Implication and contrapositive reasoning: “If Rajesh studies, he will pass his exam.” Rajesh fails the exam. Based only on this information, determine the logically valid conclusion about whether Rajesh studied or not.

Introduction / Context:
Many verbal reasoning questions test the truth-preserving rules of implication. The statement “If Rajesh studies, he will pass” is a conditional of the form P -> Q. We are also told Rajesh failed (not Q). The task is to decide what follows about whether he studied (P).

Given Data / Assumptions:

• Rule: If study then pass (study -> pass).
• Observed outcome: Rajesh fails (not pass).
• No extra information beyond these two statements.

Concept / Approach:
Use modus tollens, the contrapositive rule: from (P -> Q) and (not Q) we infer (not P). The contrapositive of “study -> pass” is “not pass -> not study.” This is logically equivalent to the original conditional and always valid.

Step-by-Step Solution:

Verification / Alternative check:
Try truth-table intuition: The only way a true conditional (P -> Q) can coexist with an observed failure of Q is when P is false. If P were true while Q is false, the conditional would be violated. Hence P must be false.

Why Other Options Are Wrong:

• “Rajesh studied…” would make P true while Q is false, contradicting the conditional.
• “May or may not have studied” ignores the contrapositive; it is too weak.
• “None of these” is incorrect because a definite conclusion exists (not P).

Common Pitfalls:
A frequent error is treating conditionals as biconditionals (“pass only if and if and only if study”). We are not told that passing implies studying; we are told that studying implies passing. However, failing does imply not studying via the contrapositive.

Final Answer:
Rajesh did not study for the exam.