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Question

The quadratic equation whose roots are three times the roots of the equation $$2{ x }^{ 2 }+3x+5=0$$, is


A
2x2+9x+45=0
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B
2x2+9x45=0
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C
5x2+9x+45=0
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D
2x29x+45=0
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E
2x2+9x+49=0
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Solution

The correct option is B $$5{ x }^{ 2 }+9x+45=0$$
Let the roots of the given equation $$2{ x }^{ 2 }+3x+5=0$$ be $$\alpha$$ and $$\beta$$.

Then, $$\alpha +\beta =-\dfrac { 3 }{ 2 } $$ and $$\alpha \beta =\dfrac { 5 }{ 2 } $$

Let the roots of required equation be $${ \alpha  }^{ ' }$$ and $${ \beta  }^{ ' }$$.

It is given that,

$${ \alpha  }^{ ' }=3\alpha $$ and $${ \beta  }^{ ' }=3\beta $$

Now, $${ \alpha  }^{ ' }+{ \beta  }^{ ' }=3\alpha +3\beta $$

$$=3\left( \alpha +\beta  \right) $$

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

Also, $${ \alpha  }^{ ' }\cdot { \beta  }^{ ' }=\left( 3\alpha  \right) \left( 3\beta  \right) =9\alpha \beta $$

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

Hence, required equation is

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

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

Mathematics

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