EquationsPYQ Sept 25Question 4406 of 155
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If α\displaystyle \alpha and β\displaystyle \beta are the roots of the equation 2x2−6x+3=0\displaystyle 2x^2-6x+3=0, then the equation with the roots αβ\displaystyle \frac{\alpha}{\beta} and βα\displaystyle \frac{\beta}{\alpha} is

Options

Ax2+4x+1=0\displaystyle x^2+4x+1=0
Bx2−4x+1=0\displaystyle x^2-4x+1=0
Cx2−4x−1=0\displaystyle x^2-4x-1=0
Dx2+4x−1=0\displaystyle x^2+4x-1=0
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Correct Answer

✅ Option b — x2−4x+1=0\displaystyle x^2-4x+1=0

All Options:

  • Ax2+4x+1=0\displaystyle x^2+4x+1=0
  • Bx2−4x+1=0\displaystyle x^2-4x+1=0
  • Cx2−4x−1=0\displaystyle x^2-4x-1=0
  • Dx2+4x−1=0\displaystyle x^2+4x-1=0

Detailed Solution & Explanation

Given quadratic equation: 2x2−6x+3=0\displaystyle 2x^2 - 6x + 3 = 0. Since α\displaystyle \alpha and β\displaystyle \beta are the roots: α+β=−−62=3\alpha + \beta = -\frac{-6}{2} = 3 αβ=32\alpha\beta = \frac{3}{2}
We need to find the quadratic equation whose roots are r1=αβ\displaystyle r_1 = \frac{\alpha}{\beta} and r2=βα\displaystyle r_2 = \frac{\beta}{\alpha}. The sum of the new roots (S\displaystyle S) is: S=r1+r2=αβ+βα=α2+β2αβS = r_1 + r_2 = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}
Using the relation α2+β2=(α+β)2−2αβ\displaystyle \alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta: α2+β2=32−2(32)=9−3=6\alpha^2 + \beta^2 = 3^2 - 2\left(\frac{3}{2}\right) = 9 - 3 = 6
Now compute S\displaystyle S: S=63/2=4S = \frac{6}{3/2} = 4
The product of the new roots (P\displaystyle P) is: P=r1⋅r2=αβ⋅βα=1P = r_1 \cdot r_2 = \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1
The required quadratic equation is given by: x2−Sx+P=0  ⟹  x2−4x+1=0x^2 - Sx + P = 0 \implies x^2 - 4x + 1 = 0
Hence, **Option B** is the correct answer.

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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