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 two-digit number? I. Sum of the digits is 6. II. Digit in the ten's place is double the digit in the unit's place.

Concept & Logic To find a specific two-digit number, we need to uniquely identify both its tens digit and its units digit. A single linear equation with two variables will yield multiple possibilities, but a system of two distinct linear equations will pinpoint the exact values. Step-by-Step Solution * **Variables:** Let the tens digit be $x$ and the units digit be $y$. The two-digit number is expressed as $10x + y$. * **Evaluating Statement I:** The sum of the digits is 6. $$x + y = 6$$ Possible $(x, y)$ pairs are $(1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0)$. This gives multiple possible numbers (15, 24, 33, 42, 51, 60). Thus, Statement I alone is not sufficient. * **Evaluating Statement II:** The digit in the ten's place is double the digit in the unit's place. $$x = 2y$$ Possible $(x, y)$ pairs are $(2, 1), (4, 2), (6, 3), (8, 4)$. This gives multiple possible numbers (21, 42, 63, 84). Thus, Statement II alone is not sufficient. * **Evaluating Both Statements Together:** We have a system of two equations: 1) $x + y = 6$ 2) $x = 2y$ Substitute the value of $x$ from equation 2 into equation 1: $$2y + y = 6$$ $$3y = 6 \implies y = 2$$ Substitute $y = 2$ back into the second equation: $$x = 2(2) \implies x = 4$$ The tens digit is 4 and the units digit is 2. The number is exactly 42. Both statements are required. Exam Strategy & Shortcut (MANDATORY) In Data Sufficiency involving two unknown digits, look for two independent mathematical constraints. Statement I gives a sum, and Statement II gives a ratio. Sum and ratio together guarantee a unique solution. You can mark (e) immediately without calculating the final number 42. Common Pitfall (MANDATORY) A common mistake is testing only a few numbers and prematurely assuming a statement is sufficient. Always systematically check bounds (digits 0-9) to verify if multiple solutions exist before concluding a single statement is enough. Final Answer **Therefore, the correct answer is that the data in both Statements I and II together are necessary to answer the question.**