Question

If one root of $3x^2=8x+(2k+1)$ is seven times the other, then the value of $k$ is

• A. $5/3$
• B. $-5/3$
• C. $2/3$
• D. $-3/2$

Answer 1

Answer: (C)

$2/3$

The given polynomial is $p\left(x\right)=3x^2–8x+2k+1$

Let $\alpha$ and $\beta$ are be the zeroes if the given equation.

Given, $\alpha =7\beta ...\left(1\right)$

Now, $\alpha +\beta =-\frac{\text{Coefficient of}x}{\text{Coefficient of}{x}^2}=-\frac{(-8)}{3}=\frac{8}{3}$

$\Rightarrow 7\beta +\beta =\frac{8}{3}$[Using $\left(1\right)$]

$\Rightarrow 8\beta =\frac{8}{3}$

$\Rightarrow \beta =\frac{1}{3}$

then, $\alpha =\frac{7}{3}$

and, $\alpha \beta =\frac{\text{Constant term}}{\text{Coefficient of}{x}^2}=\frac{2k+1}{3}$

$\Rightarrow \frac{7}{3}\times \frac{1}{3}=\frac{2k+1}{3}$

$\Rightarrow 2k=\frac{7}{3}-1$

$\Rightarrow 2k=\frac{7-3}{3}$

$\Rightarrow 2k=\frac{4}{3}$

$\Rightarrow k=\frac{2}{3}$

So, $\text{C}$ is the correct option.