ISI Admission Test 2023 Sample Paper M.Tech(CS), M.Tech(CrS) PCA

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ISI Admission Test 2023
Sample Paper ·
M.Tech(CS),
M.Tech(CrS) PCA
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2023 Sample Paper M.Tech(CS), M.Tech(CrS) PCA

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Sample Questions (PCA)
1. A person throws a pair of fair dice. If the sum of the numbers on

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the dice is a perfect square, then the probability that the number 3

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appeared on at least one of the dice is

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(A) 1/9 (B) 4/7 (C) 1/18 (D) 7/36

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l2. Consider the system of linear equations: x+y+z = 5, 2x+2y+3z = 4. a
a Then
(A) the system is inconsistent
(B) the system has a unique solution
(C) the system has infinitely many solutions
(D) none of the above is true

om x g(x ) − RR xg(x )dx + C
3. If g ′ (x) = f (x) then x3 f (x2 )dx is given by
R

c. (B)
em (D) x g(x ) − xg(x )dx + C
1 2
(A) x2 g(x2 ) − xg(x2 )dx + C 2 2
R

s
2

a
1
(C) 2x2 g(x2 ) − xg(x2 )dx + C 2 2 2
R

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2

4. If (n C n n n
a g
n n n n n
0 + C1 )( C1 + C2 ) · · · ( Cn−1 + Cn ) = k C0 C1 · · · Cn−1 ,
then k is equal to

(n+1)n nn (n+1)n (n+1)n+1
(A) n! (B) n! (C) nn! (D) n!

5. Let S be a set of n elements. The number of ways in which n
distinct non-empty subsets X1 , ..., Xn of S can be chosen such that

m
X1 ⊆ X2 · · · ⊆ Xn , is

m c. o
c. o m
n

n

· · · nn (D) 2n


(A) (B) 1 (C) n!

e
1 2

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6. Let A be a 4 × 4 matrix such that both A and Adj(A) are non-null. If

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det A = 0, then the rank of A is

a (A) 1 (B) 2 (C) 3 (D) 4

7. Suppose a, b, c are in A.P. and a2 , b2 , c2 are in G.P. If a < b < c and
a + b + c = 32 , then the value of a is

1 1 1 √1 1 √1
(A) √
2 2
(B) − 2√ 2
(C) 2 − 3 (D) 2 − 2

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8. The set of all a satisfying the inequality
Z a
1 3√ 1 
√ x + 1 − √ dx < 4
a 1 2 x

is equal to the interval

(A) (−5, −2) (B) (1, 4) (C) (0, 2) (D) (0, 4)

9. Let C0 be the set of all continuous functions f : [0, 1] → R and C1
be the set of all differentiable functions g : [0, 1] → R such that the
derivative g ′ is continuous. (Here, differentiability at 0 means right
differentiability and differentiability at 1 means left differentiability.)
If T : C1 → C0 is defined by T (g) = g ′ , then

(A) T is one-to-one and onto
(B) T is one-to-one but not onto
(C) T is onto but not one-to-one
(D) T is neither one-to-one nor onto.

10. The number of distinct even divisors of
5
Y
k!
k=1

is

(A) 24 (B) 32 (C) 64 (D) 72

11. A straight line passes through the intersection of the lines given by
3x − 4y + 1 = 0 and 5x + y = 1 and makes equal intercepts of the same
sign on the coordinate axes. The equation of the straight line is

(A) 23x − 23y + 11 = 0 (B) 23x − 23y − 11 = 0
(C) 23x + 23y + 11 = 0 (D) 23x + 23y − 11 = 0

12. Suppose A and B are two square matrices such that the largest eigen-
value of (AB − BA) is positive. Then the smallest eigen value of
(AB − BA)

(A) must be positive (B) must be negative
(C) must be 0 (D) is none of the above

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13. A fair die is rolled five times. What is the probability that the largest

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number rolled is 5?

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(A) 5/6
om (B) 1/6 (C) 1 − (1/6)6 (D) (5/6)5 − (2/3)5
e m .
emseries X 3 · 6 · 9 · · · 3n x , x > 0 l a s
la s ag
14. The

g
n

a (A) converges for 0 < x ≤ 1 and diverges for x > 1
7 · 10 · 13 · · · (3n + 4)
n

(B) converges for all x > 0
(C) converges for 0 < x < 21 and diverges for x ≥ 12
(D) converges for 12 < x < 1 and diverges for 0 < x ≤ 12 , x ≥ 1.

15. Given a real number α ∈ (0, 1), define a sequence {xn }n≥0 by the

m
following recurrence relation:

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mof ℓ is
xn+1 = αxn + (1 − α)xn−1 , n ≥ 1.

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If limn→∞ x = ℓ then the value
n

αx0 +x1 (1−α)x0 +x1
(A) 1−α (B) 2−α
αx0 +x1 (1−α)x1 +x0
(C) 2−α (D) 2−α

16. Suppose that a 3 × 3 matrix A has an eigen value −1. If the matrix
A + I is equal to  
1 0 −2
 0 0 0 
 

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c. o
0 0 0

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then the eigen vectors of A corresponding to the eigenvalue −1 are in

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the form,

las ag2t 
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 
2t
(A)  0  , t ∈ R (B)  s  , s, t ∈ R
 
t t
   
t t
(C)  0  , t ∈ R (D)  s  , s, t ∈ R
   
−2t 2t

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3

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17. The number of real roots of the polynomial x3 − 2x + 7 is

(A) 0 (B) 1 (C) 2 (D) 3

18. Two rows of n chairs, facing each other, are laid out. The number of
different ways that n couples can sit on these chairs such that each
person sits directly opposite to his/her partner is

(A) n! (B) n!/2 (C) 2n n! (D) 2n!.

19. Consider two real valued functions f and g given by

x
f (x) = x−1 for x > 1, and g(x) = 7 − x3 for x ∈ R.

Which of the following statements about inverse functions is true?
(A) Neither f −1 nor g −1 exists
(B) f −1 exists, but not g −1
(C) f −1 does not exist, but g −1 does
(D) Both f −1 and g −1 exist.

20. Let  
a 1 1
A =  b a 1 .
 
1 1 1
The number of elements in the set
n o
(a, b) ∈ Z2 : 0 ≤ a, b ≤ 2021, rank(A) = 2

is

(A) 2021 (B) 2020 (C) 20212 − 1 (D) 2020 × 2021

21. Amir, Bhola, Chaitali and Deepak are invited to a party. If Bhola
and Chaitali attend, then Deepak will attend too. If Bhola does not
attend, then Amir will not attend. If Deepak does not attend, which
of the following is necessarily true?

(A) Chaitali does not attend (B) Amir does not attend
(C) Either (a) or (b), or both (D) None of the above

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22. Let G = (V, E) be an undirected simple graph, and s be a designated

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vertex in G. For each v ∈ V , let d(v) be the length of a shortest path

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between s and v. For an edge (u, v) in G, what can not be the value

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of d(u) − d(v)?

(A)m2
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(B) −1 (C) 0 (D) 1

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a 5?
23. Which of the following degree sets is NOT possible for a graph of order

(A) 0, 1, 2, 2, 3 (B) 2, 2, 2, 2, 2 (C) 2, 2, 2, 3, 3 (D) 2, 2, 3, 3, 3

24. Transposing the adjacency matrix of an undirected graph with n con-
nected components gives the adjacency matrix of a graph with

m
o (D) n components
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(A) n/2 components (B) (n − 1) components

m
(C) (n + 1) components

s e to take part in a cultural festival at a
neighbouring institution. l
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25. Your college has sent a contingent

Each event takes placeathrough the day with many elimination rounds.
Several team events are part of the program.

Your contingent is multi-talented and each individual has the skills
to take part in a subset of the events. However, the same individual
cannot be part of the team for two different events because of a possible
clash in timings. Your aim is to create teams to take part in as many
events as possible.
To do this, you decide to model the problem as a graph where the nodes

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are the events and edges represent pairs of events where the team that

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you plan to send shares a member. In this setting, the graph theoretic

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question to be answered is:

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(A) Find a spanning tree with maximum number of edges.

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(B) Find an independent set of maximum cardinality.

a (C) Find a vertex cover of minimum cardinality.
(D) Find a vertex colouring with minimum number of colours.

26. Which of the words below matches the regular expression a(a + b)∗ b +
b(a + b)∗ a?

(A) aba (B) bab (C) abba (D) aabb

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27. Let the regular expression of all the strings generated from English
lowercase letters, which starts with a vowel, and has odd number of
letters can be expressed as [a + e + i + o + u][a + b + · · · + z]n . Which
of the following expressions is correct about n?

(A) n ≥ 0 and n is odd (B) n ≥ 0 and n is even
(C) n > 0 and n is odd (D) n > 0 and n is even

28. Which of the following properties is always applicable to the numbers
generated by the regular expression 1[3 + 6 + 9]∗ 8?

(A) Odd (B) Divisible by 3 (C) Divisible by 8 (D) Divisible by 9

29. Assume that the Boolean operator & performs bitwise AND on the 8-
bit unsigned variables associated with it. If for some n > 0, we have n
& (n − 1) = 0, which of the following statements is necessarily true?

(A) n is even (B) n is odd
(C) n is a power of 2 (D) n is a power of 3

30. What does the following function compute in terms of n and d, for
integer values of n and d, n > 1, d > 1? Note that a//b denotes the
quotient (integer part) of a ÷ b, for integers a and b. For instance 7//3
is 2.

function foo(n,d){
x := 0;
while (n >= 1) {
x := x+1;
n := n//d;
}
return(x);
}

(A) The number of ways of choosing d elements from a set of size n.
(B) The number of ways of rearranging d elements from a set of size
n.
(C) The number of digits in the base d representation of n.
(D) The number of ways of partitioning n elements into groups of size
d.

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31. Suppose a machine generates an integer uniformly at random from

m
{0, 1, . . . , n} and whenever it is less than an integer k it prints “OK”.

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If it prints “OK” with a probability 0.5, what is the guaranteed relation

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between n and k?

(A)m k = n/2
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(B) k = (n − 1)/2

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a 32. Let the variables x and y vary over the set of natural numbers, N.
(C) k = (n + 1)/2 (D) Nothing can be inferred

Consider the formulas expressing properties about N: (i) ∀x∃y(x < y),
and (ii) ∃y∀x(x ≤ y). Which of the following statements is correct?
(A) Both (i) and (ii) hold.
(B) (i) holds but (ii) does not hold.
(C) (ii) holds but (i) does not hold.

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(D) Both (i) and (ii) do not hold.

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33. Let L be a regular language and F be a finite language over {0, 1}.

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Consider the statements: (i) L ∪ F is regular, and (ii) L ∪ F c is reg-

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ular, where F c denotes the complement of F. Which of the following
statements is correct?
a
(A) Both (i) and (ii) are true.
(B) (i) is true but (ii) is false.
(C) (ii) is true but (i) is false.
(D) Both (i) and (ii) are false.
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34. Let a, b, c be members of a Boolean algebra B. Define a → b = ¬a ∨ b.

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Which of the following statements is correct?

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(A) a → (b → c) = b → (a → c)

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(B) a → (b → c) = a → (c → b)

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(C) a → (b → c) = b → (c → a)

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(D) a → (b → c) = c → (a → b)

35. Let P be a set of propositional variables. Let Γ = {p → q, ¬r →
¬q, ¬s → ¬r, s → t | p, q, r, s, t ∈ P } be a set of propositional logic for-
mulas. Which of the following formulas is derivable from Γ in classical
propositional logic?

(A) s → q (B) r → p (C) p → t (D) s → p

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36. Let us consider the language {ϵ, a, a2 , . . . , a10 }, where ϵ denotes the
empty string, and an denotes aa · · · a}. What is the minimum number
| {z
n times
of states required by a DFA to accept this language?

(A) 5 (B) 10 (C) 11 (D) 12

37. Let A be the adjacency matrix of a directed acyclic graph G. Then
Akij (the value at position (i, j) of the matrix Ak ) denotes:
(A) the number of paths from i to j in G of length exactly k
(B) the number of paths from i to j in G of length at most k
(C) the number of vertices whose removal breaks all paths of length
at most k from i to j
(D) the number of vertices that are both reachable from i as well from
whom j is reachable using paths of length at most k

38. The diameter of a tree on n vertices in which every vertex has degree
either 1 or 3 is:

(A) at least 2 log3 (n + 1) and at most n2 .
(B) at least 2 log2 ( n+2 n
3 ) and at most 3 .
(C) at least 2 log2 ( n+2 n
3 ) and at most 2 .
(D) at least 2 log3 (n + 1) and at most n3 .

39. Consider the following program fragment:

1. i = 1;
2. while (i <= n) do
3. begin
4. sum = sum + a[i];
5. i = i + 1;
6. end

Let (i) A represent the initialization in line 1, (ii) T represent the
test implied by line 2, (iii) B represent the statement in line 4, and
(iv) I represent the increment in line 5. Which of the following
regular expressions represents all possible sequences of steps taken by
this program?

(A) AT (BIT )+ (B) AT (BIT )∗ (C) A(T BI)+ (D) A(T BI)∗

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40. What is the minimum number of elementary Boolean operations (AND

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/ OR / NOT) required to construct an equivalent Boolean expression

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of AB + AB + AC (X denotes the complement of X)?

(A) 1.co m
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(B) 3 (C) 5 (D) 7

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