Question

If roots of the equation $x^3-b{x}^2+cx-d=0$ are in a G.P., then?

• A. $b^3=c^{3}d$
• B. $b^3{c}^3=d$
• C. $c^3=b^{3}d$
• D. $bc=d^3$

Answer 1

Answer: (C)

$c^3=b^{3}d$

Given

roots of $x^3-b{x}^2+cx-d=0$ are in $G\cdot P$

Let take roots $a,ar,a{r}^2$

Sum of roots=b

$a+ar+a{r}^2=b$

$a(1+r+r^2)=b-\left(i\right)$

$a\times ar+a\times a{r}^2+ar\times a{r}^2=c$

$a^{2}r+a^2{r}^2+a^2{r}^3=c$

$a^{2}r(1+r+r^2)=c-\left(i\right)$

Divide $\left(ii\right)by\left(i\right)$

$ar=\frac{a}{b}$

Product of roots d

$2\times ar\times a{r}^2=d$

$a^3{r}^3=d$

$(ar{)}^3=d$

$\frac{c^3}{b^3}=d$

$c^3=b^{3}d$

Option (C) is correct