Question : If a + b + c = 5 and ab + bc + ca = 7, then the value of a3 + b3 + c3 – 3abc is:
Option 1: 15
Option 2: 20
Option 3: 25
Option 4: 30
Correct Answer: 20
Solution :
Here, a + b + c = 5
Squaring both sides, we get,
⇒ (a + b + c)
2
= 25
⇒ a
2
+ b
2
+ c
2
+ 2ab + 2bc + 2ca = 25
As, ab + bc + ca = 7
⇒ a
2
+ b
2
+ c
2
+ 2(7) = 25
⇒ a
2
+ b
2
+ c
2
= 25 – 14 = 11
As we know,
a
3
+ b
3
+ c
3
= (a + b + c)(a
2
+ b
2
+ c
2
– ab – bc – ca) + 3abc
As, a + b + c = 5 and ab + bc + ca = 7
⇒ a
3
+ b
3
+ c
3
= (5)(11 – 7) + 3abc
$\therefore$ a
3
+ b
3
+ c
3
– 3abc = 20
Hence, the correct answer is 20.
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