JEE Advanced Maths: Algebra (Surds and Indices) questions with solutions

Q1. If x = (7 + 5*sqrt(2))^(1/3) - 1/(7 + 5*sqrt(2))^(1/3), find the value of x³ + 3x - 14.

1. 1
2. 0
3. 2
4. 4

Answer: 0

Using the algebraic identity (a - 1/a)³ = a³ - 1/a³ - 3(a - 1/a), we get x³ + 3x = a³ - 1/a³. Since 1/(7+5sqrt(2)) = -(7-5sqrt(2)) (as the denominator 49-50 = -1), we get a³ - 1/a³ = 14. Therefore x³ + 3x - 14 = 0.