Question

If x² + y² + z² = 133, xy + yz + zx = 114, then the value of x³ + y³ + z³ - 3xyz is:
1. 256
2. 178
3. 361
4. 462

Answer 1

Correct Answer - Option 3 : 361

Given: 

x² + y² + z² = 133      

xy + yz + zx = 114      

Formula used:

x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2  – xy – yz – zx)   

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Calculation:

x2 + y2 + z2 = 133  ----(i)   

xy + yz + zx = 114  ----(ii)

We know that,

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

⇒ (x + y + z)² = 133 + (2 × 114)

⇒ (x + y + z) = √361

⇒ (x + y + z) = 19      ----(iii)    

Putting the value of (i), (ii), (iii) in equation 

x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2  – xy – yz – zx)  

⇒ x³ + y³ + z³ – 3xyz = (19) × (133 – 114)

∴ x³ + y³ + z³ – 3xyz = 361