If `x -y =1,` then `x ^(2) - y ^(3) - 3 (xy -2)` is equal to :

Options

A. A)7
B. B)6
C. C)10
D. D)4

Answer

To solve the equation \( x^2 - y^3 - 3(xy - 2) \) given that \( x - y = 1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Express \( x \) in terms of \( y \)**:
From the equation \( x - y = 1 \), we can express \( x \) as:
\[ x = y + 1 \]

2. **Substitute \( x \) in the expression**:
Now, we will substitute \( x = y + 1 \) into the expression \( x^2 - y^3 - 3(xy - 2) \):
\[ (y + 1)^2 - y^3 - 3((y + 1)y - 2) \]

3. **Calculate \( (y + 1)^2 \)**:
Expanding \( (y + 1)^2 \):
\[ (y + 1)^2 = y^2 + 2y + 1 \]

4. **Calculate \( xy \)**:
Now, calculate \( xy \):
\[ xy = (y + 1)y = y^2 + y \]

5. **Substitute back into the expression**:
Now substitute these values back into the expression:
\[ (y^2 + 2y + 1) - y^3 - 3(y^2 + y - 2) \]

6. **Expand the expression**:
Expanding \( -3(y^2 + y - 2) \):
\[ -3(y^2 + y - 2) = -3y^2 - 3y + 6 \]

7. **Combine all terms**:
Now combine all the terms:
\[ y^2 + 2y + 1 - y^3 - 3y^2 - 3y + 6 \]
This simplifies to:
\[ -y^3 + (y^2 - 3y^2) + (2y - 3y) + (1 + 6) \]
Which further simplifies to:
\[ -y^3 - 2y^2 - y + 7 \]

8. **Choose a value for \( y \)**:
To find a specific value, we can choose \( y = 0 \) (as suggested in the video):
\[ -0^3 - 2(0)^2 - 0 + 7 = 7 \]

Thus, the final value of the expression \( x^2 - y^3 - 3(xy - 2) \) when \( x - y = 1 \) is:
\[ \boxed{7} \]