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 2
MCQ (Single Correct Answer)
+1
-0.33

Despite his initial hesitation, Rehman's __________ to contribute to the success of the project never wavered.

Select the most appropriate option to complete the above sentence.

A
ambivalence
B
  satisfaction
C
resolve
D
revolve
4
GATE CSE 2025 Set 2
MCQ (Single Correct Answer)
+1
-0.33

Bird : Nest :: Bee : ________

Select the correct option to complete the analogy.

A
Kennel
B
Hammock
C
Hive
D
Lair
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