(x + 1)/(x - 1) - (x - 1)/(x + 1) = {(x + 1)² - (x - 1)²} / {(x - 1) (x + 1)}
= (x² + 1 + 2x) - (x² + 1 - 2x) / (x² - 1)
= 4x / (x² - 1) = 4(√3 + √2) / {(√3 + √2)² - 1}

Correct Option: B

(x + 1)/(x - 1) - (x - 1)/(x + 1) = {(x + 1)² - (x - 1)²} / {(x - 1) (x + 1)}
= (x² + 1 + 2x) - (x² + 1 - 2x) / (x² - 1)
= 4x / (x² - 1) = 4(√3 + √2) / {(√3 + √2)² - 1}
= 4(√3 + √2) / (3 + 2 + 2√6 - 1) = 4(√3 + √2) / (4 + 2 √6)
= 4(√3 + √2) / 2 √2 (√3 + √2)
= √2

∛8 and ∛1000 = 8^1/3 and 1000^1/3

Correct Option: B

∛8 and ∛1000 = 8^1/3 and 1000^1/3
Since 1000 > 8
∴ ∛1000 > ∛8

√81 and √1 = 81^1/2 and 1^1/2

Correct Option: B

√81 and √1 = 81^1/2 and 1^1/2
Since, 1 < 81
∴ √1 < √81

Let the number be x.
According to the question,
x² + (14)³ = 4425

Correct Option: C

Let the number be x.
According to the question,
x² + (14)³ = 4425
⇒ x² = 4425 - (14)³
⇒ x² = 4425 - 2744
⇒ x² = 1681
∴ x = √1681 = 41

∜10 and ∛8 = 10^1/4 and 8^1/3
Since, LCM of 4 and 3 is 12.
∴ 10^1/4 = 10^3/12 = (10³)^1/12 = (1000)^1/12 = ¹²√1000
and 8^1/3 = 8^4/12 = (8⁴)^1/12 (4096)^1/2 = ¹²√4096
∴ 4096 > 1000

Correct Option: B

∜10 and ∛8 = 10^1/4 and 8^1/3
Since, LCM of 4 and 3 is 12.
∴ 10^1/4 = 10^3/12 = (10³)^1/12 = (1000)^1/12 = ¹²√1000
and 8^1/3 = 8^4/12 = (8⁴)^1/12 (4096)^1/2 = ¹²√4096
∴ 4096 > 1000
Hence, 8^1/3 >10^1/4
∴ ∛8 > ∜10