If a + b + c = 13 and ab + bc + ca = 22, then the value of a3 + b3 + c3 − 3abc is _________.
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. 1339.
Given that, a + b + c = 13 ...(i)
And ab + bc + ca = 22 ...(ii)
a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) ...(iii)
According to the given question,
Squarying both sides of equation (i),
⇒ (a + b + c)2 = 132
⇒ a2 + b2 + c2 + 2(ab + bc + ca) = 169
⇒ a2 + b2 + c2 + 2(22) = 169
⇒ a2 + b2 + c2 + 44 = 169
⇒ a2 + b2 + c2 = 125 ...(iv)
Using the values of equation (i), (ii) and (iv) in equation (iii),
a3 + b3 + c3 − 3abc = 13(125 - 22)
⇒ a3 + b3 + c3 − 3abc = 13 × 103
⇒ a3 + b3 + c3 − 3abc = 1339
Hence the required answer is 1339.