First slide

Theory of equations

Question

If the roots of the equation $12 x^{2} - mx + 5 = 0$ are in the ratio $2 : 3$ , then $m =$

Moderate

A

$5 \sqrt{10}$
B

$3 \sqrt{10}$
C

$2 \sqrt{10}$
D

None of these

Solution

Let roots are $α, β$ so, $\frac{α}{β} = \frac{2}{3} \Rightarrow α = \frac{2 β}{3}$
$∴ α + β = \frac{m}{12}$
$\Rightarrow \frac{2 β}{3} + β = \frac{m}{12} \Rightarrow \frac{5 β}{3} = \frac{m}{12} . . . (i)$
and $αβ = \frac{5}{12} \Rightarrow \frac{2 β}{3} . β = \frac{5}{12} \Rightarrow β^{2} = \frac{5}{8}$
$\Rightarrow β = \sqrt{5 / 8}$
Put the value of $β$ in (i), $(i), \frac{5}{3} . \sqrt{\frac{5}{8}} = \frac{m}{12} \Rightarrow m = 5 \sqrt{10}$

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