Question : If $a+b=1$, find the value of $a^{3}+b^{3} - ab-(a^{2}-b^{2})^{2}$.
Option 1: –1
Option 2: 1
Option 3: 0
Option 4: 2
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Correct Answer: 0
Solution : Given: $a+b=1$ We know that $a^{3}+b^{3}=(a+b)^{3}-3ab(a+b)$ So, $a^{3}+b^{3}-ab-(a^{2}-b^{2})^{2}=(a+b)^{3}-3ab(a+b)-ab-(a+b)^{2}(a-b)^{2}$ Putting the value of $a+b=1$, we get, $a^{3}+b^{3}-ab-(a^{2}–b^{2})^{2}$ $=(1)^{3}-3ab(1)-ab-(1)^{2}[(1)^{2}-4ab]$ $= 1-3ab-ab-1+4ab$ $=0$ Hence, the correct answer is 0.
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