The following questions are accompanied by two statements (I) and (II). You have to determine which statements(s) is/are sufficient/ necessary to answer the questions.
What will be the area of square?
I. Side of square is equal to breadth of rectangle, whose area and perimeter are 416 cm2 and 84cm respectively and length is greater than breadth.
II Ratio of length to breadth of rectangle is 13: 8.

Options

A. Statement (I) alone is sufficient to answer the question but statement (II) alone is not sufficient to answer the question
B. Statement (II) alone is sufficient to answer the question but statement (I) alone is not sufficient to answer the question.
C. Both the statements taken together are necessary to answer the question, but neither of the statements alone is sufficient to answer the question.
D. Either statement (I) or statement (II) by itself is sufficient to answer the question.

Answer

To solve the question regarding the area of the square, we need to analyze the provided statements step by step.

### Step 1: Understanding the Problem
We need to find the area of a square, and we are given two statements about a rectangle whose dimensions are related to the square.

### Step 2: Analyzing Statement I
**Statement I:** The side of the square is equal to the breadth of a rectangle, whose area is 416 cm² and perimeter is 84 cm, and the length is greater than the breadth.

1. **Area of the rectangle:**
\[ \text{Area} = \text{Length} \times \text{Breadth} = 416 \text{ cm}^2 \]

2. **Perimeter of the rectangle:**
\[ \text{Perimeter} = 2(\text{Length} + \text{Breadth}) = 84 \text{ cm} \]
This simplifies to:
\[ \text{Length} + \text{Breadth} = 42 \text{ cm} \]

### Step 3: Setting Up Equations
Let the length be \( L \) and the breadth be \( B \).

From the area:
\[ L \times B = 416 \quad (1) \]
From the perimeter:
\[ L + B = 42 \quad (2) \]

### Step 4: Solving the Equations
From equation (2), we can express \( L \) in terms of \( B \):
\[ L = 42 - B \]
Substituting \( L \) in equation (1):
\[ (42 - B) \times B = 416 \]
Expanding this gives:
\[ 42B - B^2 = 416 \]
Rearranging:
\[ B^2 - 42B + 416 = 0 \]

### Step 5: Finding the Roots
Using the quadratic formula \( B = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ B = \frac{42 \pm \sqrt{42^2 - 4 \times 1 \times 416}}{2 \times 1} \]
Calculating the discriminant:
\[ 42^2 = 1764, \quad 4 \times 416 = 1664 \quad \Rightarrow \quad 1764 - 1664 = 100 \]
Thus,
\[ B = \frac{42 \pm 10}{2} \]
Calculating the two possible values for \( B \):
1. \( B = \frac{52}{2} = 26 \)
2. \( B = \frac{32}{2} = 16 \)

### Step 6: Finding Length
Using \( B = 16 \):
\[ L = 42 - 16 = 26 \]
Using \( B = 26 \):
\[ L = 42 - 26 = 16 \]
Thus, the breadth \( B \) can be 16 cm, and the length \( L \) can be 26 cm.

### Step 7: Finding the Side of the Square
Since the side of the square is equal to the breadth of the rectangle:
\[ \text{Side of the square} = 16 \text{ cm} \]

### Step 8: Calculating the Area of the Square
The area of the square is:
\[ \text{Area} = \text{Side}^2 = 16^2 = 256 \text{ cm}^2 \]

### Conclusion
Thus, the area of the square is 256 cm².