Question

answer the 1st question

Answer 1

the val ue of k is 2

Answer 2

$hi..$
$here's \: \: your \: \: answer..$

One of the root is 2 / 3
So..
$\alpha = \frac{2}{3}$

Now ,
Comparing the quadratic equation
$6 {x}^{2} - x - k = 0$
with
$a {x}^{2} + bx + c = 0$
Here ..

a = 6 , b = -1 , c = -k

We know that ,

$\alpha + \beta = \frac{ - b}{a}$

$\frac{2}{3} + \beta = \frac{ - ( - 1)}{6}$

$\frac{2}{3} + \beta = \frac{1}{6}$

$\beta = \frac{1}{6} - \frac{2}{3}$

$\beta = \frac{1 - 4}{6} \: ................( l.c.m)$

$\beta = \frac{ - 3}{6}$

$\beta = \frac{ - 1}{2}$

Also , we know that ,

$\alpha \beta = \frac{c}{a}$

$( \frac{2}{3} )( \frac{ - 1}{2} ) = \frac{ - k}{6}$

$\frac{ - 2}{6} = \frac{ - k}{6}$

[Multiplying both sides by 6]

$- 2 = - k$

$k = 2$
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Therefore ,
The value of k is 2 .
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$hope \: \: \: this \: \: \: helps..$