EquationsMCQPYQ Dec 23Question 1109 of 221
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The roots of the equation x3+x2−x−1=0\displaystyle x^3 + x^2 - x - 1 = 0 are

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

✅ Option d — x=1,x=1,x=1\displaystyle x = 1, x = 1, x = 1

All Options:

  • Ax=1,x=−1,x=−1\displaystyle x = 1, x = -1, x = -1
  • Bx=1,x=−1,x=1\displaystyle x = 1, x = -1, x = 1
  • Cx=−1,x=−1,x=−1\displaystyle x = -1, x = -1, x = -1
  • Dx=1,x=1,x=1\displaystyle x = 1, x = 1, x = 1

Detailed Solution & Explanation

Let us solve the cubic equation x3+x2−x−1=0\displaystyle x^3 + x^2 - x - 1 = 0 by factoring by grouping:
x3+x2−x−1=0x^3 + x^2 - x - 1 = 0
x2(x+1)−1(x+1)=0x^2(x + 1) - 1(x + 1) = 0
(x2−1)(x+1)=0(x^2 - 1)(x + 1) = 0
(x−1)(x+1)(x+1)=0(x - 1)(x + 1)(x + 1) = 0

This gives the three roots:
x−1=0  ⟹  x=1\displaystyle x - 1 = 0 \implies x = 1
x+1=0  ⟹  x=−1\displaystyle x + 1 = 0 \implies x = -1
x+1=0  ⟹  x=−1\displaystyle x + 1 = 0 \implies x = -1

Thus, the roots are x=1,x=−1,x=−1\displaystyle x = 1, x = -1, x = -1, which corresponds exactly to Option (a). (Note: The official key marks the correct option as Option (d) due to a typographical error, but mathematically Option (a) is the correct set of roots).

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

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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