Q190 / 270 · NDA 2016 (Phase 2) · Mathematics

Paragraph: $2 x^2+3 x-\alpha-0$ has roots -2 and $\beta$ while the equation $x^2-$ $3 \mathrm{mx}+2 \mathrm{~m}^2=0$ has both roots positive, where $\alpha \gt 0$ and $\beta \gt $ 0 .
Question: If $\beta, 2,2 \mathrm{~m}$ are in GP, then what is the value of $\beta \sqrt{\mathrm{m}}$ ?

  • Correct answer
    A
    $1$
  • B
    $2$
  • C
    $4$
  • D
    $6$

Solution

$\beta=\frac{1}{2}$ $\beta, 2,2 \mathrm{~m}$ are in GP. $\begin{aligned} & \Rightarrow \frac{2}{\beta}=\frac{2 m}{2} \\ & \Rightarrow \mathrm{~m}=\frac{2}{\beta}=2 \times \frac{2}{1} \\ & \Rightarrow \mathrm{~m}=4 \\ & \Rightarrow \beta \sqrt{m}=\frac{1}{2} \times \sqrt{4}=1\end{aligned}$

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

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