If ab, cd, ba and dc are two digit numbers then the maximum value of `(ab xx cd) - (ba xx dc)` is , where a,b,c,d are distinct non-zoro integers:

Options

A. a. 7938
B. b. 7128
C. c. 6930
D. d. none of these

Answer

To solve the problem, we need to find the maximum value of the expression \((ab \times cd) - (ba \times dc)\), where \(ab\), \(cd\), \(ba\), and \(dc\) are two-digit numbers formed by distinct non-zero integers \(a\), \(b\), \(c\), and \(d\).

### Step-by-Step Solution:

1. **Define the Two-Digit Numbers**:
- Let \(ab = 10a + b\)
- Let \(cd = 10c + d\)
- Let \(ba = 10b + a\)
- Let \(dc = 10d + c\)

2. **Write the Expression**:
We need to calculate:
\[ (ab \times cd) - (ba \times dc) \]
Substituting the definitions:
\[ ((10a + b) \times (10c + d)) - ((10b + a) \times (10d + c)) \]

3. **Expand Both Products**:
- For \(ab \times cd\):
\[ (10a + b)(10c + d) = 100ac + 10ad + 10bc + bd \]
- For \(ba \times dc\):
\[ (10b + a)(10d + c) = 100bd + 10bc + 10ad + ac \]

4. **Combine the Expressions**:
Now, substituting back into the expression:
\[ (100ac + 10ad + 10bc + bd) - (100bd + 10bc + 10ad + ac) \]
Simplifying this gives:
\[ 100ac - 100bd + bd - ac \]
Which can be rearranged to:
\[ 99ac - 99bd \]
Factoring out \(99\):
\[ 99(ac - bd) \]

5. **Maximize \(ac - bd\)**:
To maximize the expression, we need to maximize \(ac - bd\). Since \(a\), \(b\), \(c\), and \(d\) are distinct non-zero integers, we will try different combinations of these integers.

6. **Choose Values for \(a\), \(b\), \(c\), and \(d\)**:
Let's try \(a = 9\), \(b = 1\), \(c = 8\), \(d = 2\):
\[ ac = 9 \times 8 = 72 \]
\[ bd = 1 \times 2 = 2 \]
Thus,
\[ ac - bd = 72 - 2 = 70 \]

7. **Calculate the Maximum Value**:
Now substituting back:
\[ 99(ac - bd) = 99 \times 70 = 6930 \]

8. **Conclusion**:
The maximum value of \((ab \times cd) - (ba \times dc)\) is \(6930\).