Question

The value of $P$ for which the equation $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$ has both the roots at infinity is

• A. $1$
• B. $2$
• C. $0$
• D. $-2$

Answer 1

The correct option is A $1$

If $a{x}^2+bx+c=0$ has both the roots at infinity, then it implies that $a=0,b=0$ and $c\ne 0$.
Given $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$,
Comparing both, we get
$P^3-3P^2+2P=0$
$\Rightarrow P(P^2-3P+2)=0$
$\Rightarrow P(P^2-2P-P+2)=0$
$\Rightarrow P(P-1)(P-2)=0$
$\Rightarrow P=0,1,2\cdots \cdots \left(i\right)$

Also,
$P^3-P=0$
$\Rightarrow P(P-1)(P+1)=0$
$\Rightarrow P=0,\pm 1\cdots \cdots \left(ii\right)$

Also,
$P^3+3P^2+2P\ne 0$
$\Rightarrow P(P^2+3P+2)=0$
$\Rightarrow P(P^2+2P+P+2)\ne 0$
$\Rightarrow P(P+1)(P+2)\ne 0$
$\Rightarrow P\ne 0,-1,-2\cdots \cdots \left(iii\right)$
$\therefore$ From $\left(i\right),\left(ii\right),\left(iii\right)$, we get
$P=1$