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Question

If $$\alpha ,\beta $$ are the roots of the equation $$2{ x }^{ 2 }-5x+16=0\quad $$ then the value of $$\quad { \left( \cfrac { { \alpha  }^{ 2 } }{ \beta  }  \right)  }^{ \cfrac { 1 }{ 3 }  }+{ \left( \cfrac { { \beta  }^{ 2 } }{ \alpha  }  \right)  }^{ \cfrac { 1 }{ 3 }  }$$ is :


A
14
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B
54
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C
13
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D
512
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Solution

The correct option is B $$\cfrac { 5 }{ 4 } $$
$$\alpha +\beta =\cfrac { 5 }{ 2 } ;\alpha \beta =8\quad $$
$$\quad { \left( \cfrac { { \alpha  }^{ 2 } }{ \beta  }  \right)  }^{ \cfrac { 1 }{ 3 }  }+{ \left( \cfrac { { \beta  }^{ 2 } }{ \alpha  }  \right)  }^{ \cfrac { 1 }{ 3 }  }\Rightarrow \cfrac { { \alpha  }^{ \cfrac { 2 }{ 3 }  } }{ { \beta  }^{ \cfrac { 1 }{ 3 }  } } +\cfrac { { \beta  }^{ \cfrac { 2 }{ 3 }  } }{ { \alpha  }^{ \cfrac { 1 }{ 3 }  } } =\cfrac { \alpha +\beta  }{ { \left( \alpha \beta  \right)  }^{ \cfrac { 1 }{ 3 }  } } =\cfrac { \cfrac { 5 }{ 2 }  }{ { (8) }^{ \cfrac { 1 }{ 3 }  } } =\cfrac { 5 }{ 4 } $$

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