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Examples 7 to 8

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 (7) If   x  +  y = – z ,  then find the value of  \frac{\textup{x}^{2}}{\textup{yz}} + \frac{\textup{y}^{2}}{\textup{zx}} + \frac{\textup{z}^{2}}{\textup{xy}} .

Soln. Here,  x  +  y  =   – z

\therefore  x  +  y  +  z   =   0

\therefore  x3 +  y3 +  z3 =  3xyz  –––––––––  ( 1 )

 Now, \frac{\textup{x}^{2}}{\textup{yz}} + \frac{\textup{y}^{2}}{\textup{zx}} + \frac{\textup{z}^{2}}{\textup{xy}}

=   \frac{x^{3}+y^{3}+z^{3}}{xyz}

= \frac{3 \textup{xyz}}{\textup{xyz}}  (  From  equation ( 1 ) )

=  3

(8) If    a2 – 3a  + 1  =  0,  then find the value of   a2 + \frac{1}{a^{3}}.

Soln. Here,  a2 – 3a  + 1  =  0

\therefore  a2  + 1  =   3a

\therefore  a  +  \frac{1}{a}   =   3 \large \Rightarrow  a3 + \frac{1}{a^{3}} =  \left (a+\frac{1}{a} \right )^{3}  – 3\left (a+\frac{1}{a} \right )

=   ( 3 ) 3– 3 ( 3 )

=   18

 

Best of Luck

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