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. The area of a rectangle is equal to the area of a circle. What is the length of the rectangle ? I. The radius of the circle is equal to the breadth of the rectangle. II. The perimeter of the rectangle is 14 cm more than that of the circle.

### Concept & Algebraic Formulation Translate the geometric properties into algebraic equations to check if the target variable (length of the rectangle, $l$) can be isolated to a single numerical value. Let the rectangle have length $l$ and breadth $b$, and the circle have radius $r$. ### Step-by-Step Solution * **Given:** Area of rectangle = Area of circle. $$ l \times b = \pi r^2 $$ * **Analyze Statement I:** Radius ($r$) = Breadth ($b$). Substitute this into our given equation: $l \times r = \pi r^2 \implies l = \pi r$. This gives the length in terms of $r$, but not a definite numerical value. I alone is insufficient. * **Analyze Statement II:** Perimeter of rectangle = Perimeter of circle + 14. $$ 2(l + b) = 2\pi r + 14 \implies l + b = \pi r + 7 $$ This is a single equation with three variables. II alone is insufficient. * **Combine I and II:** From Statement I, we established $b = r$ and $l = \pi r$. Substitute these into the simplified equation from Statement II: $$ \pi r + r = \pi r + 7 \implies r = 7 $$ * Now that we have the radius $r = 7$, we can find the length: $l = \pi(7) = \frac{22}{7} \times 7 = 22$ cm. * Since we can find a unique value by combining both pieces of information, both are necessary. ### Exam Strategy & Shortcut When dealing with multiple unknown variables, count your distinct equations. You have three variables ($l, b, r$). The prompt gives one equation, Statement I gives a second, and Statement II gives a third. Three independent linear/quadratic relationships for three variables generally guarantee a solution. ### Common Pitfall Stopping at Statement I thinking that relating length to radius is enough, forgetting that a final answer must be an exact numerical value unless the question asks for a ratio. ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question**.