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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2021 · Mathematics · Quadratic Equation

If α and β are the roots of the equation 4 x^2+2 x-1=0 , then which...

If α\alpha and β\beta are the roots of the equation 4x2+2x−1=04 x^2+2 x-1=0, then which one of the following is correct?

Options

  1. Aβ=−2α2−2α\beta=-2 \alpha^2-2 \alpha
  2. Bβ=4α2−3α\beta=4 \alpha^2-3 \alpha
  3. Cβ=α2−3α\beta=\alpha^2-3 \alpha
  4. Dβ=−2α2−2α\beta=-2 \alpha^2-2 \alpha

Correct answer

A. β=−2α2−2α\beta=-2 \alpha^2-2 \alpha

Step-by-step solution

α\alpha and β\beta are roots of 4x2+2x−1=04 x^2+2 x-1=0. ∴\therefore Sum of roots (α+β)=−24=−12∴β=−12−α=−1−2α2(\alpha+\beta)=-\frac{2}{4}=-\frac{1}{2}\therefore \beta=-\frac{1}{2}-\alpha=\frac{-1-2 \alpha}{2} and 4α2+2α−1=0⇒4α2=1−2α\begin{aligned} & \text{ and } 4 \alpha^2+2 \alpha-1=0 \\ & \Rightarrow 4 \alpha^2=1-2 \alpha\end{aligned}$Now,Now,-2 \alpha^2-2 \alpha=\frac{2 \alpha-1}{2}-2 \alpha=\frac{-2 \alpha-1}{2}=\beta$.
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