EquationsMCQMTP May 19 Series IIQuestion 1069 of 221
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If α\displaystyle α\displaystyle \alpha and β\displaystyle β\displaystyle \beta be the roots of the equation $2x^2 - 4x - 3 = 0$ the value of $α2+β2\displaystyle \alpha^2 + \beta^2$ is

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

✅ Option b — 7

All Options:

  • A5
  • B7
  • C7
  • D4

Detailed Solution & Explanation

Given the quadratic equation:
2x2−4x−3=02x^2 - 4x - 3 = 0
Using Vieta's formulas:
Sum of roots: α+β=−ba=−−42=2\displaystyle α+β\displaystyle \alpha + \beta = -ba\displaystyle \frac{b}{a} = -−42=2\displaystyle \frac{-4}{2} = 2
Product of roots: αβ=ca=−32\displaystyle α\displaystyle \alphaβ=ca\displaystyle \beta = \frac{c}{a} = -32\displaystyle \frac{3}{2}

We need to find the value of α2+β2\displaystyle α2+β2\displaystyle \alpha^2 + \beta^2. Using the algebraic identity:
α2+β2=(α+β)2−2αβα2+β2\displaystyle \alpha^2 + \beta^2 = (α+β\displaystyle \alpha + \beta)^2 - 2α\displaystyle \alphaβ\displaystyle \beta
Substitute the values:
α2+β2=(2)2−2(−32)α2+β2\displaystyle \alpha^2 + \beta^2 = (2)^2 - 2\left(-32\displaystyle \frac{3}{2}\right)
α2+β2=4+3=7α2+β2=4+3=7\displaystyle \alpha^2 + \beta^2 = 4 + 3 = 7
Therefore, the value of α2+β2\displaystyle α2+β2\displaystyle \alpha^2 + \beta^2 is 7\displaystyle 7, which corresponds to Option b (and Option c).
**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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