Question

If the roots of $x^4+5x^3-30x^2+40x+64=0$, are in G.P. then the roots of $x^4-5x^3-30x^2-40x+64=0$ are in

• A. A.P
• B. G.P
• C. H.P
• D. A.G.P

Answer 1

Answer: (B)

G.P

Let ${\alpha }_1:{\alpha }_4$ be the roots of $x^4+5x^3-30x^2+40x+64=0$.

$\therefore \sum {\alpha }_i=-5,\sum {\alpha }_i{\alpha }_j=-30,\sum {\alpha }_i{\alpha }_j{\alpha }_k=-40$ and $\prod {\alpha }_i=64$.

Now, let ${\beta }_1:{\beta }_4$ be the roots of $x^4-5x^3-30x^2-40x+64=0$.

$\therefore \sum {\beta }_i=5,\sum {\beta }_i{\beta }_j=-30,\sum {\beta }_i{\beta }_j{\beta }_k=40$ and $\prod {\beta }_i=64$.

It is evident from the equations that $\sum {\beta }_i=-\sum {\alpha }_i,\sum {\beta }_i{\beta }_j=\sum {\alpha }_i{\alpha }_j,\sum {\beta }_i{\beta }_j{\beta }_k=-\sum {\alpha }_i{\alpha }_j{\alpha }_k$ and $\prod {\beta }_i=\prod {\alpha }_i$.

Hence, ${\beta }_i=-{\alpha }_i$

So, if ${\alpha }_i$ are in GP so are ${\beta }_i$.