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Question

The equation $$\displaystyle x^{3} - \frac{3}{4} x = - \frac{\sqrt{3}}{8}$$ is satisfied by


A
x=cos(5π18)
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B
x=cos(7π18)
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C
x=cos(23π18)
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D
x=cos(17π18)
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Solution

The correct options are
A $$\displaystyle x = \cos \left ( \frac{5 \pi}{18} \right )$$
C $$\displaystyle x = \cos \left ( \frac{7 \pi}{18} \right )$$
D $$\displaystyle x = \cos \left ( \frac{17 \pi}{18} \right )$$
$$\displaystyle { x }^{ 3 }-\frac { 3 }{ 4 } x=-\frac { \sqrt { 3 }  }{ 8 } $$
Let $$x\Rightarrow \cos { \theta  } $$
$$\displaystyle \cos ^{ 3 }{ \theta  } -\frac { 3 }{ 4 } \cos { \theta  } =\frac { -\sqrt { 3 }  }{ 8 } $$
$$\displaystyle \Rightarrow 4\cos ^{ 3 }{ \theta  } -3\cos { \theta  } =\frac { -\sqrt { 3 }  }{ 2 } $$
$$\displaystyle \cos { 3\theta  } =\frac { -\sqrt { 3 }  }{ 2 } \Rightarrow 3\theta =2n\pi +\frac { 5\pi  }{ 6 } $$ or $$\displaystyle 2n\pi +\frac { 7\pi  }{ 6 } $$
$$\displaystyle \Rightarrow \theta =\frac { 1 }{ 18 } \left( 12\pi n\pm 5\pi  \right) $$
$$\displaystyle \therefore \theta =\frac { 5\pi  }{ 18 } ,\frac { 7\pi  }{ 18 } ,\frac { 17\pi  }{ 18 } $$

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