ISI Admission Test 2025 Question Paper M.Tech(CS), M.Tech(CrS) PCA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2025
Question Paper ·
M.Tech(CS),
M.Tech(CrS) PCA
EXAM YEAR TYPE

ISI Admission Test 2025 Question Paper
SUBJECT

M.Tech(CS), M.Tech(CrS) PCA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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The following notations are used throughout the question paper.

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• R denotes the set of real numbers.

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• Z denotes the set of integers.
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a s
• N denotes the set of natural numbers.
l • C denotes the set of complex numbers. ag
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• For a real number x, �x� denotes the greatest integer ≤ x.

• For a real number x, �x� denotes the least integer ≥ x.

1. The roots of the equation x4 + x3 + x2 + x + 1 = 0 are

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(A) the vertices of a square. . c
(B) (some of) the verticesse
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g la of a regular hexagon.
of a regular pentagon.

a
(C) (some of) the vertices
(D) (some of) the vertices of a regular heptagon.

� � � �
2. Let C = { 12 , 21 } be a basis of R2 and T : R2 → R2 be
� � � x+y �
defined by T xy = x−2y . If T (C) represents the matrix of
T with respect to the basis C, then which of the following is true?

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� −3 −2 � � 3 −2 �

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(A) T (C) = (B) T (C) =

se
3 1
� −3 � � −3 1
�

em la
3 −1
(C) T (C) = 3 −1 (D) T (C) = −3 2

s
2

la g
a are not
ag 3. Which of the following pairs of regular expressions
equivalent?

(A) xx� and x� x (B) x(yx)� and (xy)� x
(C) x(xx)� and (xx)� x (D) (xy)� and x� y � .

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4. Consider the disc on the (x, y) plane with centre at (0, 1) and
radius 1 unit. Let A be the highest point on the disk, that is,
(0, 2). If this disk rolls on the x-axis such that its centre is now
at (1, 1), then the new coordinates of A are

� �
(A) 1 + sin(1/2π), 1 + cos(1/2π)
� �
(B) 1 + sin(1), 1 + cos(1)
� �
(C) 1 + sin(1/2π), 1 − cos(1/2π)
(D) (1, 2)

5. Let f : [0, 1] → (0, ∞) be a continuous function. Define
�x
g(x) = 0 f (t) dt for 0 ≤ x ≤ 1. If g(1) = 1, the equation
g(x) = 1/2

(A) may not have a solution.
(B) may have more than one solution.
(C) will have at least two solutions.
(D) will have a unique solution.

6. Let A = �a1 a0 � and B = �b1 b0 � be two 2-bit numbers and
C = �c3 c2 c1 c0 � be a 4-bit number, where c0 = a0 , c1 = a1 ⊕ b0 ,
c2 = b1 ⊕ a1 b0 , and c3 = a1 b0 b1 (‘⊕’ denotes the XOR
operation). If X + Y and XY denote the arithmetic sum and
product (of X and Y ) respectively, which of the following is true?

(A) C = A + B (B) C = AB
(C) C = 2A + B (D) C = A + 2B

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7. Let A = {a1 , a2 , . . . , an } be a set of n natural numbers and

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let n be odd. For any finite set X ⊆ N, Max(X) and Min(X)

m .co
denote the maximum and minimum elements in X respectively.

c o
. returns the (|X| + 1)/2-th smallest element in X.
Let |X| denote the number of elements in X. If |X| is odd,
e m
m a s
seConsider the following pseudo-code, which takes as input the l
Median(X)

la set A and outputs a set S. ag
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1: function fun(A)
2: Y := {a1 , a2 , a3 }
3: m := Median(Y )
4: Y := Y \ {m}
5: for i = 4 to n do

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6: Y := Y ∪ {ai }

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7: m := Median(Y )
8: Y := Y \ {m}
l a s
9:
10:
S := Y
Return S a g
Which of the following is not correct?
(A) Median(A) = Median(A \ S)
(B) Max(A) = Max(S)

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(C) Min(A) = Min(S)

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(D) There is an x ∈ S, such that Min(A) < x < Max(A).
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em la se
s
√ √ √

la g
2 + 3 3 + ··· + n n
a
1+

g
8. The lim

a
n→∞ n

(A) is equal to 0. (B) is equal to 2.
(C) is equal to 1. (D) does not exist.

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9. Let A and B be two arrays of size n, each containing the
numbers {1, 2, . . . , n} in some order. The operations allowed on
the entries of the arrays are comparisons between two entries
and swapping two entries. Consider the best possible algorithm
to transform the array A to array B using the above two
operations only. The total number of operations in which the
above algorithm can perform this task in the worst case is

(A) a constant (B) n
(C) n! (D) none of these

10. For w ∈ C, let Re(w) and Im(w) denote the real and imaginary
parts of w, respectively. Suppose z ∈ C is such that |z| = 1.
Which of the following is necessarily true?

(A) Re(z) is rational implies that Re(z 2 ) is rational.
(B) Re(z 2 ) is rational implies that Re(z) is rational.
(C) Im(z) is rational implies that Im(z 2 ) is rational.
(D) Im(z 2 ) is rational implies that Im(z) is rational.

11. Suppose A is an n × n matrix for some n ≥ 2. Then An = 0 if
and only if

(A) A = 0 (B) all the eigenvalues of A are zero.
(C) A is singular. (D) Rank(A) ≤ 1

12. Let A1 and A2 be two arbitrarily chosen subsets of {1, 2, . . . , n}.
The number of ways such that A1 ∩ A2 = ∅ is

(A) 2n (B) 4n − 3n (C) 3n (D) 3n − 1

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13. The following program Check takes as input a graph and an

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integer and outputs True or False.

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1: function Check(G = (V, E), k)

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if for all vertices u and v in V , edge (u, v) ∈ E then
e
2:

a s g l
a
if the number of vertices is at least k then
l
3:

ag 5:
4: Return True
else
6: Return False
7: Let u and v be vertices in V such that (u, v) �∈ E
8: A := V \ {v}
9: Return Check(A, k)

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c
Which of the following is true about the program?

m . G and an integer k such that
s ek) never halts.
(A) There exists an input graph

a
gl G and integers k, the program
the program Check(A,

a
(B) For all input graphs
Check(A, k) outputs True if and only if there are at least k
vertices in the graph G.
(C) For all input graphs G and integers k, the program
Check(A, k) outputs True if and only if there is a subset
S of k vertices in the graph and all the vertices in S are

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connected to each other by an edge.

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(D) For all input graphs G and integers k, the program

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em k vertices in the graph, all the vertices in S areaconnected
Check(A, k) outputs True if and only if for all subsets

s g l
gla a
a
to each other by an edge.

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14. The maximum value of the function f (x) = x2 − 1+x
1
2 on the set

{x ∈ R : x(x − 1) ≤ 12} is

(A) 0 (B) 33/17 (C) −8/5 (D) −1

15. If P and Q are two propositions, then the expression

(P ∧ (P → Q)) → Q is

(A) always true. (B) true for only one assignment.
(C) always false. (D) false for only one assignment.

16. Let n ≥ 2. Suppose α1 , α2 , · · · , αn are real numbers such that
α1 < α2 < · · · < αn . Let

P (x) = (x − α1 )2 (x − α2 ) · · · (x − αn ), x ∈ R .

If P � is the derivative of P , then P � (x) has
(A) no real root.
(B) n real roots, the smallest of which is α1 .
(C) n − 1 real roots, the smallest of which is α1 .
(D) n real roots, the smallest of which is strictly larger than α1 .

17. Let p(x) be a polynomial in x of degree k ≥ 0. For n = 1, 2, . . . ,
define an = p(n + 1) − p(n). If {an }∞
n=1 is in AP, then we must
have
(A) k ≤ 1 (B) k ≤ 2 (C) k ≥ 3 (D) k ≥ 4

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18. Let Sn = k=1 k . Then

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(A) S2n ≥ n2 for every n ≥ 1

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(B) Sn is a bounded sequence.

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a a
(C) |S2n − S2n−1 | → 0 as n → ∞

g l (D)
a
Sn
n
→ 1 as n → ∞

19. If
20212024 = 100 m + n ,

for some m ∈ {0, 1, 2, . . .} and n ∈ {0, 1, . . . , 99}, then the value
of n is

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c. (C)
em
(A) 21 (B) 31 (D) 81

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20. Let G be a simple undirected graph. For a vertex u ∈ V (G),
let NG (u) denote the set of neighbours of u in G, i.e.,
NG (u) = {v : uv ∈ E(G)}. Let H be the undirected graph
having V (H) = V (G) and E(H) = {uv : NG (u)∩NG (v) �= ∅}. If
G is a connected bipartite graph, then the number of connected
components of H:

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(A) is exactly one (B) can be either one or two

c. o (C) is exactly two
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(D) can be any positive integer
s
s em 21. What is the maximum number of nonzero g la in the
gla a values

a adjacency matrix of a directed graph that has 36 vertices, but
no self-loops?

(A) 36 (B) 180 (C) 630 (D) 1260

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22. Let A and B be two real symmetric square matrices with the
property that both A and B have the same set of eigenvectors.
Then, which of the following is always true?

(A) A = B (B) AB = BA
−1
(C) A = B (D) AAT = BB T

23. Consider the following pseudo-code.

1: function f (n, m)
2: if m = 0 then
3: return 1
4: if m = 1 then
5: return n
6: Return f (n, � m2 �) · f (n − � m2 �, � m2 �)

Which of the following is true?

(A) f (n, m) = nm (B) f (n, m) = (n + 1 − m)m
� �m
n!
(C) f (n, m) = (n−m)! (D) f (n, m) = n(n−1)
n−m

24. The value of
� 1� 1
3/2
ey dy dx
0 x2

is
2
(A) (e − 1) (B) e − 1
3

2e 2e 3
(C) (D) −
3 3 2

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25. Consider the curves

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1 1

co
y1 (x) = x + x5 ,

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2 2

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1 3 1 7

s e
y2 (x) = x + x,

m a
2 2

s e 1 3 1 7
g l
l a a
y3 (x) = 1 + x + x .

g
2 2

a Which of the following is true?
(A) y1 and y2 meet at 2 points, y1 and y3 meet at 1 point.
(B) y1 and y2 meet at 3 points, y1 and y3 meet at 2 points.
(C) y1 and y2 meet at more than 3 points, y1 and y3 meet at 1

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point.

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(D) y1 and y2 meet at 3 points, y1 and y3 meet at 1 point.

s em expressions is equivalent to the
g a
lautomaton
26. Which of the following regular

a
non-deterministic finite shown in the figure below?

𝒂, 𝒃
𝒃 𝒃

𝒂

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𝒂 𝒃

las a g
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𝒂

(A) (a∗ b∗ a)∗ (� ∪ aa∗ b∗ ) (B) (aa∗ bb∗ a)(� ∪ aa∗ ∪ bb∗ )
(C) (aa∗ bb∗ a)(� ∪ aa∗ ∪ aa∗ bb∗ ) (D) (aa∗ bb∗ a)∗ (� ∪ aa∗ b∗ )

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27. The 26 letters of the alphabet are randomly permuted. What
is the probability that at least two of the three letters I, N, D
come next to each other?

72 3 3 3
(A) (B) (C) (D)
325 26 13 25 × 26

28. There are two coins in a box. One of the coins is a fair coin and
for the other, the probability of head is 23 . A coin is chosen at
random from the box and tossed 5 times. Given that at least 4
heads were obtained, what is the conditional probability that
the fair coin was chosen?

729 1 2 243
(A) 2521
(B) 2
(C) 3
(D) 275

29. For x, y ≥ 0, consider the regions A = {(x, y) ∈ R2 : x2 +y 2 ≤ 1}
and B = {(x, y) ∈ R2 : x4 + y 4 > 1}. Then

(A) A ⊂ B (B) B ⊂ A
(C) A ∩ B = ∅ (D) A ∪ B = R2

� √ �
30. Let S = { −ba√5 b a 5 : a, b ∈ Z}. The number of A ∈ S such
that A−1 is also in S is

(A) 2 (B) 4
(C) more than 4 but finite (D) infinite

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