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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2018 · Mathematics · Quadratic Equation

If α and β(≠ 0) are the roots of the quadratic equation x^2+α x-β=0...

If α\alpha and β(≠0)\beta(\neq 0) are the roots of the quadratic equation x2+αx−β=0x^2+\alpha x-\beta=0, then the quadratic expression −x2+αx+β-x^2+\alpha x+\beta where x∈x \in R has

Options

  1. ALeast value −14-\frac{1}{4}
  2. BLeast value −94-\frac{9}{4}
  3. CGreatest value \frac{1}{4}
  4. DGreatest value \frac{9}{4}

Correct answer

D. Greatest value \frac{9}{4}

Step-by-step solution

α+β=−α,αβ=−β⇒αβ+β=0⇒(α+1)β=0⇒α=−1(∵β≠0)\begin{aligned} & \alpha+\beta=-\alpha, \alpha \beta=-\beta \\ \Rightarrow & \alpha \beta+\beta=0 \\ \Rightarrow & (\alpha+1) \beta=0 \\ \Rightarrow & \alpha=-1(\because \beta \neq 0)\end{aligned}⇒(2α+β)=0⇒β=2∴−x2+αx+β=−x2−x+2Greatest value =−1+8−4=94\begin{aligned} & \Rightarrow \quad(2 \alpha+\beta)=0 \\ & \Rightarrow \quad \beta=2 \\ & \therefore-x^2+\alpha x+\beta=-x^2-x+2 \\ & \text{Greatest value }=-\frac{1+8}{-4}=\frac{9}{4}\end{aligned}$
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