UPSC CSE Prelims 2022

Correct Option (Option 4):

The question asks whether x is an integer. To determine sufficiency, we must ascertain if the given statements, individually or combined, provide a definitive 'yes' or 'no' answer to this question.

• Analysis of Statement-1: $\frac{x}{3}$ is not an integer.

  • If x = 7, then $\frac{x}{3}=\frac{7}{3}$, which is not an integer. In this instance, x is an integer.
  • If x = $\frac{7}{3}$, then $\frac{x}{3}=\frac{7}{9}$, which is not an integer. In this instance, x is not an integer.

  Since Statement-1 permits scenarios where x is an integer and where x is not an integer, it is not sufficient to answer the question.

• Analysis of Statement-2: 3x is an integer.

  • Let 3x = k, where k represents an integer. This implies that $x=\frac{k}{3}$.
  • If k = 21 (an integer), then $x=\frac{21}{3}=7$. In this instance, x is an integer.
  • If k = 7 (an integer), then $x=\frac{7}{3}$. In this instance, x is not an integer.

  Since Statement-2 also allows for both possibilities, it is not sufficient to answer the question.

• Combined Analysis of Statement-1 and Statement-2:

  • From Statement-2, we establish that $x=\frac{k}{3}$ for some integer k.
  • Substituting this into Statement-1 yields $\frac{x}{3}=\frac{k/3}{3}=\frac{k}{9}$.
  • Statement-1 asserts that $\frac{k}{9}$ is not an integer. This condition implies that k is not a multiple of 9.
  • Now, we evaluate whether x (which is $\frac{k}{3}$) is an integer, given that k is an integer and not a multiple of 9.
    • Consider k = 21. Here, k is an integer and not a multiple of 9. Then $x=\frac{21}{3}=7$, which is an integer.
    • Consider k = 7. Here, k is an integer and not a multiple of 9. Then $x=\frac{7}{3}$, which is not an integer.

  As both statements combined still lead to scenarios where x can be an integer or not an integer, they are not sufficient to provide a definitive answer to the question.

Therefore, both Statement 1 and Statement 2 are not sufficient to answer the question.

Incorrect Options:

• Option 1: Statement 1 alone is sufficient to answer the question.

  This is incorrect. As demonstrated, Statement 1 allows for x to be an integer (e.g., x=7) and for x not to be an integer (e.g., x=$\frac{7}{3}$). Thus, it does not provide a conclusive answer.

• Option 2: Statement 2 alone is sufficient to answer the question.

  This is incorrect. As demonstrated, Statement 2 allows for x to be an integer (e.g., x=7) and for x not to be an integer (e.g., x=$\frac{7}{3}$). Thus, it does not provide a conclusive answer.

• Option 3: Both Statement 1 and 2 are sufficient to answer the question.

  This is incorrect. Even when both statements are considered together, counterexamples exist (x=7 and x=$\frac{7}{3}$) that satisfy both conditions but yield different answers to the question. Therefore, the combined statements are not sufficient.