with a distinct weight. You are given
a common balance using which you can compare weights of any pair of balls with
a single measurement. Describe how you would identify both the heaviest and the
lightest balls by using at most 100 measurements in total.

N3. Let A = {a1 , a2 , . . . , an } and B = {b1 , b2 , . . . bm } be two sorted arrays of n and m
numbers, respectively. Devise an algorithm to count the number of pairs (i, j), 1 ≤
i ≤ n, 1 ≤ j ≤ m, so that ai > bj . The efficiency of your algorithm will be counted
in terms of the number of comparisons you perform between the elements of A and
B. For full credit, your algorithm should perform O(n + m) comparisons.

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N4. Let A = (aij ) be an m × n matrix such that for i ∈ {1, 2, . . . , m} and j ∈
{1, 2, . . . , n},

(a) aij s are real valued and independent random variables;
(b) E[aij ] = 0 and Var(aij ) = 1.

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Let x ∈ Rn . Show that E[∥Ax∥2 ] = m∥x∥2 . Note that

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∥x∥2 = xT x.

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N5. For a polynomial of the form a0 + a1 x + · · · + am xm having degree m > 1, it is

l a s
s g
ai−1 i

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known that the coefficients satisfy the relation = . Show that the

g
ai−1 + ai n+1

N6.aA and B are playing a game which involves tossing a fair coin repeatedly. A wins if
polynomial has at least one real root with multiplicity greater than 1.

the sequence HHT appears before the sequence HT H. Compute the probability
that A wins.

N7. For any undirected connected graph G, let χ(G) be the minimum number of colours
needed to colour all the vertices of G in such a way that no two adjacent vertices
have the same colour. Prove that for every vertex v of G, there exists a path of

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length χ(G) − 1 starting from v. A path of length k is a sequence of k + 1 distinct

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vertices v1 , v2 , . . . , vk+1 such that (vi , vi+1 ) is an edge for 1 ≤ i ≤ k.

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N8. Consider the alphabet Σ = {0, 1}.

a s
l Σ }.
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(a) Design a finite automaton accepting the language

a
k k
A = {0 w0 | k ≥ 1 and w ∈ ∗

(b) Show that no finite automaton accepts the language
B = {0k 1w0k | k ≥ 1 and w ∈ Σ∗ }. [5+5]

N9. Let f : (−1, 1) → R be twice differentiable such that f (1/n) = 0 for all n ∈ N.
Show that (i) f ′ (0) = 0, and (ii) f ′′ (0) = 0.

N10. Design a synchronous counter that has an Enable input. When Enable = 1, it
generates a binary sequence equivalent to 0, 3, 1, 2, 0, 3, 1, 2, . . . in decimal. When

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Enable = 0, the counter remains in its current state. Use J-K flip flops only and a

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minimal number of AND-OR-NOT gates in your design. Argue why your gate

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count is minimal. Modify your circuit without changing the input-output be-

em N11. An engine operates at 75% of the efficiency of a Carnot engine aoperating between
haviour, so that it uses universal gates only.

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gla two temperatures T and T (T > T ). This engine has aapower output of 100 W
a
1 2 1 2
and discharges heat at the rate of 300 J/s into the low-temperature reservoir with
◦
T = 27 C. Compute the temperature T of the high-temperature reservoir.
2 1

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N12. In the circuit shown below, determine the resonant frequency and the resonant
impedance between terminals A and B, when R = 100 kΩ, C = 10 µF , and
L = 100 mH. Find the condition in terms of R, C and L, under which the
impedance between A and B at resonance equals R.

N13. A small block starts sliding down an inclined plane which forms an angle of θ with
the horizontal. The coefficient of friction µ depends on the distance x covered by
the block as µ = ax, where a is a constant. Find (a) the distance covered by the
block till it stops, and (b) it’s maximum velocity over the distance traversed.

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Sample Questions for Engineering Sciences
(PCB, Non-CS Group)
Here are some more sample questions for the topics in Engineering Sciences (PCB, Non-
CS Group). To see sample questions for PCA and the rest of the topics of PCB, please

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refer to the question papers of the last few years, accessible through the admission

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website:

c. o http://www.isical.ac.in/~admission. s em
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1. A wooden plank of thickness h stands perpendicular to the surface. A bullet of
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mass m enters the plank perpendicular to it at a height of z above the surface
with velocity v0 , as shown in Figure 1. The plank offers a resistive force to the
bullet as F = −kv 2 , where v is the instantaneous velocity of the bullet, and k is
a known constant. Assume that the bullet comes out of the plank perpendicular
to the plank at the same height z from the surface and subsequently drops to the
surface. Let x be the distance of the point where the bullet drops from the base
of the plank. Assuming air drag to be zero and acceleration due to gravity as g,
find x in terms of m, v0 , k, z, h, and g.

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c. o 2. Design a synchronous sequential circuit which detects the occurrence m
Figure 1: Figure for Question E1.

m s e of at least
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three 1's arriving at the input. The output will go high after it receives three 1's

a
(not necessarily in consecutive clock periods) at the input, and
receives three 0's (not necessarily in consecutive clock periods) at the input. When
it detects three 0's, the circuit should return to the reset state and start looking
again for three 1's. Use J-K flip-flops and other basic logic gates as necessary.

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3. A metal rod of length 1 m and uniform cross-section is suspended from ceiling
by one of its ends. Assume that the density of the metal is 9000 kg/m3 , Young’s
modulus is 1.5×1011 N/m2 , and Posisson’s ratio is 0.3. Given that the gravitational
acceleration g = 10 m/s2 , calculate the fractional change in the volume of the rod
(i.e. the volume strain) due to its own weight.

4. (a) A Carnot engine absorbs 480 J heat per cycle. The engine operates between
a hot source at 320 K and a cold sink at 280 K. What is the work done per
cycle output of the engine?
(b) Assume now that the above engine operates as a refrigerator. Also assume
that 700 J heat per cycle is to be removed from the refrigerator. What is the
work done per cycle supplied to the refrigerator?
(c) The inlet and outlet temperatures of a pipe carrying a liquid are 250 K and
260 K respectively. The flow rate of the liquid is 5 kg/s. The pipe is in
an ambient temperature of 300 K. The temperature increase at the outlet
of the pipe is due to poor insulation and there is no other loss. Calculate
irreversible energy associated with the heat leakage due to poor insulation.
Assume specific heat of the liquid is 3 kJ/kg K and ln(1.04) = 0.04.

5. A thin and rigid horizontal rod, fixed at both ends, passes through a small hole
made in a uniform disc of radius r. The disc remains suspended vertically from the
rod and oscillates with small angular amplitude about the rod. Derive expressions
for:

(a) the distance of the hole from the center of the disc for which the time period
of such small oscillations is minimum;
(b) the corresponding minimum time period.

6. A piston can move slowly inside a horizontal cylinder closed at both ends. Initially
the piston divides the space inside the cylinder into two equal parts each of volume
10 liters and containing an ideal gas at the same pressure 10 kPa. What amount
of work has to be done by an external agent in order to isothermally increase the
volume of one part of the gas to 9 times the other part, by slowly moving the
piston? Assume that ln(5/3) = 0.511.

7. Using J-K flip-flops, design a 4-bit Möbius counter which begins with all the output
bits as 0’s. Starting from the left, these bits toggle to 1, one after another, till all
the output bits are 1’s. Then these bits become 0, one after another, from the left
till all are 0’s.

(a) Provide the state transition table.
(b) Derive the excitation functions (in the minimized form) of the flip-flops.
(c) Draw the corresponding circuit.

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8. In the figure below, n non-identical cells of emfs e, 2e, · · · , ne and the respective
internal resistances r, r/2, · · · , r/n are connected in parallel to deliver a current i
to an external resistance R. Derive an expression for the current i. What is the
value of i when n >> 1 and R = 2n + 1?

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9. (a) A sequential circuit has two input lines x1 and x2 and one output line y. The
value of y is 1 if the sequence 011 is fed on the x1 line while x2 remains 1

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for all 3 cycles. Once y is 1, it remains so until x2 becomes 0. Construct a

(b) (i) Draw the Karnaugh-map of an. irreducible four-variable Boolean func-
minimum-row state table for this circuit.

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tion f (x , x , x , x ) whosessum-of-products
a
l of minterms.
representation consists of the

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1 2 3 4

a Boolean function with n variables,
maximum possible number
(ii) Hence prove that no when expressed
in sum-of-products form, requires more than 2 n−1
product terms.

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