Question

If t be a real number satisfying the equation $2t^3-9t^2+30-a=0$, then the value of the parameter $a$ for which the equation $x+\frac{1}{x}=t$, gives six real and distinct values of $x$.

• A. $-11$
• B. $10$
• C. $21$
• D. No value exists

Answer 1

The correct option is D No value exists

We have $2t^3-9t^2+30-a=0$
Any real root t${}_0$ of this equation gives two real and distinct values of x if $\left|t_0\right|>2$.
Thus, we need to find the condition for the equation in t to have three real and distinct roots, none of which lies in $[-2,2].$
Let $f\left(t\right)=2t^3-9t^2+30-a$
$f^{\prime }\left(t\right)=6t^2-18t=0\Rightarrow t=0,3.$
So, the equation $f\left(t\right)=0$ has three real and distinct roots if ,$f\left(0\right)\cdot f\left(3\right)<0$
$\Rightarrow (30-a)\cdot (54-81+30-a)<0$
$\Rightarrow (30-a)(3-a)<0$
$\Rightarrow (a-3)(a-30)<0\Rightarrow a\in (3,30)\cdots \left(i\right)$
From graph of polynomial :
If $\alpha ,\beta$ and $\gamma$ are roots of cubic polynomial,
and no root lies in $[-2,2]$
So, $(-2,2)$ will lie between roots $\alpha$ and $\beta$
$\Rightarrow f(-2)>0\&f\left(2\right)>0$
$\Rightarrow$ $(-16-36+30-a)>0\&(16-36+30-a)>0$
$\Rightarrow a<-22\&a<10$
$\Rightarrow a<-22\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right):a\in \varphi$