EquationsMCQPYQ Dec 21Question 1049 of 221
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If one root is half of the other of a quadratic equation and the difference in roots is a\displaystyle a, then the equation is

Options

Ax2+ax+2a2=0\displaystyle x^2 + ax + 2a^2 = 0
Bx2−3ax−2a2=0\displaystyle x^2 - 3ax - 2a^2 = 0
Cx2−3ax+2a2=0\displaystyle x^2 - 3ax + 2a^2 = 0
Dx2+3ax−2a2=0\displaystyle x^2 + 3ax - 2a^2 = 0
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Correct Answer

✅ Option c — x2−3ax+2a2=0\displaystyle x^2 - 3ax + 2a^2 = 0

All Options:

  • Ax2+ax+2a2=0\displaystyle x^2 + ax + 2a^2 = 0
  • Bx2−3ax−2a2=0\displaystyle x^2 - 3ax - 2a^2 = 0
  • Cx2−3ax+2a2=0\displaystyle x^2 - 3ax + 2a^2 = 0
  • Dx2+3ax−2a2=0\displaystyle x^2 + 3ax - 2a^2 = 0

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Detailed Solution & Explanation

Let the roots of the quadratic equation be α\displaystyle \alpha and β\displaystyle \beta.
According to the question, one root is half of the other, so α=2β\displaystyle \alpha = 2\beta.
The difference between the roots is a\displaystyle a:
∣α−β∣=a  ⟹  ∣2β−β∣=a  ⟹  ∣β∣=a|\alpha - \beta| = a \implies |2\beta - \beta| = a \implies |\beta| = a
Thus, β=a\displaystyle \beta = a or β=−a\displaystyle \beta = -a.
If β=a\displaystyle \beta = a, the roots are a\displaystyle a and 2a\displaystyle 2a. The quadratic equation is:
(x−a)(x−2a)=0  ⟹  x2−3ax+2a2=0(x - a)(x - 2a) = 0 \implies x^2 - 3ax + 2a^2 = 0
If β=−a\displaystyle \beta = -a, the roots are −a\displaystyle -a and −2a\displaystyle -2a. The quadratic equation is:
(x+a)(x+2a)=0  ⟹  x2+3ax+2a2=0(x + a)(x + 2a) = 0 \implies x^2 + 3ax + 2a^2 = 0
Thus, the equation is x2−3ax+2a2=0\displaystyle x^2 - 3ax + 2a^2 = 0.
**Option c**

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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