If α, β e C are the distinct roots, of the
equation x2−x+1=0, then α101+β107 is
equal to :
(1) 0
(2) 1
(3) 2
(4) −1
![begin mathsize 16px style straight x squared minus straight x plus 1 equals 0
straight x equals fraction numerator 1 plus-or-minus square root of 1 minus 4 end root over denominator 2 end fraction equals fraction numerator 1 plus-or-minus square root of 3 straight i over denominator 2 end fraction
straight x equals fraction numerator 1 plus square root of 3 straight i over denominator 2 end fraction comma fraction numerator 1 minus square root of 3 straight i over denominator 2 end fraction comma
Also space complex space conjugates space are space fraction numerator negative 1 minus square root of 3 straight i over denominator 2 end fraction equals negative straight omega squared comma fraction numerator negative 1 plus square root of 3 straight i over denominator 2 end fraction equals negative straight omega
straight alpha to the power of 101 plus straight beta to the power of 107 equals open parentheses negative straight omega squared close parentheses to the power of 101 plus open parentheses negative straight omega close parentheses to the power of 107
equals open parentheses negative straight omega to the power of 202 minus straight omega to the power of 107 close parentheses
equals negative open parentheses straight omega to the power of 3 cross times 67 end exponent cross times straight omega plus straight omega to the power of 3 cross times 35 end exponent cross times straight omega squared close parentheses
equals negative open parentheses straight omega plus straight omega squared close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
equals space minus left parenthesis negative 1 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
equals 1 end style](https://images.topperlearning.com/topper/tinymce/integration/showimage.php?formula=2ca4f90f94693292e5bf8c308dd28823.png&d=0)
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