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If x=51/3+5−1/3\displaystyle x = 5^{1/3} + 5^{-1/3}, then $5x^3 - 15x$ is given by

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — $26$

All Options:

  • A$25$
  • B$26$
  • C$27$
  • D$30$

Detailed Solution & Explanation

Given x=51/3+5−1/3\displaystyle x = 5^{1/3} + 5^{-1/3}.

Let us cube both sides using the identity (a+b)3=a3+b3+3ab(a+b)\displaystyle (a+b)^3 = a^3 + b^3 + 3ab(a+b):
x3=(51/3+5−1/3)3x^3 = \left(5^{1/3} + 5^{-1/3}\right)^3
x3=(51/3)3+(5−1/3)3+3(51/3)(5−1/3)(51/3+5−1/3)x^3 = \left(5^{1/3}\right)^3 + \left(5^{-1/3}\right)^3 + 3\left(5^{1/3}\right)\left(5^{-1/3}\right)\left(5^{1/3} + 5^{-1/3}\right)

Since 51/3⋅5−1/3=50=1\displaystyle 5^{1/3} ⋅\displaystyle \cdot 5^{-1/3} = 5^0 = 1 and 51/3+5−1/3=x\displaystyle 5^{1/3} + 5^{-1/3} = x, we simplify:
x3=5+5−1+3(1)(x)x^3 = 5 + 5^{-1} + 3(1)(x)
x3=5+15+3xx^3 = 5 + 15+3x\displaystyle \frac{1}{5} + 3x
x3=265+3xx^3 = 265+3x\displaystyle \frac{26}{5} + 3x

Multiply the entire equation by 5\displaystyle 5 to eliminate the fraction:
5x3=26+15x5x^3 = 26 + 15x
5x3−15x=265x^3 - 15x = 26

Hence, the correct option is **Option (b)**.

About This Chapter: Equations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Linear, Quadratic and Cubic Equations

This chapter covers Linear, Quadratic and Cubic Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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