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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2016 · Mathematics · Quadratic Equation

Let α and β(α 0 and c 0 2. α^2 β+β^2 α > 0 Which of the above...

Let α\alpha and β(α<β)\beta(\alpha < \beta) be the roots of the equation x2+bx+c=0\mathrm{x}^2+\mathrm{bx}+\mathrm{c}=0, where b>0\mathrm{b} > 0 and c<0\mathrm{c} < 0. Consider the following: 1. α+β+αβ>02.α2β+β2α>0\alpha+\beta+\alpha \beta > 0 2. \alpha^2 \beta+\beta^2 \alpha > 0 Which of the above is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

B. 2 only

Step-by-step solution

Sum of roots =α+β=−b=\alpha+\beta=-b Multiplication of roots =αβ=c=\alpha \beta=c Hence α+β+αβ=−b+cα2β+β2=αβ(α+β)=−bc∵b>0c<0∴−b+c<0−bc>0\begin{aligned} & \alpha+\beta+\alpha \beta=-b+c \quad \alpha^2 \beta+\beta^2=\alpha \beta(\alpha+\beta)=-b c \\ & \because \mathrm{ b} > 0 \quad & \quad \mathrm{c} < 0 \\ & \therefore-\mathrm{b}+\mathrm{c} < 0 \quad & \quad-\mathrm{bc} > 0 \end{aligned}
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