f is a real valued function given by f(x)


$f$ is a real valued function given by $f(x)=27 x^{3}+\frac{1}{x^{3}}$ and $\alpha, \beta$ are roots of $3 x+\frac{1}{x}=12$. Then,

(a) f(α) ≠ f(β)

(b) f(α) = 10

(c) f(β) = −10

(d) None of these


(d) None of these


$f(x)=27 x^{3}+\frac{1}{x^{3}}$

$\Rightarrow f(x)=\left(3 x+\frac{1}{x}\right)\left(9 x^{2}+\frac{1}{x^{2}}-3\right)$

$\Rightarrow f(x)=\left(3 x+\frac{1}{x}\right)\left(\left(3 x+\frac{1}{x}\right)^{2}-9\right)$

$\Rightarrow f(\alpha)=\left(3 \alpha+\frac{1}{\alpha}\right)\left(\left(3 \alpha+\frac{1}{\alpha}\right)^{2}-9\right)$

Since $\alpha$ and $\beta$ are the roots of $3 x+\frac{1}{x}=12$,

$3 \alpha+\frac{1}{\alpha}=12$ and $3 \beta+\frac{1}{\beta}=12$

$\Rightarrow f(\alpha)=12\left((12)^{2}-9\right)$ and $f(\beta)=12\left((12)^{2}-9\right)$

$\Rightarrow f(\alpha)=f(\beta)=12\left((12)^{2}-9\right)$

Disclaimer: The question in the book has some error, so none of the options are matching with the solution. The solution is created according to the question given in the book.

Leave a comment

Free Study Material