Let L_1 and L_2 be two languages over a finite alphabet, such that L_1 L_2 and L_2 are regular languages. Which of the following statements is/are always true?

GATE CSE 2026 Session 1 Q51: What follows from L1 ∩ L2 and L2 being regular

Let $L_1$ and $L_2$ be two languages over a finite alphabet, such that $L_1 \cap L_2$ and $L_2$ are regular languages.

Which of the following statements is/are always true?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: C (official GATE 2026 final answer key)

Concept used: Regular languages are context-free; knowing $L_1 \cap L_2$ says nothing about $L_1$ outside $L_2$ .

First step: Try $L_2 = \emptyset$ : then both conditions hold for any $L_1$ , even an undecidable one.

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Source: official GATE 2026 Session 1 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key