Question : If $x+y=1+xy$, then $x^3+y^3-x^3y^3$ is equal to:
Option 1: 0
Option 2: 1
Option 3: –1
Option 4: 2
Correct Answer: 1
Solution :
Given: $x+y=1+xy$ (equation 1)
We know that the algebraic identity is $(x^3+y^3)=(x+y)^3-3xy(x+y)$
$x^3+y^3-x^3y^3=(x+y)^3-3xy(x+y)-x^3y^3$
Substitute the value from equation 1 in the above equation and we get,
$x^3+y^3-x^3y^3=(1+xy)^3-3xy(1+xy)-x^3y^3$
⇒ $x^3+y^3-x^3y^3=1+x^3y^3+3xy+3x^2y^2-3xy-3x^2y^2-x^3y^3$
⇒ $x^3+y^3-x^3y^3=1$
Hence, the correct answer is 1.
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