Q160 / 270 · NDA 2016 (Phase 2) · Mathematics

If both the roots of the equation $x^2-2 k x+k^2-4=0$ lie between -3 and 5 , then which one of the following is correct?

  • A
    $-2 \lt \mathrm{k} \lt 2$
  • B
    $-5 \lt \mathrm{k} \lt 3$
  • C
    $-3 \lt k \lt 5$
  • Correct answer
    D
    $-1 \lt \mathrm{k} \lt 3$

Solution

$\mathrm{x}^2-2 \mathrm{kx}+\mathrm{k}^2-4=0$ $\begin{array}{ll}\Rightarrow & (\mathrm{x}-\mathrm{k})^2-2^2=0 \\ \Rightarrow & (\mathrm{x}-\mathrm{k}-2)(\mathrm{x}-\mathrm{k}+2)=0 \\ \Rightarrow & \mathrm{x}=\mathrm{k}+2, \mathrm{k}-2 \\ \Rightarrow & \mathrm{k}+2 \lt 5 \& \mathrm{k}-2 \gt -3 \\ \Rightarrow & \mathrm{k} \lt 3 \& \mathrm{k} \gt -1 \\ \Rightarrow & -1 \lt \mathrm{k} \lt 3\end{array}$

Hence, the correct answer is option: 'D'.

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