Question

If $\alpha ,\beta ,\gamma$ are the roots of the equation$x^3+a{x}^2+bx+2=0$ such that $\alpha \beta$ = –1, then the value of 2a + b will be .

• A. 1
• B. 0.0
• C. 5
• D. -5

Answer 1

The correct option is D -5

$\mathrm{Let}\alpha ,\beta ,\mathrm{and}\gamma \mathrm{be}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}{\mathrm{Ax}}^3+{\mathrm{Bx}}^2+\mathrm{Cx}+D=0,\mathrm{then}\mathrm{\alpha \beta \gamma }=\frac{-D}{A}\mathrm{For}\mathrm{the}\mathrm{given}\mathrm{equation},\mathrm{\alpha \beta \gamma }=-2\&\mathrm{and}\mathrm{given}\mathrm{that}\mathrm{\alpha \beta }=-1\Rightarrow -1\times \gamma =-2\Rightarrow \gamma =2\because x=2\mathrm{is}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{equation}.\therefore \mathrm{Putting}x=2\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{equation},\mathrm{we}\mathrm{get}:{\left(2\right)}^3+a{\left(2\right)}^2+2b+2=0\Rightarrow 8+4a+2b+2=0\Rightarrow 4a+2b=-10\Rightarrow 2a+b=-5$