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241.
If $straight x space plus space 1 over straight x space equals space 4 comma space$ then what is the value of $straight x to the power of 6 space plus space 1 over straight x to the power of 6 ?$

52

256

1026

2702

2702

If $straight x space plus space 1 over straight x space equals space straight a comma space space$ then, $straight x cubed space plus space 1 over straight x cubed space equals space straight a cubed space minus space 3 straight a$
$therefore space space straight x cubed space plus space 1 over straight x cubed space equals space left parenthesis 4 right parenthesis cubed space minus space 3 space cross times space 4 space equals space 64 space minus space 12 space equals space 52$
If $straight x space plus space 1 over straight x space equals space straight a comma space space space then space straight x squared space plus space 1 over straight x squared space equals space straight a squared space minus space 2$
$therefore space space space left parenthesis straight x cubed right parenthesis squared space plus space fraction numerator 1 over denominator left parenthesis straight x cubed right parenthesis squared end fraction space equals space left parenthesis 52 right parenthesis squared space minus space 2 space space rightwards double arrow space space space straight x to the power of 6 space plus space 1 over straight x to the power of 6 space equals space 2704 space minus space 2 space equals space 2702$

242.

If $straight y space equals space fraction numerator 2 minus straight x over denominator 1 plus straight x end fraction comma$ then what is the value of $fraction numerator 1 over denominator straight y space plus space 1 end fraction space plus space fraction numerator 2 straight y space plus space 1 over denominator straight y squared minus 1 end fraction ?$

$fraction numerator left parenthesis 1 space plus space straight x right parenthesis thin space left parenthesis 2 space minus space straight x right parenthesis over denominator 2 straight x minus 1 end fraction$
$fraction numerator left parenthesis 1 space minus space straight x right parenthesis thin space left parenthesis 2 space plus space straight x right parenthesis over denominator straight x space minus space 1 end fraction$
$fraction numerator left parenthesis 1 space plus space straight x right parenthesis thin space left parenthesis 2 space minus space straight x right parenthesis over denominator 1 space minus space 2 straight x end fraction$
$fraction numerator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 1 minus 2 straight x right parenthesis over denominator 2 minus straight x end fraction$