Q71 / 200 · Manipal (MET) 2010 · Mathematics

If $a\gt1$, roots of the equation $(1-a) x^2+3 a x-1=0$ are

  • A
    one positive and one negative
  • B
    both negative
  • Correct answer
    C
    both positive
  • D
    both non-real complex

Solution

Let $\alpha, \beta$ are the roots of the equation $(1-a) x^2+3 a x-1=0$, then $\alpha+\beta=\frac{3 a}{a-1}$, $\alpha \beta=\frac{1}{a-1}$ As $a\gt1, \alpha+\beta\gt0$ and $\alpha \beta\gt0$ and $D=9 a^2+4(1-a)$ $=9\left(a^2-\frac{4}{9} a+\frac{4}{9}\right)=9\left\{a\left(a-\frac{4}{9}\right)+\frac{4}{9}\right\}\gt0 \text {, }$ as $a\gt1$ $\therefore$ The equation has real and positive roots.

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

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