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AP EAMCET202123 Aug 2021Evening ShiftMathematicsComplex NumberActual

Let = cis ( 2 3 )= ( 2 3 )+i ( 2 3 ) and f(x)=x^7-2 x^4-4 x^3+8 . Which of the following option is correct?

Options

  1. A2^ 1 2 , 2^ 1 3 , , 2^ 1 3 is a subset of the solution set of f(x) .
  2. B2^ 1 2 ,-2^ 1 3 , 2^ 1 3 ^2, 2^ 1 2 i is a subset of the solution set of f(x) .
  3. C2^ 1 3 , 2^ 1 2 -2^ 1 2 , i, 2^ 1 3 ^2 is not a subset of the solution set of f(x) .
  4. D2^ 1 3 , 2^ 1 3 , , 2^ 1 2 i,-2^ 1 2 is a subset of the solution set of f(x) .

Correct answer

D. 2^ 1 3 , 2^ 1 3 , , 2^ 1 2 i,-2^ 1 2 is a subset of the solution set of f(x) .

Step-by-step solution

Given, = cis ( 2 3 )= ( 2 3 )+i ( 2 3 )f(x)=x^7-2 x^4-4 x^3+8 = -1 2 +i 3 2 = -1+i 3 2 f(x)=x^7-2 x^4-4 x^3+8 x^4 (x^3-2 )-4 (x^3-2 )f(x)= (x^3-2 ) (x^4-4 ) For finding solution, let f(x)=0 (x^3-2 ) (x^4-4 )=0 x^3-2=0 ...(i) or x^4-4=0x=2^ 1 / 3 or x^4=4x^2= 2 When x^2=2 and when x^2=-2x= 2^ 1 / 2 and x=2^ 1 / 2 i From Eq. (i), we get array r x^3-2=0 x^3-( [3] 2 )^3=0 array (x-2^ 1 / 3 ) (x^2+2^ 2 / 3 +2^ 1 / 3 x )=0 x-2^ 1 / 3 =0 or x^2+2^ 1 / 3 x+2^ 2 / 3 =0 x=2^ 1 / 3 or x= -2^ 2 / 3 2^ 2 / 3 -4 2^ 2 / 3 2 align

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