First slide

Theory of equations

Question

The value of $k$ for which one of the roots of $x^{2} - x + 3 k = 0$ is double of one of the roots of $x^{2} - x + k = 0$ is

Moderate

A

1
B

$- 2$
C

2
D

None of these

Solution

Let $α$ be a root of $x^{2} - x + k = 0$ ,
then $2 α$ is a root of $x^{2} - x + 3 k = 0$ .
Then the equations are $α^{2} - α + k = 0$ ….. ( 1 )
and $4 α^{2} - 2 α + 3 k = 0$ ….. ( 2 )
solve ( 1 ) & ( 2 )
we get,
$\Rightarrow$ $\frac{α^{2}}{- 3 k + 2 k} = \frac{α}{4 k - 3 k} = \frac{1}{- 2 + 4}$
$\Rightarrow$ $α^{2} = - \frac{k}{2}$ and $α = \frac{k}{2}$
Now, $α^{2} = {(α)}^{2} \Rightarrow - \frac{k}{2} = {(\frac{k}{2})}^{2}$
$\Rightarrow$ $k^{2} + 2 k = 0 \Rightarrow k = - 2$ (k not equals to 0)

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