ISRO CS 2025 Q.32: solution and answer

Official answer: ASource: ISRO's final answer key of 15 May 2026

1. A grammar is ambiguous when some string in its language has more than one parse tree. So statement (i), two different parse trees for the same string, is true.

2. Each parse tree corresponds to exactly one leftmost derivation. Two different trees for the same string therefore give two different leftmost derivations, so statement (ii) is true.

3. Each parse tree also corresponds to exactly one rightmost derivation. Two different trees therefore give two different rightmost derivations, so statement (iii) is true.

4. Statement (iv) does not hold in general for ambiguous grammars. Take $S\to A\mid B$, $A\to a$, $B\to a$, which is ambiguous on the string $a$. Both trees have a leftmost derivation identical to their rightmost derivation, since each step has only one variable to expand. So the leftmost and rightmost derivations need not differ, and (iv) is not a property of ambiguity. It is also not a test for it, because the two derivations of one tree usually differ even in unambiguous grammars. So (iv) is false.

5. So (i), (ii) and (iii) are true and (iv) is false, which is option (a). Option (b) wrongly denies (ii) and (iii). Option (c) wrongly denies (ii) and (iii) and wrongly accepts (iv). Option (d) denies (i), the defining property.