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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2022 · Mathematics · Quadratic Equation

Let α and β(α > β) be the roots of the equation x^2-8 x+q=0 . If...

Let α\alpha and β(α>β)\beta(\alpha > \beta) be the roots of the equation x2−8x+q=0x^2-8 x+q=0. If α2−β2=16\alpha^2-\beta^2=16, then what is the value of qq ?

Options

  1. A-15
  2. B-10
  3. C10
  4. D15

Correct answer

D. 15

Step-by-step solution

Given that α\alpha and β(α>β)\beta(\alpha > \beta) are roots of the equation x2−8x+q=0∴α+β=8 and α⋅β=q∴(α−β)2=(α+β)2−4αβ=64−4q(α−β)=64−4q(∵α>β)\begin{array}{ll} & x^2-8 x+q=0 \\ \therefore & \alpha+\beta=8 \text{ and } \alpha \cdot \beta=q \\ \therefore & (\alpha-\beta)^2=(\alpha+\beta)^2-4 \alpha \beta=64-4 q \\ & (\alpha-\beta)=\sqrt{64-4 q} \quad(\because \alpha > \beta)\end{array} Now α2−β2=16⇒ and (α+β)−(α−β)=16⇒ and 864−4q=16⇒64−4q=2⇒ and 64−4q=4⇒q=15\begin{aligned}& \text{ Now } \alpha^2-\beta^2=16 \\ \Rightarrow \text{ and } (\alpha+\beta)-(\alpha-\beta)=16 \\ \Rightarrow \text{ and } 8 \sqrt{64-4 q}=16 \quad \Rightarrow \sqrt{64-4 q}=2 \\ \Rightarrow \text{ and } 64-4 q=4 \quad \Rightarrow q=15\end{aligned}$
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