Directions: Each of the questions below consists of a statement and/or a question and two statements labelled I and II given below it. You have to decide whether the data provided in the statements 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; and Give answer (e) if the data in both Statements I and II are necessary to answer the question. Is $x^2 : y^2 < 1$? (M.A.T., 2006) I. $(y - x) (x + y) = 40\%$ of $60 - 120\%$ of $20$ II. $x < y$

### Concept & Algebraic Identities and Inequalities The core question asks if $\frac{x^2}{y^2} < 1$, which simplifies to asking whether $x^2 < y^2$. A definitive "Yes" or a definitive "No" both count as sufficient data. ### Step-by-Step Solution * **Analyzing Statement I:** * Given equation: $(y - x)(x + y) = 40\%$ of $60 - 120\%$ of $20$. * Simplify the left side using the difference of squares identity: $y^2 - x^2$. * Simplify the right side: $(0.40 \times 60) - (1.20 \times 20) = 24 - 24 = 0$. * Equating both sides: $y^2 - x^2 = 0 \implies y^2 = x^2$. * If $x^2 = y^2$, then the ratio $\frac{x^2}{y^2} = 1$ (assuming $y \neq 0$). * The question asks if the ratio is strictly less than 1. Since it equals 1, the answer is a definitive "NO". * Because we can definitively answer the question, Statement I is sufficient. * **Analyzing Statement II:** * Given inequality: $x < y$. * Let's test with positive numbers: If $x = 2$ and $y = 3$, then $x^2 = 4$ and $y^2 = 9$. Here, $\frac{4}{9} < 1$ (Answer is Yes). * Let's test with negative numbers: If $x = -5$ and $y = 2$, then $x^2 = 25$ and $y^2 = 4$. Here, $\frac{25}{4} > 1$ (Answer is No). * Since we get contradictory answers depending on the values chosen, Statement II is not sufficient. ### Exam Strategy & Shortcut For Data Sufficiency Yes/No questions, remember that providing a solid, mathematical "NO" means the statement is 100% sufficient. You do not need the statement to prove the condition true, you only need it to resolve the question with certainty. ### Common Pitfall A classic trap in inequalities is forgetting about negative numbers. Statement II ($x < y$) tempts you to assume $x^2$ will always be less than $y^2$, which is only true if $|x| < |y|$. Plugging in a large negative number for $x$ quickly disproves this. ### Final Answer Therefore, the correct answer is **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**.