Directions : Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. What is the value of the two-digit number $ab$? I. The difference between its digits is 2. II. The sum of its digits is 4.

Concept & Logic In a two-digit number $ab$, knowing the sum and absolute difference of the digits allows you to find the values of the individual digits. However, unless the order or relative size of the digits is specified, you cannot determine which digit belongs in the tens place and which belongs in the units place. Step-by-Step Solution * **Evaluating Statement I:** The difference between the digits is $2$. This means $|a - b| = 2$. It could be $a - b = 2$ or $b - a = 2$. Possible pairs include $(3,1), (1,3), (4,2), (2,4) \dots$ This yields multiple possible numbers ($31, 13, 42, 24 \dots$). Statement I is not sufficient. * **Evaluating Statement II:** The sum of the digits is $4$. $$a + b = 4$$ Possible pairs include $(4,0), (3,1), (1,3), (2,2)$. This yields multiple numbers ($40, 31, 13, 22$). Statement II is not sufficient. * **Evaluating Both Statements Together:** We know $a + b = 4$ and $|a - b| = 2$. If $a > b$, we solve $a + b = 4$ and $a - b = 2$. Adding them gives $2a = 6 \implies a = 3$ and $b = 1$. The number would be $31$. If $b > a$, we solve $a + b = 4$ and $b - a = 2$. Adding them gives $2b = 6 \implies b = 3$ and $a = 1$. The number would be $13$. Because the problem does not specify whether $a$ or $b$ is larger, the number could be $31$ or $13$. Exam Strategy & Shortcut (MANDATORY) When a problem asks for an exact two-digit number but only provides symmetrical equations (like sum and absolute difference), you will always end up with two interchangeable digit values. Without an asymmetrical condition (e.g., "the units digit is larger" or "the number is even"), the data will never be sufficient. Spotting this symmetry saves you from doing the algebra. Common Pitfall (MANDATORY) The most common mistake is assuming that $a - b = 2$ is the only interpretation of "the difference between its digits". Students often forget the $b - a = 2$ case, wrongly concluding that $31$ is the single unique answer and incorrectly marking (e) instead of (d). Final Answer **Therefore, the correct answer is that the data even in both Statements I and II together are not sufficient to answer the question.**