Question

Evaluate: \(\frac{(8.53 \space-\space 6.07)^2}{(6.07\space -\space 2.15)(2.15\space-\space8.53)} \space+\space\frac{(6.07\space-\space2.15)^2}{(8.53\space-\space6.07)(2.15\space-\space8.53)}+\space\frac{(2.15\space-\space8.53)^2}{(8.53\space-\space6.07)(6.07\space-\space2.15)}\)
1. 138.42
2. 142.863
3. 3
4. 0

Answer 1

Correct Answer - Option 3 : 3

Concept used:
a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c²  – ab – bc – ac
If a + b + c = 0, then a3 + b3 + c3 = 3abc.

Calculation:
\(\frac{(8.53 \space-\space 6.07)^2}{(6.07\space -\space 2.15)(2.15\space-\space8.53)} \space+\space\frac{(6.07\space-\space2.15)^2}{(8.53\space-\space6.07)(2.15\space-\space8.53)}+\space\frac{(2.15\space-\space8.53)^2}{(8.53\space-\space6.07)(6.07\space-\space2.15)}\)
Let (8.53 – 6.07), (6.07 – 2.15) and (2.15 – 8.53) be a, b and c respectively.  
⇒ a + b + c = 8.53 – 6.07 + 6.07 – 2.15 + 2.15 – 8.53 = 0
⇒ (a²)/(b × c) + (b²)/(a × c) + (c²)/(a × b)
⇒ [(a²  × a) + (b² × b) + (c² × c)]/(a × b × c)
⇒ (a³ + b³ + c³)/(abc)
⇒ (3abc)/(abc)
⇒ 3
∴ The value is 3.