x = 2 - 2^{1 / 3} + 2^{2 / 3}
⇒ x – 2 = 2^{2 / 3} - 2^{1 / 3}
On cubing both sides ,
⇒ x³ - 3x² × 2 + 3x × 4 – 8 = ( 2^{2 / 3} )³ - ( 2^{1 / 3} )³ - 3 . 2^{2 / 3} . 2^{1 / 3}( 2^{2 / 3} - 2^{1 / 3} )
⇒ x³ - 6x² + 12x – 8 = = 4 – 2 – 6 (x – 2)
⇒ x³ - 6x² + 12x – 8 = = 2 – 6x + 12
⇒ x³ - 6x² + 18x + 18 = 2 + 12 + 8 + 18 = 40

Correct Option: C

x = 2 - 2^{1 / 3} + 2^{2 / 3}
⇒ x – 2 = 2^{2 / 3} - 2^{1 / 3}
On cubing both sides ,
⇒ x³ - 3x² × 2 + 3x × 4 – 8 = ( 2^{2 / 3} )³ - ( 2^{1 / 3} )³ - 3 . 2^{2 / 3} . 2^{1 / 3}( 2^{2 / 3} - 2^{1 / 3} )
⇒ x³ - 6x² + 12x – 8 = = 4 – 2 – 6 (x – 2)
⇒ x³ - 6x² + 12x – 8 = = 2 – 6x + 12
⇒ x³ - 6x² + 18x + 18 = 2 + 12 + 8 + 18 = 40

Using Rule 21,
a³ + b³ + c³ – 3abc = 0
If a + b + c = 0 , a³ - b³ - c³ – 3abc = 0
⇒ a - b - c = 0
⇒ a = b + c

Correct Option: D

Using Rule 21,
a³ + b³ + c³ – 3abc = 0
If a + b + c = 0 , a³ - b³ - c³ – 3abc = 0
⇒ a - b - c = 0
⇒ a = b + c

Using Rule 21,
Here, p – q + q – r + r – p = 0
∴ (p – q)³ + (q – r)³ + (r – p)³ = 3(p – q)(q – r)(r – p)
[Formula : If a + b + c = 0 , then a³ + b³ + c³ = 3abc]

Correct Option: B

Using Rule 21,
Here, p – q + q – r + r – p = 0
∴ (p – q)³ + (q – r)³ + (r – p)³ = 3(p – q)(q – r)(r – p)
[Formula : If a + b + c = 0 , then a³ + b³ + c³ = 3abc]

Using Rule 21,
a + b + (–c) = 2.361 + 3.263 – 5.624 = 0
∴ a³ + b³ + [ -c³ - 3ab(-c) ] = 0
i.e. a³ + b³ – c³ + 3abc = 0

Correct Option: B

Using Rule 21,
a + b + (–c) = 2.361 + 3.263 – 5.624 = 0
∴ a³ + b³ + [ -c³ - 3ab(-c) ] = 0
i.e. a³ + b³ – c³ + 3abc = 0

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 36 = 14 + 2(ab + bc + ca)
⇒ ab + bc + ca = (36 – 14) ÷ 2
⇒ ab + bc + ca = 11 ....(i)
∴ a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
⇒ 36 – 3abc = 6 (14 – 11) [By (i)]
⇒ 36 – 3abc = 84 – 66 = 18
⇒ 3abc = 36 – 18 = 18
⇒ abc = 6

Correct Option: B

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ 36 = 14 + 2(ab + bc + ca)
⇒ ab + bc + ca = (36 – 14) ÷ 2
⇒ ab + bc + ca = 11 ....(i)
∴ a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
⇒ 36 – 3abc = 6 (14 – 11) [By (i)]
⇒ 36 – 3abc = 84 – 66 = 18
⇒ 3abc = 36 – 18 = 18
⇒ abc = 6