Let ABCD be a parallelogram. Let m and n be positive integers such that `n lt mlt 2n`. Let AC =2 mn and `BD =m^2 - n^2`. Let `AB = (m^2 + n^2) /2`.
Statement I `AC gt BD`
Statement II ABCD is a rhombus. Which one of the following is correct in respect of the above statements ?

Options

A. Both statements I and II are true and statement II is the correct explanation of statement I
B. Both statements I and II are true but statement I is not the correctexplanation of statement ii
C. Statement I is true but statement II is false
D. Statement II is true but statement I is false

Answer

To solve the problem, we need to analyze the given statements about the parallelogram ABCD and the conditions provided.

### Step 1: Understand the Properties of a Parallelogram
- In a parallelogram, the diagonals bisect each other.
- The lengths of the diagonals can be calculated using the formula:
\[ AC^2 + BD^2 = 2(AB^2 + BC^2) \]
- Since ABCD is a parallelogram, we can use this property to analyze the given lengths.

### Step 2: Substitute the Given Values
- We have the following values:
- \( AC = 2mn \)
- \( BD = m^2 - n^2 \)
- \( AB = \frac{m^2 + n^2}{2} \)

### Step 3: Calculate \( AC^2 \) and \( BD^2 \)
- Calculate \( AC^2 \):
\[ AC^2 = (2mn)^2 = 4m^2n^2 \]
- Calculate \( BD^2 \):
\[ BD^2 = (m^2 - n^2)^2 = m^4 - 2m^2n^2 + n^4 \]

### Step 4: Use the Parallelogram Diagonal Formula
- Substitute these into the diagonal formula:
\[ 4m^2n^2 + (m^4 - 2m^2n^2 + n^4) = 2\left(\left(\frac{m^2 + n^2}{2}\right)^2 + \left(\frac{m^2 + n^2}{2}\right)^2\right) \]
- Simplifying the right side:
\[ 2\left(\frac{(m^2 + n^2)^2}{4} + \frac{(m^2 + n^2)^2}{4}\right) = \frac{(m^2 + n^2)^2}{2} \]

### Step 5: Equate and Simplify
- Now we equate both sides:
\[ 4m^2n^2 + m^4 - 2m^2n^2 + n^4 = \frac{(m^2 + n^2)^2}{2} \]
- This simplifies to:
\[ m^4 + 2m^2n^2 + n^4 = \frac{(m^2 + n^2)^2}{2} \]
- Which means:
\[ (m^2 + n^2)^2 = 2(m^2 + n^2)^2 \]
- This holds true under the conditions given.

### Step 6: Analyze the Statements
- **Statement I**: \( AC > BD \)
- We need to check if \( 2mn > m^2 - n^2 \).
- Rearranging gives \( m^2 - 2mn + n^2 < 0 \), which factors to \( (m - n)^2 < 0 \), which is not possible.

- **Statement II**: ABCD is a rhombus.
- For ABCD to be a rhombus, all sides must be equal, which we have shown is true from our calculations.

### Conclusion
- Statement I is false, while Statement II is true.
- Therefore, the correct answer is that Statement II is true, but Statement I is not a correct explanation for Statement II.