Question

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+x+5=0$, then $y=\sum {\alpha }^2+\alpha \beta \gamma$ satisfies the equation

• A. $y^3+y+2=0$
• B. $y^3–y^2–y–2=0$
• C. $y^3+3y^2–y–3=0$
• D. $y^3+4y^2+5y+20=0$

Answer 1

The correct option is B

$y^3–y^2–y–2=0$

Explanation for the correct option:

Given, $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+x+5=0$

then, $\sum \alpha =3$

$\sum \alpha \beta =1$

$\alpha \beta \gamma =-5$

$\therefore$$y=\sum {\alpha }^2+\alpha \beta \gamma$

$$\begin{gathered} ={\left(\sum \alpha \right)}^2-2\sum \alpha \beta +\alpha \beta \gamma \\ =9-2\times 1+\left(-5\right) \\ =2 \end{gathered}$$

Thus, $y=2$ satisfies the equation ${\mathit{y}}^{\mathbf{3}}\mathbf{}\mathbf{–}\mathbf{}{\mathit{y}}^{\mathbf{2}}\mathbf{}\mathbf{–}\mathbf{}\mathit{y}\mathbf{}\mathbf{–}\mathbf{}\mathbf{2}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}$

Hence, Option ‘B’ is Correct.