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

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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

Detailed Solution & Explanation

Let the roots of the quadratic equation be α\displaystyle α\displaystyle \alpha and β\displaystyle β\displaystyle \beta. According to the question, one root is half of the other, so α=2β\displaystyle α=2\displaystyle \alpha = 2β\displaystyle \beta. The difference between the roots is a\displaystyle a: ∣α−β∣=a  ⟹  ∣2β−β∣=a  ⟹  ∣β∣=a|α−β\displaystyle \alpha - \beta| = a \implies |2β−β\displaystyle \beta - \beta| = a \implies |β\displaystyle \beta| = a Thus, β=a\displaystyle β=a\displaystyle \beta = a or β=−a\displaystyle β\displaystyle \beta = -a. If β=a\displaystyle β=a\displaystyle \beta = a, the roots are a\displaystyle a and $2a$. 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 β\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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