Theory Of Equations Ques 59
- If one real root of the quadratic equation $81 x^{2}+k x+256=0$ is cube of the other root, then a value of $k$ is
(2019 Main, 11 Jan I)
(a) 100
(b) 144
(c) -81
(d) -300
Show Answer
Answer:
Correct Answer: 59.(d)
Solution:
- Given quadratic equation is
$ 81 x^{2}+k x+256=0 $
Let one root be $\alpha$, then other is $\alpha^{3}$.
Now, $\alpha+\alpha^{3}=-\frac{k}{81}$ and $\alpha \cdot \alpha^{3}=\frac{256}{81}$
$\left[\because\right.$ for $a x^{2}+b x+c=0$, sum of roots $=-\frac{b}{a}$
and product of roots $=\frac{c}{a}$ ]
$ \begin{aligned} & \Rightarrow \quad \alpha^{4}=(\frac{4}{3})^{4} \Rightarrow \alpha= \pm \frac{4}{3} \\ & \therefore \quad k=-81\left(\alpha+\alpha^{3}\right) \\ & =-81 \alpha\left(1+\alpha^{2}\right) \\ & =-81 \quad (\pm \frac{4}{3}) \quad (1+\frac{16}{9})= \pm 300 \end{aligned} $
BETA
