Correct Answer - Option 2 : –2800
Given:
x + y + z = 20 and x2 + y2 + z2 = 40
Formula used:
(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx)
x3 + y3 + z3 – 3xyz = (x + y + z){x2 + y2 + z2 – (xy + yz + zx)}
Calculation:
(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx)
⇒ (20)2 = 40 + 2(xy + yz + zx)
⇒ 400 = 40 + 2(xy + yz + zx)
⇒ 2(xy + yz + zx) = 360
⇒ (xy + yz + zx) = 180
Then, x3 + y3 + z3 – 3xyz = (x + y + z){x2 + y2 + z2 – (xy + yz + zx)}
⇒ 20 × (40 – 180)
⇒ 20 × (– 140)
⇒ –2800
∴ The value of x3 + y3 + z3 – 3xyz is –2800.
