Question

If $\alpha ,\beta$ are the roots of the quadratic equation $x^2-px+q=0$, then
$(\alpha +\beta )x-({\alpha }^2+{\beta }^2)\frac{x^2}{2}+({\alpha }^3+{\beta }^3)\frac{x^3}{3}-\dots \infty =$

• A. $\log (1-px+q{x}^2)$
• B. $\log (1-qx+p{x}^2)$
• C. $log(1+qx+p{x}^2)$
• D. $\log (1+px+q{x}^2)$

Answer 1

The correct option is D $\log (1+px+q{x}^2)$

Here, $\alpha +\beta =p,\alpha .\beta =q$
$\therefore (\alpha +\beta )x-({\alpha }^2+{\beta }^2)\frac{x^2}{2}+({\alpha }^3+{\beta }^3)\frac{x^3}{3}-\dots \infty$
$=(\alpha x-\frac{(\alpha x{)}^2}{2}+\frac{(\alpha x{)}^3}{3}-....\dots \infty )+(\beta x-\frac{(\beta x{)}^2}{2}+\frac{(\beta x{)}^3}{3}-....\dots \infty )$
$=\log (1+\alpha x)+\log (1+\beta x)=\log (1+(\alpha +\beta )x+\alpha .\beta x^2)=\log (1+px+q{x}^2)$