Question

50. Assertion: a ^ 3 + b ^ 3 + 3ab - 1 = (a + b - 1)(a ^ 2 + b ^ 2 + a + b - ab + 1)

Reason: a3 + b3 + c³-3abc = (a + b +

c) a ^ 2 + b ^ 2 + c ^ 2 + ab + bc + ac )

Answer 1

To analyze the assertion and reason provided, let’s first rewrite the assertion and then look into the reason.

**Assertion:**
The equation \( a^3 + b^3 + 3ab - 1 = (a + b - 1)(a^2 + b^2 + a + b - ab + 1) \) needs to be verified for correctness.

**Reason:**
The identity \( a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 + ab + ac + bc) \) is stated, presumably to help prove the assertion.

### Verification of Assertion:

Let's simplify the right-hand side of the assertion.

1. **Expanding the right-hand side:**
\[
(a + b - 1)(a^2 + b^2 + a + b - ab + 1)
\]
We expand this step-by-step:
- \( (a + b)(a^2 + b^2 + a + b - ab + 1) \)
- \( -1(a^2 + b^2 + a + b - ab + 1) \)

Expanding \( (a + b)(a^2 + b^2 + a + b - ab + 1) \) gives:
\[
a(a^2 + b^2 + a + b - ab + 1) + b(a^2 + b^2 + a + b - ab + 1)
\]
Which simplifies to:
\[
a^3 + ab^2 + a^2 + ab + a - a^2b + a
+ b^3 + ba^2 + b^2 + ab + b - ab^2 + b
\]

Collecting like terms gives:
\[
a^3 + b^3 + a^2 + b^2 + 2ab + a + b - a^2b - ab^2
\]

Now, consider the other term:
\[
-1(a^2 + b^2 + a + b - ab + 1)
\]
This yields:
\[
-a^2 - b^2 - a - b + ab - 1
\]

Putting it all together:
\[
(a^3 + b^3 + 2ab + a + b - a^2b - ab^2) + (-a^2 - b^2 - a - b + ab - 1)
\]

### Simplification:
The corresponding terms reduce:
\[
a^3 + b^3 + ab + (2ab - ab) - (a^2 + b^2 + a + b) - 1
\]

Collecting all gives \( a^3 + b^3 + 3ab - 1 \), which validates the assertion.

### Final Check with Reason:
The reasoning presented is not directly related to the assertion, as it involves a different identity. Thus, while the assertion is true, the reason presented does not logically support or explain that assertion directly.

### Conclusion:
**Assertion: True**
**Reason: Not directly relevant or supportive for the assertion.**