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AP EAMCET202018 Sep 2020Evening ShiftMathematicsQuadratic EquationActual

If ( , ) are the roots of (x^2-2 x+4=0 ), for (n N ), what is the value of ( ^n+ ^n= )

Options

  1. A(2^ n+2 ( n 3 ) )
  2. B(2^ n+1 ( n 3 ) )
  3. C(2^ n+1 ( n 6 ) )
  4. D(2^ n+2 ( n 6 ) )

Correct answer

B. (2^ n+1 ( n 3 ) )

Step-by-step solution

Given, ( , ) are roots of (x^2-2 x+4=0 ) ( aligned & x= 2 4-4 1 4 2 1 & x= 2 -12 2 & x=1 3 i & x=1+ 3 i (o r) 1- 3 i & x=2 ( 1 2 + 3 2 i )=2 ( 3 +i 3 ) & =2 e^ i 3 aligned ) Similarly, ( =2 e^ -i 3 ) ( aligned ^n+ ^n & = (2 e^ i 3 )^n+ (2 e^ -i 3 )^n=2^n (e^ i n 3 +e^ -i n 3 ) & =2^n ( n 3 +i n 3 + n 3 -i n 3 ) & =2^n 2 n 3 =2^ n+1 n 3 aligned ) Hence, option (b) is correct.

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