Question : If a + b + c = 15 and a2 + b2 + c2 = 83 then the value of a3 + b3 + c3 – 3abc:
Option 1: 200
Option 2: 180
Option 3: 190
Option 4: 210
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Correct Answer: 180
Solution :
Given: a
2
+b
2
+c
2
= 83, a+b+c = 15
We know that,
(a + b + c)
2
= a
2
+ b
2
+ c
2
+ 2(ab + bc + ca)
⇒ (15)
2
= 83 + 2(ab + bc + ca)
⇒ 225 – 83 = 2(ab + bc + ca)
⇒$\frac{142}{2}$ = ab + bc + ca
$\therefore$ ab + bc + ca = 71
Now, a
3
+ b
3
+ c
3
–3abc = (a + b + c)(a
2
+ b
2
+ c
2
– ab – bc – ca)
Putting the values, we get:
= (15)(83 – 71)
= 15 × 12
= 180
Hence, the correct answer is 180.
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