Question

If $\alpha ,\beta$ are roots of the equation $2x^2-35x+2=0$, then the value of ${(2\alpha -35)}^3\times {(2\beta -35)}^3$ is equal to?

• A. $10$
• B. $100$
• C. $64$
• D. $1$

Answer 1

The correct option is C

$64$

Explanation of the correct answer:

$\alpha ,\beta$ are roots of the equation $2x^2-35x+2=0$

Sum of roots $(\alpha +\beta )=\frac{-b}{a}=\frac{35}{2}$

Product of roots $\left(\alpha \beta \right)=\frac{c}{a}=1$

When $\alpha$ is a root

$$\begin{gathered} f\left(\alpha \right)=2{\alpha }^2-35\alpha +2=0 \\ f\left(\alpha \right)=2\alpha -35=\frac{-2}{\alpha } \end{gathered}$$

When $\beta$ is a root

$$\begin{gathered} f\left(\beta \right)=2{\beta }^2-35\beta +2=0....\left(i\right) \\ f\left(\beta \right)=2\beta -35=\frac{-2}{\beta }....\left(ii\right) \end{gathered}$$

Putting $\left(i\right)$ and $\left(ii\right)$ in ${(2\alpha -35)}^3\times {(2\beta -35)}^3$

$$\begin{gathered} {(2\alpha -35)}^3\times {(2\beta -35)}^3 \\ ={\left(\frac{-2}{\alpha }\right)}^3\times {\left(\frac{-2}{\beta }\right)}^3 \\ =\frac{64}{\alpha \beta } \\ =\frac{64}{1}=64 \end{gathered}$$

Hence, option (C) is correct.