If a and b are the roots of the quadratic equation ${6 x}^{2} - x - 2 = 0$ from an equation whose roots are $a^{2}$ and $b^{2}$ ?

{36 x}^{2} - 25 x + 4 = 0

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{36 x}^{2} + 25 x + 4 = 0

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{6 x}^{2} - 25 x + 4 = 0

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None of these

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Solution

The correct option is B ${36 x}^{2} - 25 x + 4 = 0$
Given equation is: $6 x^{2} - x - 2 = 0$
$a$ and $b$ are the roots of this equation.
$a + b = \frac{1}{6}$
$a b = - \frac{2}{6} = - \frac{1}{3}$
$\Rightarrow a^{2} b^{2} = \frac{1}{9}$
$a^{2} + b^{2} = {(a + b)}^{2} - 2 a b$
$= {(\frac{1}{6})}^{2} - 2 \times - \frac{1}{3}$
$= \frac{1}{36} + \frac{2}{3}$
$= \frac{25}{36}$
Thus, required equation
$= x^{2} - \frac{25}{36} x + \frac{1}{9} = 0$
$= {36 x}^{2} - 25 x + 4 = 0$

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