EquationsMCQPYQ Dec 23Question 1063 of 221
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If α,β\displaystyle α\displaystyle \alpha, β\displaystyle \beta are the roots of the equation x2−4x+1=0\displaystyle x^2 - 4x + 1 = 0, then value of α3+β3\displaystyle α3+β3\displaystyle \alpha^3 + \beta^3 will be

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

✅ Option d — 52

All Options:

  • A-76
  • B76
  • C-52
  • D52

Detailed Solution & Explanation

Given the quadratic equation:
x2−4x+1=0x^2 - 4x + 1 = 0
Using Vieta's formulas:
Sum of roots: α+β=4\displaystyle α+β=4\displaystyle \alpha + \beta = 4
Product of roots: αβ=1\displaystyle α\displaystyle \alphaβ=1\displaystyle \beta = 1

We need to find the value of α3+β3\displaystyle α3+β3\displaystyle \alpha^3 + \beta^3. We use the identity:
α3+β3=(α+β)3−3αβ(α+β)α3+β3\displaystyle \alpha^3 + \beta^3 = (α+β\displaystyle \alpha + \beta)^3 - 3α\displaystyle \alphaβ\displaystyle \beta(α+β\displaystyle \alpha + \beta)
Substitute the sum and product:
α3+β3=(4)3−3(1)(4)α3+β3\displaystyle \alpha^3 + \beta^3 = (4)^3 - 3(1)(4)
α3+β3=64−12=52α3+β3=64−12=52\displaystyle \alpha^3 + \beta^3 = 64 - 12 = 52
Therefore, the value of α3+β3\displaystyle α3+β3\displaystyle \alpha^3 + \beta^3 is $52,whichmatchesOptiond.<br><br>∗Note:∗Acommonsignerrorintheformula,i.e.,using\displaystyle , which matches Option d.<br><br>*Note:* A common sign error in the formula, i.e., using+3α\displaystyle \alphaβ\displaystyle \beta(α+β\displaystyle \alpha+\beta)\displaystyle , leads to &#36;64 + 12 = 76 (Option b). The mathematically correct answer is $52$ (Option d).
**Option d**

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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