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KEAM 2015 Math Paper

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Question : 35 of 120

Marks: +1, -0

The quadratic equation whose roots are three times the roots of the equation

2x^2+3x+5=0,

is

$2x^{2}+9x+45=0$
$2x^{2}+9x-45=0$
$5x^{2}+9x+45=0$
$2x^{2}-9x+45=0$
$2x^{2}+9x+49=0$

Solution:

Let the roots of the given equation

2x^{2}+3x+5=0

be

\alpha

and

\beta

Then,

\alpha+\beta=-\frac{3}{2}

and

\alpha\beta=\frac{5}{2}

Let the roots of required equation be

\alpha'

and

\beta'

It is given that,

\alpha'=3\alpha

and

\beta'=3\beta

Now,

\alpha'+\beta'=3\alpha+3\beta

=3(\alpha+\beta)

=3\left(-\frac{3}{2}\right)=-\frac{9}{2}

Also,

\alpha'\cdot\beta'=(3\alpha)(3\beta)=9\alpha\beta

=9\left(\frac{5}{2}\right)=\frac{45}{2}

Hence, required equation is

x^{2}-\left(-\frac{9}{2}\right)x+\frac{45}{2}=0

\Rightarrow 2x^{2}+9x+45=0

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