Question

Let $\alpha ,\beta$ are roots of $x^2-3x+5=0$ then value of $\frac{5\alpha \cdot {\beta }^{2016}+3{\alpha }^{2017}\cdot \beta +\alpha \cdot {\beta }^{2018}}{{\alpha }^{2016}+{\beta }^{2016}}$

• A. 5
• B. 3
• C. 15
• D. $5/3$

Answer 1

Answer: (C)

$\alpha +\beta =3$ and $\alpha \beta =5$
$\frac{5\alpha {\beta }^{2016}+3{\alpha }^{2017}\beta +\alpha {\beta }^{2018}}{{\alpha }^{2016}+{\beta }^{2016}}=\frac{{\alpha }^2{\beta }^{2017}+{\alpha }^{2018}\beta +{\alpha }^{2017}{\beta }^2+\alpha {\beta }^{2018}}{{\alpha }^{2016}+{\beta }^{2016}}$
$=\frac{{\alpha }^2\beta \left({\alpha }^{2016}+{\beta }^{2016}\right)+{\beta }^2\alpha \left({\alpha }^{2016}+{\beta }^{2016}\right)}{{\alpha }^{2016}+{\beta }^{2016}}={\alpha }^2\beta +{\beta }^2\alpha =15$