We know that, a³ + b³ + c³ - 3abc
= (a + b + c) (a² + b² + c² - ab - bc - ca)

Correct Option: C

We know that, a³ + b³ + c³ - 3abc
= (a + b + c) (a² + b² + c² - ab - bc - ca)
⇒ (a + b + c) = (a³ + b³ + c³ - 3abc/a² + b² + c² - ab - bc - ca) ....(i)
Given that,
(2.247)³ + (1.730)³ + (1.023)³ - 3 x 2.247 x 1.730 x 1.023/ (2.247)² + (1.730)² + (1.023)² - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023)
= 2.247 + 1.730 x 1.023) - (2.247 x 1.023)
= (2.247 + 1.730 + 1.023) [from Eq.(i)]
= 5.000 = 5

a² + 1 = a ⇒ a + 1/a = 1
On squaring both sides, we get
a² + 1/a² + 2 = 1
⇒ a² + 1/a² = -1
On cubing both sides, we get
(a² + 1/a²)³ = (-1)³
⇒ a⁶ + 1/a⁶ + 3a² x 1/a²(a² + 1/a²) = -1

Correct Option: B

a² + 1 = a ⇒ a + 1/a = 1
On squaring both sides, we get
a² + 1/a² + 2 = 1
⇒ a² + 1/a² = -1
On cubing both sides, we get
(a² + 1/a²)³ = (-1)³
⇒ a⁶ + 1/a⁶ + 3a² x 1/a²(a² + 1/a²) = -1
⇒ a⁶ + 1/a⁶ + 3 x (-1) = -1
Now, a⁶ + 1/a⁶ + 1 = 3
As, a¹² + a⁶ + 1 can also be written as a⁶ + 1/a⁶ + 1
∴ a¹² + a⁶ + 1 = 3