Question

If roots of equation $2x^4-3x^3+2x^2-7x-1=0$ are $\alpha ,\beta ,\gamma \text{and}\delta$ then value of $\sum \frac{\alpha +1}{\alpha }$ is equal to ?

• A. -3
• B. 3
• C. $\frac{11}{2}$
• D. -11

Answer 1

The correct option is A -3

Sum of the roots$=\sum \alpha =\frac{3}{2}$

Sum of the roots taken two at a time$=\sum \alpha \beta =\frac{2}{2}=1$

Sum of the roots taken three at a time$=\sum \alpha \beta \gamma =-\frac{-7}{2}=\frac{7}{2}$

Product of the roots$=\alpha \beta \gamma \delta =-\frac{1}{2}$

$\sum \frac{\alpha +1}{\alpha }$

$=\sum 1+\frac{1}{\alpha }$

$=\sum 1+\sum \frac{1}{\alpha }$

$=4+\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }+\frac{1}{\delta }$

$=4+\frac{\beta \gamma \delta +\alpha \gamma \delta +\alpha \beta \delta +\alpha \beta \gamma }{\alpha \beta \gamma \delta }$

$=4+\frac{1}{\alpha \beta \gamma \delta }\sum \alpha \beta \gamma$

$=4-2\times \frac{7}{2}=4-7=-3$