If `a o+ b = a^3 + b^2 + 1`
`a © b = (a - b)^2 + 1`
`a ** b = a^3 - b^2 + 1`
The value of `P = (((1 o+ 2) ** 2) © 1)` is :

Options

A. 40001
B. 10804
C. 40401
D. None of these

Answer

To solve the problem step by step, we need to evaluate the expression \( P = (((1 \, o+ \, 2) \, ** \, 2) \, © \, 1) \) using the given operations:

1. **Evaluate \( 1 \, o+ \, 2 \)**:
- According to the operation defined, \( a \, o+ \, b = a^3 + b^2 + 1 \).
- Here, \( a = 1 \) and \( b = 2 \).
- So, \( 1 \, o+ \, 2 = 1^3 + 2^2 + 1 \).
- Calculate:
- \( 1^3 = 1 \)
- \( 2^2 = 4 \)
- Therefore, \( 1 \, o+ \, 2 = 1 + 4 + 1 = 6 \).

2. **Now evaluate \( 6 \, ** \, 2 \)**:
- According to the operation defined, \( a \, ** \, b = a^3 - b^2 + 1 \).
- Here, \( a = 6 \) and \( b = 2 \).
- So, \( 6 \, ** \, 2 = 6^3 - 2^2 + 1 \).
- Calculate:
- \( 6^3 = 216 \)
- \( 2^2 = 4 \)
- Therefore, \( 6 \, ** \, 2 = 216 - 4 + 1 = 213 \).

3. **Now evaluate \( 213 \, © \, 1 \)**:
- According to the operation defined, \( a \, © \, b = (a - b)^2 + 1 \).
- Here, \( a = 213 \) and \( b = 1 \).
- So, \( 213 \, © \, 1 = (213 - 1)^2 + 1 \).
- Calculate:
- \( 213 - 1 = 212 \)
- Therefore, \( 213 \, © \, 1 = (212)^2 + 1 \).

4. **Calculate \( (212)^2 + 1 \)**:
- First, calculate \( 212^2 \):
- \( 212 \times 212 = 44944 \) (you can use the formula \( (a+b)^2 = a^2 + 2ab + b^2 \) where \( a = 200 \) and \( b = 12 \)).
- Now add 1:
- \( 44944 + 1 = 44945 \).

So, the final value of \( P \) is \( 44945 \).