ISI Admission Test 2024 Question Paper M.Tech(CS), M.Tech(CrS) PCA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2024
Question Paper ·
M.Tech(CS),
M.Tech(CrS) PCA
EXAM YEAR TYPE

ISI Admission Test 2024 Question Paper
SUBJECT

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

Notes · Sample Papers · Previous Year Papers · Mock Tests

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The following notation is used throughout the question paper.

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• For a real number x, [x] denotes the greatest integer  x. For

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example, [ 5] = 2
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. the set of real numbers. s e
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• R denotes
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a•s Q denotes the set of rational numbers. ag
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a • N denotes the set of natural numbers.
• |S| denotes the cardinality of a set S.

2 ↵3 3
1. Let ↵, be roots of the equation x +3x+1. Then + equals

om321.
3 ↵3

(A) 320.
c. (D)
(B)
(C) 322.
s em 323.
l a
agstatements is the negation of “for all sets
2. Which of the following
X and for all sets Y , if X ✓ Y , then X \ Y = ;”?

(A) For all sets X, there is a set Y such that if X ✓ Y then
X \ Y = ;.
(B) There is a set X such that for all sets Y , if X ✓ Y then

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X \ Y = ;.

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(C) There are sets X and Y for which X ✓ Y and X \Y = ;.
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(D) There are sets X and Y for which X ✓ Y and X \Y 6= ;.

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3. Let n = aaaaaaaaabcd be a 12-digit number divisible

a where a 6= 0 and a, b, c, d need not be distinct. How many such
numbers are possible?

(A) 180 (B) 189
(C) 198 (D) 207

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4. The rank of the matrix
0 1
0 1 0 0 0
B C
B0 1 2 0 0 C
B C
B0 0 0 3 0 C
B C
B C
@0 1 0 a 0 A
1 0 0 0 b

(A) depends on the value of a but not on the value of b.
(B) depends on the value of b but not on the value of a.
(C) depends on the values of both a and b.
(D) is independent of the values of both a and b.

5. Express the output f(n) of the following code fragment as a
mathematical function of n (where n is a natural number).

f(n){
if n is 0
return 0;
return 2 * f(n-1) + 1;
}

(A) f(n) = 2n+1 2 (B) f(n) = 2n 1
n 1
(C) f(n) = n! 1 (D) f(n) = n · 2

6. Let p(x) := ax2 + bx + c be a polynomial where a, b, c are drawn
uniformly at random with replacement from the set {0, 1, 2}.
What is the probability that the equation p(x) = 0 has two
distinct real roots?
2 4
(A) 27
(B) 27
1 21
(C) 27
(D) 27

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7. Let a < 500 be a positive integer. Let there be a box containing

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balls numbered a, a + 1, . . . , 500. Suppose that for each x, a 

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x  500, the probability that the ball numbered x is picked is

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s e m
m
e a
.

s l
(500 + a)(500 a + 1)

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l Then the value of a is ag
g
a (A) 1 (B) 2
(C) 3 (D) 4

8. For a function f : R ! R and a subset A of R, define

f (A) = {f (x) | x 2 A} and

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f 1 (A) = {x 2 R | f (x) 2 A}.

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l\af (Y ) for all subsets X, Y of R.
Consider the following statements.

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(I) f (X \ Y ) = f (X)
(II) f 1 (X \ Y ) = f 1 (X) \ f 1 (Y ) for all subsets X, Y of R.

Which of the following is correct?

(A) (I) is always true; (II) need not be true.
(B) (II) is always true; (I) need not be true.
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(C) Both (I) and (II) are always true.

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(D) Neither (I) nor (II) is always true.

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gla 9. Let A be a 4 ⇥ 4 upper triangular matrix with adiagonal entries
a ( 1, 1, 12 , 1
2
). Then A 1 equals

(A) 5A 4A3 (B) 5A 4A3
(C) 4A + A2 (D) A2 + A3

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10. A connected component of a simple undirected graph G = (V, E)
is a subset of vertices U ✓ V such that between any pair of
vertices in U there is a path between these two vertices. Let
|V | = n and |E| = m with 0  m  (n 1). Then the number
of connected components in G is

(A) at least (n m)
(B) at most (n m)
(C) exactly (n m)
(D) none of the above

11. Consider the following routine for driving a robot:

Drive(n) {
if n < 2
STOP
else {
Move NORTH n kms.
Move WEST n kms.
Move SOUTH n kms.
Move EAST (n-1) kms.
Move NORTH 1 km.
Drive(n-2)
}
}

Let M(n) denote the distance (in kms) traversed by the robot
when we call Drive(n). What is the value of M(n) when n 2 N?
⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤
(A) 4 · n2 · n+32
(B) 4 · n2 · n+2 2
⇥ n ⇤ ⇥ n+2 ⇤ ⇥ n ⇤ ⇥ n+3 ⇤
(C) 3 · 2 · 2 (D) 3 · 2 · 2

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12. Let a, b and c be the sides of a triangle such that c2 = a2 +b2 ab.

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Then which of the following is always true?

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(A) a 6 c and b 6 c

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m a 6 c 6 b or b 6 c 6 a s
(B) a > c and b > c

se(D) None of the above. g l a
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(C)

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13. For any non-negative integers x and y, consider the following
function

f(x,y){
if y is 0

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return 0;
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else
if y is even
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return 2f(x, [y/2]);

return 2f(x, [y/2]) + x;
}

Which of the following is true:

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(A) f(x+1, y+1) = 2f(x,y) + x + y

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(B) f(x+1, y+1) = 2f(x,y) + x + y + 1

c. o (C) f(x+1, y+1) = f(x,y) + x + y
se
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(D) f(x+1, y+1) = f(x,y) + x + y + 1

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ag a 1
!
14. If the matrix A = has 1 as an eigenvalue, then the
2 3
determinant of A is

(A) 2 (B) 3
(C) 4 (D) 5

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15. Let f (x) = ax + b for some a, b 2 R. Define fn (x) inductively
by setting
f1 (x) = f (x)
and
fn+1 (x) = f (fn (x)) for n > 1.
If f7 (x) = 128x + 381, then ab equals
1 1
(A) 32
(B) 8
(C) 8 (D) 32

R a+1
16. For a real number a, the value of a
[x]dx is

(A) 1 (B) a
(C) [a] (D) [a] + 1

17. Diplomats of 4 countries C1, C2, C3 and C4 are having some
serious negotiation with a fifth country C5. If country C1 buys
fighter jets from country C5, then country C2 cannot buy heli-
copters from country C5. If country C3 sells cars to country C5,
then country C5 will buy petroleum from country C4. Now if
country C1 does not buy fighter jets from country C5 or country
C5 buys petroleum from country C4 then the people of country
C4 are happy.
Which of the following is necessarily TRUE?

(A) If the people of country C4 are happy then it means
country C5 buys petroleum from country C4.
(B) If country C5 buys petroleum from country C4 then
country C1 buys fighter jets from country C5.
(C) If the people of country C4 are not happy then coun-
try C2 does not buy helicopters from country C5 and
country C3 does not sell cars to country C5.
(D) None of the above.

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18. The coefficient of a6 b4 c2 in the expansion of (ac + bd)6 (1 + ad )8 is

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12⇥6!8! 11!
(A) (B)

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11!
(C) 1551 (D)
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3!4!5!

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l at the coordinates (0, 0), (1, 4) and (4, 0), respectively. A pave-
19. Let ABC be a triangular park with vertices A, B and C located
a g
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a ment of uniform width 1 unit has to be built all around the
park as shown in the figure below. What is the total area of the
pavement?

B

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1 unit

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a
1 unit
1 unit
A 1 unit
C

p p
(A) (9 + 17) + ⇡ (B) (9 + 17) + 2⇡
p p

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(C) (17 + 17) + ⇡ (D) (17 + 17) + 2⇡

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20. Let a 2 R be such that

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1 2a |a| 3+ 3 |a|

g
n=
a
4+a 3 a
is an integer. Then the value of n is

(A) 1 (B) 1
(C) 5 (D) 7

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21. Let f : R ! R be the function defined by
(
x2 if x 2 Q
f (x) =
3x 2 if x 2 /Q
Then

(A) f is not continuous at any real number.
(B) f is continuous exactly at x = 1 and x = 2.
(C) f is continuous at all rational numbers.
(D) f is continuous exactly at x = 1 and x = 2.

22. Suppose P is a real polynomial of degree 5 such that the equation
P (x) = 0 has exactly 4 distinct real roots ↵1 , ↵2 , ↵3 , ↵4 . Let P 0
denote the derivative of P . The equation

P 0 (x) = 0

(A) has exactly 3 distinct real solutions.
(B) has exactly 4 distinct real solutions, none of which belongs
to {↵1 , ↵2 , ↵3 , ↵4 }.
(C) has exactly 4 distinct real solutions, at least two of which
belong to {↵1 , ↵2 , ↵3 , ↵4 }.
(D) has exactly 4 distinct real solutions, exactly one of which
belongs to {↵1 , ↵2 , ↵3 , ↵4 }.

23. Let A be the centre of the circle x2 + y 2 4x 6y 12 = 0.
Tangents are drawn at the points B = (2, 8) and D = (5, 1)
on the circle. Those tangents meet at C. Then the area of the
quadrilateral ABCD is

(A) 60. (B) 75.
(C) 90. (D) 105.

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24. The sum of all 3-digit positive numbers which are divisible by 7

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but not by 14 is

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(A) 62 ⇥ 548. (B) 63 ⇥ 547.

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(C) 64 ⇥ 546. (D) 65 ⇥ 545.

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l25. Let a < b be real numbers. Let f be a real-valued continuous a
a function on [a, b] which is di↵erentiable on (a, b) and which sat-
isfies the condition f (a) = f (b) = 0. Which of the following is
true?

(A) For every ↵ 2 R, there exists x0 2 (a, b) such that ↵f (x0 ) +
f 0 (x0 ) = 0.

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(B) For ↵ 2 (0, 1), there does not exist x0 2 (a, b) such that

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↵f (x0 ) + f 0 (x0 ) = 0.

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then ↵ must be a lnon-zero integer.
0
(C) If there exists x 2 (a, b) for which ↵f (x ) + f (x ) = 0,

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0 0 0

a
(D) If there exists x 2 (a, b) and ↵ 2 R for which ↵f (x ) +
0 0
0
f (x0 ) = 0, then ↵ must be zero.

26. Consider an equilateral triangle ABC where each side is 1 unit

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long. Let be the collection of points lying inside or on the

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boundary of ABC. Which of the following statements is true?

c. o (A) Any set of 5 points in will have two points whose e
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distance

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(B) Any set of 5 points in will have two pointsawhose distance
is at most 1/2 unit.

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is strictly less than 1/2 unit.
(C) Any set of 6 points in will have two points whose distance
is strictly less than 1/2 unit.
(D) None of the above.

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27. Consider the regular expressions R1 = a⇤ (ba⇤ ba⇤ )⇤ and
R2 = a⇤ (ba⇤ b)⇤ a⇤ over the alphabet {a, b}. Let L(Ri ) denote the
languages corresponding to the regular expressions Ri , i = 1, 2.
Consider the string w = bbaaabb. Which of the following
statements is correct?

(A) w 2 L(R1 ) and w 2 L(R2 ).
(B) w 62 L(R1 ) and w 2 L(R2 ).
(C) w 2 L(R1 ) and w 62 L(R2 ).
(D) w 62 L(R1 ) and w 62 L(R2 ).

28. What is the minimum value of ✏ such that the inequality
x+✏ 2x
1 x ✏ 1 x
holds for all real x 2 [0, 1/2]?
p
(A) 1/2 (B) 3 2 2
p
(C) 0 (D) 3 + 2 2

p p
x+8 8x + 1
29. The value of limx!1 p p is
5 x 7x 3
1 2
(A) 2
(B) 3
7
(C) 12
(D) undefined

30. Let T (n) be the count of n-digit numbers (using digits 0, 1, . . . ,
9, with first digit non-zero) such that no three consecutive digits
are the same. Then which of the following is a correct recurrence
relation for T (n)? Note that T (1) = 9 and T (2) = 90.

(A) T (n) = 9T (n 1) + 9T (n 2)
(B) T (n) = 9T (n 1) + 8T (n 2) + 9
(C) T (n) = 9T (n 1) + 9
(D) T (n) = 9T (n 1) 9T (n 2)

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