1
GATE CSE 2025 Set 2
MCQ (More than One Correct Answer)
+2
-0

Let $\Sigma=\{a, b, c\}$. For $x \in \Sigma^{\star}$, and $\alpha \in \Sigma$, let $\#_\alpha(x)$ denote the number of occurrences of a in $x$. Which one or more of the following option(s) define(s) regular language(s)?

A
$\left\{a^m b^n \mid m, n \geq 0\right\}$
B
$\{a, b\}^* \cap\left\{a^m b^n c^{m-n} \mid m \geq n \geq 0\right\}$
C
$\left\{w \mid w \in\{a, b\}^*, \#_a(w) \equiv 2(\bmod 7)\right.$, and $\left.\#_b(w) \equiv 3(\bmod 9)\right\}$
D
$\left\{w \mid w \in\{a, b\}^*, \#_a(w) \equiv 2(\bmod 7)\right.$, and $\left.\#_a(w)=\#_b(w)\right\}$
2
GATE CSE 2025 Set 2
Numerical
+2
-0

Let $\Sigma=\{1,2,3,4\}$ For $x \in \Sigma^*$, let prod $(x)$ be the product of symbols in $x$ modulo 7 . We take $\operatorname{prod}(\varepsilon)=1$, where $\varepsilon$ is the null string.

For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$.

Define $L=\left\{x \in \Sigma^{\star} \mid \operatorname{prod}(x)=2\right\}$.

The number of states in a minimum state DFA for $L$ is _________ (Answer in integer)

Your input ____
3
GATE CSE 2025 Set 1
MCQ (Single Correct Answer)
+2
-0

Consider the following two languages over the alphabet $\{a, b\}$ :

$$\begin{aligned} & L_1=\left\{\alpha \beta \alpha \mid \alpha \in\{a, b\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \\ & L_2=\left\{\alpha \beta \alpha \mid \alpha \in\{a\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \end{aligned}$$

Which ONE of the following statements is CORRECT?

A
Both $L_1$ and $L_2$ are regular languages.
B
$L_1$ is a regular language but $L_2$ is not a regular language.
C
$L_1$ is not a regular language but $L_2$ is a regular language.
D
Neither $L_1$ nor $L_2$ is a regular language.
4
GATE CSE 2025 Set 1
MCQ (More than One Correct Answer)
+2
-0

Consider the following deterministic finite automaton (DFA) defined over the alphabet, $\Sigma=\{a, b\}$. Identify which of the following language(s) is/are accepted by the given DFA.

GATE CSE 2025 Set 1 Theory of Computation - Finite Automata and Regular Language Question 3 English

A
The set of all strings containing an even number of $b$ 's.
B
  The set of all strings containing the pattern bab.
C
The set of all strings ending with the pattern bab.
D
The set of all strings not containing the pattern aba.
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