Question

If $\alpha$and $\beta$ are the roots of the equation $x^2-6x+a=0$ and satisfy the relation $3\alpha +2\beta =16$, then the value of $a$ is

• A. $-8$
• B. $8$
• C. $16$
• D. $9$
• E. None of these

Answer 1

The correct option is B

$8$

Explanation for the correct option:

Step1. Find the value of $\alpha$and $\beta$:

Given that $\alpha$and $\beta$ are the roots of the equation $x^2-6x+a=0$ and satisfy the relation $3\alpha +2\beta =16$.

$\Rightarrow (\alpha +\beta )=6.........\left(i\right);\alpha .\beta =a$

Now, $3\alpha +2\beta =16$

$$\begin{gathered} \Rightarrow \alpha +2(\alpha +\beta )=16 \\ \Rightarrow \alpha +2\left(6\right)=16 \\ \Rightarrow \alpha +12=16 \\ \Rightarrow \alpha =16-12 \\ \Rightarrow \alpha =4 \end{gathered}$$

Putting te value of $\alpha$.we get

$$\begin{gathered} \beta =6-4 \\ \Rightarrow \beta =2 \end{gathered}$$

Step2. Find the value of $\mathit{a}$:

since $a=\alpha .\beta$

$$\begin{gathered} \Rightarrow a=4\times 2 \\ \Rightarrow a=8 \end{gathered}$$

Hence, the correct option is B.