Question

If $\alpha$ occurs $p$ times and $\beta$ occurs $q$ times in polynomial equation $f\left(x\right)=0$ of degree $n(1<p,q<n)$, then which of the following is not true? $($ where $f^r\left(x\right)$ represents $r^{th}$ derivative of $f\left(x\right)$ w.r.t. $x)$

• A. If $p<q<n$, then $\alpha$ and $\beta$ are two of the roots of the equation $f^{p-1}\left(x\right)=0$
• B. If $q<p<n$, then $\alpha$ and $\beta$ are two of the roots of the equation $f^{q-1}\left(x\right)=0$
• C. If $p<q<n$, then equations $f\left(x\right)=0$ and $f^{q-1}\left(x\right)=0$ have exactly one root common
• D. If $p<q<n$, then equations $f^q\left(x\right)=0$ and $f^p\left(x\right)=0$ have exactly one root common

Answer 1

The correct option is D If $p<q<n$, then equations $f^q\left(x\right)=0$ and $f^p\left(x\right)=0$ have exactly one root common

$f\left(x\right)={(x-\alpha )}^p.{(x-\beta )}^q.g\left(x\right)$
$f\left(\alpha \right)=f^1\left(\alpha \right)=f^2\left(\alpha \right)=........=f^{p-1}\left(\alpha \right)=0$ .........(1)
$g\left(\beta \right)=g^1\left(\beta \right)=g^2\left(\beta \right)=............=g^{q-1}\left(\beta \right)=0$ ........(2)
$\therefore$ if $p<q<n\Rightarrow \alpha ,\beta$ are the roots of $f^{p-1}\left(x\right)=0$
If $q<p<n\Rightarrow \alpha ,\beta$ are the roots of $f^{q-1}\left(x\right)=0$
If $p<q<n$ then $f\left(x\right)=0$ and $f^{q-1}\left(x\right)=0$ has exactly one root common i.e. $x=\beta$