EquationsMCQMTP Mar 21Question 1098 of 221
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If arithmetic mean between roots of a quadratic equation is 8\displaystyle 8 and the geometric mean between them is 5\displaystyle 5, the equation is _______.

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

✅ Option b — x−16x+25=0\displaystyle x - 16x + 25 = 0

All Options:

  • Ax2−16x−25=0\displaystyle x^2 - 16x - 25 = 0
  • Bx−16x+25=0\displaystyle x - 16x + 25 = 0
  • Cx2−16x+5=0\displaystyle x^2 - 16x + 5 = 0
  • DNone of these

Detailed Solution & Explanation

Let the roots of the quadratic equation be α\displaystyle α\displaystyle \alpha and β\displaystyle β\displaystyle \beta.

We are given:
1. The arithmetic mean (AM) of the roots is 8\displaystyle 8:
α+β2=8  ⟹  α+β=16α+β2=8\displaystyle \frac{\alpha + \beta}{2} = 8 \implies α+β=16\displaystyle \alpha + \beta = 16

2. The geometric mean (GM) of the roots is 5\displaystyle 5:
αβ=5  ⟹  αβ=25αβ=5\displaystyle \sqrt{\alpha \beta} = 5 \implies α\displaystyle \alpha β=25\displaystyle \beta = 25

A quadratic equation with roots α\displaystyle α\displaystyle \alpha and β\displaystyle β\displaystyle \beta is represented by the formula:
x2−(α+β)x+αβ=0x^2 - (α+β\displaystyle \alpha + \beta)x + α\displaystyle \alpha β=0\displaystyle \beta = 0
Substituting the values of the sum (α+β=16\displaystyle α+β=16\displaystyle \alpha + \beta = 16) and product (αβ=25\displaystyle α\displaystyle \alpha β=25\displaystyle \beta = 25):
x2−16x+25=0x^2 - 16x + 25 = 0

Note that Option (b) is listed with a typographical error as x−16x+25=0\displaystyle x - 16x + 25 = 0 (missing the square exponent on the first term), but is the intended choice representing x2−16x+25=0\displaystyle x^2 - 16x + 25 = 0.

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

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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