Understanding the Main Statement

The main statement is: "When she takes an examination, she clears it." This is a classic example of a conditional statement, which can be represented in the form "If P, then Q".

• Let P represent the statement: "she takes an examination".
• Let Q represent the statement: "she clears it".

Therefore, the main statement can be written logically as: If P, then Q, or $P \implies Q$.

Analyzing the Options and Implications

We need to find an ordered pair of statements (X, Y) such that X implies Y ($X \implies Y$) and both X and Y are logically consistent with the main statement ($P \implies Q$).

Let's represent the given options:

• A: "she took an examination" (P)
• B: "she did not take an examination" (Not P)
• C: "she cleared it." (Q)
• D: "she did not clear it." (Not Q)

Analyzing Option Pairs:

• Option 1: DA
  This suggests that "she did not clear it" (Not Q) implies "she took an examination" (P). The contrapositive of the main statement ($P \implies Q$) is "If Not Q, then Not P" (i.e., "If she did not clear it, then she did not take an examination"). Therefore, D does not imply A; it implies Not P. This pair is incorrect.
• Option 2: AC
  This suggests that "she took an examination" (P) implies "she cleared it" (Q). This is precisely what the main statement says ($P \implies Q$). Both P ("she took an examination") and Q ("she cleared it") are part of the original conditional statement and are thus consistent with it. This implication ($P \implies Q$) is valid and consistent with the main statement.
• Option 3: BC
  This suggests that "she did not take an examination" (Not P) implies "she cleared it" (Q). The main statement only tells us what happens *if* she takes the exam. It doesn't provide any information about whether she clears it or not if she *doesn't* take it. She might clear it for other reasons, or she might not. This implication is not guaranteed by the main statement. This pair is incorrect.
• Option 4: CD
  This suggests that "she cleared it" (Q) implies "she did not clear it" (Not Q). This is a logical contradiction (Q implies Not Q). This pair is incorrect.

Conclusion

The only ordered pair where the first statement implies the second, and both are logically consistent with the main statement "When she takes an examination, she clears it" ($P \implies Q$), is AC.

Statement A (P) implies Statement C (Q), which is the core of the given conditional statement.