Question : If $\small p^{3}-q^{3}=\left (p-q \right )\left \{ \left (p-q \right)^{2}-xpq \right \}$, then find the value of $x$.
Option 1: 3
Option 2: –3
Option 3: 1
Option 4: –1
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Correct Answer: –3
Solution : As we know $p^3 - q^3 = (p - q)(p^2 + pq + q^2)$ Comparing $\small p^{3}-q^{3}=\left (p-q \right )\left \{ \left (p-q \right)^{2}-xpq \right \}$ with the above identity, we get, $p^2 + pq + q^2 = (p-q)^2-xpq$ ⇒ $p^2 + pq + q^2 = p^2+q^2-2pq - xpq$ ⇒ $pq = -pq(2+x)$ ⇒ $2+x = -1$ ⇒ $ x=-3$ Hence, the correct answer is –3.
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