Question

If \(x^{4/3} + x^{-1/3} = 1\), \(x^5 + 3x^2 - x\) is equal to

• A. 0
• B. 1
• C. -1
• D. 2

Hint:

After substitution, always reduce higher powers using the derived identity.

Answer 1

The Correct Option is C

Concept: Let \(x^{1/3} = t\).

Step 1: Substitute.
\[ t^4 + \frac{1}{t} = 1 \] Multiply by \(t\): \[ t^5 + 1 = t \Rightarrow t^5 - t + 1 = 0 \]

Step 2: Find required expression.
\[ x = t^3 \Rightarrow x^5 + 3x^2 - x = t^{15} + 3t^6 - t^3 \] Using \(t^5 = t - 1\), reduce powers: \[ t^{15} = (t^5)^3 = (t-1)^3 \] Simplifying gives: \[ x^5 + 3x^2 - x = -1 \]