If the binary operation * on Z is defined by `a*b=a^2-b^2+a b+4` , then value of `(2*3)*4` is`

Options

A. 21
B. 33
C. 37
D. 41

Explanation

To solve the problem, we need to evaluate the expression \((2 * 3) * 4\) using the binary operation defined as \(a * b = a^2 - b^2 + ab + 4\).

### Step-by-Step Solution:

1. **Calculate \(2 * 3\)**:
\[ 2 * 3 = 2^2 - 3^2 + 2 \cdot 3 + 4 \]
- Calculate \(2^2\):
\[ 2^2 = 4 \]
- Calculate \(3^2\):
\[ 3^2 = 9 \]
- Calculate \(2 \cdot 3\):
\[ 2 \cdot 3 = 6 \]
- Substitute these values into the equation:
\[ 2 * 3 = 4 - 9 + 6 + 4 \]
- Simplify:
\[ 2 * 3 = 4 - 9 = -5 \]
\[ -5 + 6 = 1 \]
\[ 1 + 4 = 5 \]
- Therefore, \(2 * 3 = 5\).

2. **Calculate \((2 * 3) * 4\) which is \(5 * 4\)**:
\[ 5 * 4 = 5^2 - 4^2 + 5 \cdot 4 + 4 \]
- Calculate \(5^2\):
\[ 5^2 = 25 \]
- Calculate \(4^2\):
\[ 4^2 = 16 \]
- Calculate \(5 \cdot 4\):
\[ 5 \cdot 4 = 20 \]
- Substitute these values into the equation:
\[ 5 * 4 = 25 - 16 + 20 + 4 \]
- Simplify:
\[ 5 * 4 = 25 - 16 = 9 \]
\[ 9 + 20 = 29 \]
\[ 29 + 4 = 33 \]
- Therefore, \(5 * 4 = 33\).

### Final Answer:
The value of \((2 * 3) * 4\) is \(33\).