x + y + z = 6
xy + yz + zx = 10
∴ (x + y + z)² = 36
⇒ x² + y² + z² + 2xy + 2yz + 2zx = 36
⇒ x² + y² + z² + 2 × 10 = 36
⇒ x² + y² + z² = 36 – 20 = 16
∴ x³ + y³ + z³ - 3xyz = (x + y + z)( x² + y² + z² – xy – yz – zx) = 6 (16 – 10) = 6 × 6 = 36

Correct Option: A

x + y + z = 6
xy + yz + zx = 10
∴ (x + y + z)² = 36
⇒ x² + y² + z² + 2xy + 2yz + 2zx = 36
⇒ x² + y² + z² + 2 × 10 = 36
⇒ x² + y² + z² = 36 – 20 = 16
∴ x³ + y³ + z³ - 3xyz = (x + y + z)( x² + y² + z² – xy – yz – zx) = 6 (16 – 10) = 6 × 6 = 36

p³ - q³ = (p – q){(p – q)² – xpq}
⇒ (p – q)(p² + q² + pq) = (p – q)(p² + q² – 2pq – xpq)
⇒ (p² + q² + pq) = p² + q² – (2 + x) pq
∴ – (2 + x) = 1
⇒ x = –2 – 1 = –3

Correct Option: B

p³ - q³ = (p – q){(p – q)² – xpq}
⇒ (p – q)(p² + q² + pq) = (p – q)(p² + q² – 2pq – xpq)
⇒ (p² + q² + pq) = p² + q² – (2 + x) pq
∴ – (2 + x) = 1
⇒ x = –2 – 1 = –3

x – √3 – √2 = 0
⇒ x = √3 + √2
Again,
y – √3 + √2 = 0
⇒ y = √3 - √2
∴ x – y = √3 + √2 - √3 + √2 = 2√2
and xy = (√3 + √2)(√3 - √2) = 3 – 2 = 1
∴ Expression = x³ - 20√2 - y³ - 2√2 = x³ - y³ - 22√2
Expression = (x - y)³ + 3xy(x – y) – 22√2
Expression = (2√2)³ + 3(2√2) – 22√2 = 16 √2 + 6 √2 - 22 √2 = 0

Correct Option: A

x – √3 – √2 = 0
⇒ x = √3 + √2
Again,
y – √3 + √2 = 0
⇒ y = √3 - √2
∴ x – y = √3 + √2 - √3 + √2 = 2√2
and xy = (√3 + √2)(√3 - √2) = 3 – 2 = 1
∴ Expression = x³ - 20√2 - y³ - 2√2 = x³ - y³ - 22√2
Expression = (x - y)³ + 3xy(x – y) – 22√2
Expression = (2√2)³ + 3(2√2) – 22√2 = 16 √2 + 6 √2 - 22 √2 = 0

x² + y² + z² = xy + yz + zx
⇒ x² + y² + z² - xy - yz - zx = 0
⇒ 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0
⇒ x² + y² - 2xy + y² + z² - 2yz + z² + x² - 2zx = 0
⇒ (x - y)² + (y - z)² + (z - x)² = 0
∴ x – y = 0 ⇒ x = y
y – z = 0 ⇒ y = z
z – x = 0 ⇒ z = x
∴ x = y = z
[If a² + b² + c² = 0, then a = 0, b = 0, c = 0]

Correct Option: B

x² + y² + z² = xy + yz + zx
⇒ x² + y² + z² - xy - yz - zx = 0
⇒ 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0
⇒ x² + y² - 2xy + y² + z² - 2yz + z² + x² - 2zx = 0
⇒ (x - y)² + (y - z)² + (z - x)² = 0
∴ x – y = 0 ⇒ x = y
y – z = 0 ⇒ y = z
z – x = 0 ⇒ z = x
∴ x = y = z
[If a² + b² + c² = 0, then a = 0, b = 0, c = 0]

x = a^{1 / 3} + a^{-1 / 3}
y = a^{1 / 2} - a^{-1 / 2}
∴ x² – y² = 4a^{1 / 2}.a^{-1 / 2} = 4
[∴ (a + b)² – (a – b)² = 4ab]
Again, y² - x² = -4a^{1 / 2}.a^{-1 / 2} = -4
Expression = (x⁴ - x²y² – 1) + (y⁴ - x²y² + 1)
Expression = x²(x² - y²) – 1 + y²(y² - x²) + 1
Expression = 4x² - 1 - 4y² + 1
Expression = 4(x² - y²) = 4 × 4 = 16

Correct Option: A

x = a^{1 / 3} + a^{-1 / 3}
y = a^{1 / 2} - a^{-1 / 2}
∴ x² – y² = 4a^{1 / 2}.a^{-1 / 2} = 4
[∴ (a + b)² – (a – b)² = 4ab]
Again, y² - x² = -4a^{1 / 2}.a^{-1 / 2} = -4
Expression = (x⁴ - x²y² – 1) + (y⁴ - x²y² + 1)
Expression = x²(x² - y²) – 1 + y²(y² - x²) + 1
Expression = 4x² - 1 - 4y² + 1
Expression = 4(x² - y²) = 4 × 4 = 16