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 area of the rectangle ? I. The difference between the sides is 5 cm. II. The measure of its diagonal is 10 cm.

### Concept & Algebraic Identities The area of a rectangle is length $\times$ breadth ($lb$). You can find $lb$ if you know $(l-b)$ and $(l^2 + b^2)$ using the algebraic identity: $$ (l-b)^2 = l^2 + b^2 - 2lb $$ ### Step-by-Step Solution * Let the length be $l$ and breadth be $b$. We need to find $l \times b$. * **Analyze Statement I:** Difference between sides is 5 cm. $l - b = 5$. This alone is insufficient as many pairs of numbers have a difference of 5. * **Analyze Statement II:** Diagonal is 10 cm. Using Pythagoras theorem, $l^2 + b^2 = 10^2 = 100$. This alone is insufficient. * **Combine I and II:** We know $l - b = 5$ and $l^2 + b^2 = 100$. * Square the first equation: $(l - b)^2 = 5^2 \implies l^2 + b^2 - 2lb = 25$. * Substitute $l^2 + b^2 = 100$: $100 - 2lb = 25 \implies 2lb = 75 \implies lb = 37.5$. * Since we can find the area ($lb$) using both pieces of information, both statements are necessary. ### Exam Strategy & Shortcut Recognize that diagonal gives the sum of squares ($l^2 + b^2$), perimeter gives the sum ($l+b$), and side difference gives ($l-b$). Any two of these three pieces of information will always allow you to find the area ($lb$) via standard algebraic expansion identities. ### Common Pitfall Trying to solve for the exact individual values of $l$ and $b$ using the quadratic formula. It takes too much time and isn't necessary—you only need their product for the area! ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question**.