1
GATE CSE 2025 Set 1
MCQ (Single Correct Answer)
+2
-0

Let $A$ be a $2 \times 2$ matrix as given.

$$A=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$$

What are the eigenvalues of the matrix $A^{13}$ ?

A
$1,-1$
B
$2 \sqrt{2},-2 \sqrt{2}$
C
$4 \sqrt{2},-4 \sqrt{2}$
D
$64 \sqrt{2},-64 \sqrt{2}$
2
GATE CSE 2025 Set 1
MCQ (More than One Correct Answer)
+2
-0

Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: "Everyone has exactly one mother"?

The meanings of the predicates used are:

$\bullet$ mother $(y, x): y$ is the mother of $x$

$\bullet$ noteq $(x, y): x$ and $y$ are not equal

A
$\forall x \exists y \exists z$ (mother $(y, x) \wedge \neg \operatorname{mother}(z, x))$
B
$\forall x \exists y[\operatorname{mother}(y, x) \wedge \forall z(\operatorname{noteq}(z, y) \rightarrow \neg \operatorname{mother}(z, x))]$
C
$\forall x \forall y[\operatorname{mother}(y, x) \rightarrow \exists z(\operatorname{mother}(z, x) \wedge \neg \operatorname{noteq}(z, y))]$
D
$\forall x \exists y[\operatorname{mother}(y, x) \wedge \neg \exists z(\operatorname{noteq}(z, y) \wedge \operatorname{mother}(z, x))]$
3
GATE CSE 2025 Set 1
MCQ (More than One Correct Answer)
+2
-0

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in \mathrm{~F}$ we define

$$\left(f_1 \odot f_2\right)(n)=f_1(n)+f_2(n)$$

for every number $n$ in $A$. Which of the following is/are CORRECT about the mathematical structure $(\mathrm{F}, \odot)$ ?

A
$(F, \odot)$ is an Abelian group.
B
$(F, \odot)$ is an Abelian monoid.
C
$(F, \odot)$ is a non-Abelian group.
D
$(F, \odot)$ is a non-Abelian monoid.
4
GATE CSE 2025 Set 1
Numerical
+2
-0

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is __________ ( (Rounded off to three decimal places)

Your input ____
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