Math, asked by bhardwajvijaykumar08, 23 days ago

plz solve it plz


plz do part (a) and (f)​

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Answers

Answered by Yugant1913
15

 \red{\sf \: x = 2 -  \sqrt{3}  \: find \:  \bigg( x -  \frac{1}{x}  { \bigg)}^{3} }

 \sf \green{given}

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \sf \: x = 2 -  \sqrt{3}

 \sf \green{to \: find :} \bigg(x -  \frac{1}{ x}  { \bigg)}^{3}

 \sf \green{solution :}x = 2 -  \sqrt{3}

 \tt \:  \frac{1}{x}  =  \frac{1}{2 \sqrt{3} }  \times   \frac{2 +  \sqrt{3} }{2 +  \sqrt{3} }  \\

 \tt \:  \frac{1}{x}  =  \frac{2 +  \sqrt{3} }{ {(2)}^{2} - ( \sqrt{3}  {)}^{2}  }  \\

 \tt \:  \frac{1}{x}  =  \frac{2 +  \sqrt{3} }{4 - 3}  \\

 \tt \:  \:  \frac{1}{x}  = 2 +  \sqrt{3}  \\

 \sf \green{now}

 \tt \: x -  \frac{1}{x}  = 2 -  \sqrt{3}  - 2 \sqrt{3}  \\

  \tt \: x -  \frac{1}{x}  =  - 2 \sqrt{3}  \\

 \sf \green{and}

 \tt \:  { \bigg(x -  \frac{1}{x} \bigg) }^{3}

 \tt  - \bigg(2 \sqrt{3}  { \bigg)}^{2}

 \tt \therefore \boxed{-24 \sqrt{3} }

____________________________________

 \sf \red{x = 2 -  \sqrt{3}  \: find \:  {x}^{3} +  \frac{1}{ {x}^{3} }  } \\

 \sf  \green{given : }x = 2 +  \sqrt{3}

 \sf \green{ to \: find : } {x}^{3}  +  \frac{1}{  {x}^{3}  }  \\

 \sf \underline \green{ solution : }x = 2 \sqrt{3}

 \tt\:  \frac{1}{x}  =  \frac{1}{2 +  \sqrt{2} }  \times  \frac{2 -  \sqrt{3} }{2 -  \sqrt{3} }  \\

 \tt \:  \frac{1}{x}  =  \frac{2 -  \sqrt{3} }{ {(2)}^{2}  -  {(3)}^{2} }  \\

 \tt \:  \frac{1}{x}  =  \frac{2 -  \sqrt{3} }{4 - 3}  \\

 \tt \:  \frac{1}{x}  =  2 -  \sqrt{3}  \\

 \sf \green{now}

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \tt \: x +  \frac{1}{x}  \\

 \tt \implies2 +   \cancel{ \sqrt{3} } + 2 -  \cancel{  \sqrt{3}  }

 \tt \implies \: 2 + 2

 \tt \implies4

So, on cubing both sides, we have

 \sf \bigg(x +  \frac{1}{x}  { \bigg)}^{3}  =  {(4)}^{3}  \\

 \sf \:  {x}^{3}  +  \frac{1}{ {x}^{3} }  + 3 \bigg(x +  \frac{1}{x}  \bigg) = 64 \\

 \sf \:  {x}^{3}  +  \frac{1}{ {x}^{3} }  + 3 \times 4 = 64 \\

 \sf \:  {x}^{3}  +  \frac{1}{ {x}^{3} }  + 12 = 64 \\

 \sf \:  {x}^{3}  +  \frac{1}{ {x}^{3} }  = 64 - 12 \\

 \sf \:  {x}^{3}  +  \frac{1}{ {x}^{3} }  = 52    \\

 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \sf \boxed{ \green{  \sf{x}^{3}  +  \frac{1}{ {x}^{3}  } =  52}} \\

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