f is a real valued function given by f(x)=27x^3+1 x^3 and , are r… - Vidyadip

MCQ

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

A
$\text{f}(\alpha)\neq\text{f}(\beta)$
B
$\text{f}(\alpha)=10$
C
$\text{f}(\beta)=-10$
✓
None of these.

✓

Answer

Correct option: D.

None of these.

$\text{f(x)}=27\text{x}^3+\frac{1}{\text{x}^3}$
$\Rightarrow\text{f(x)}=\Big(3\text{x}+\frac{1}{\text{x}}\Big)\Big(9\text{x}^2+\frac{1}{\text{x}^2}-3\Big)$
$\Rightarrow\text{f(x)}=\Big(3\text{x}+\frac{1}{\text{x}}\Big)\Big(\Big(3\text{x}+\frac{1}{\text{x}}\Big)^2-9\Big)$
$\Rightarrow\text{f}(\alpha)=\Big(3\alpha+\frac{1}{\alpha}\Big)\Big(\Big(3\alpha+\frac{1}{\alpha}\Big)^2-9\Big)$
Since $\alpha$ and $\beta$ are the roots of $3\text{x}+\frac{1}{\text{x}}=12,$
$3\alpha+\frac{1}{\alpha}=12$ and $3\beta+\frac{1}{\beta}=12$
$\Rightarrow\text{f}(\alpha)=12\big((12)^2-9\big)$ and $\text{f}(\beta)=12\big((12)^2-9\big)$
$\Rightarrow\text{f}(\alpha)=\text{f}(\beta)=\big((12)^2-9\big)$

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