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AP EAMCET202319 May 2023Morning ShiftMathematicsComplex NumberActual

If 1, , ^2 are the cube roots of unity, then the roots of the equation 8 z^3-12 z^2+6 z-28=0 are

Options

  1. A2,2 , 3 ^2+1
  2. B2, 3 +1 2 , 3 ^2+1 2
  3. C2, 1+3 3 , 1+3 ^2 3
  4. D2, 1- 2 , 1- ^2 2

Correct answer

B. 2, 3 +1 2 , 3 ^2+1 2

Step-by-step solution

8 z^3-12 z^2+6 z-28=0 ...(i) Since z=2 satisfies equation (i). Hence z=2 is one of solution equation (i). Therefore, aligned & 8 z^2(z-2)+4 z(z-2)+14(z-2)=0 & (8 z^2+4 z+14 )(z-2)=0 & (4 z^2+2 z+7 )(z-2)=0 & z=2 or 4 z^2+2 z+7=0 & z= -2 (2)^2-4 4 7 2 4 aligned z= -1 3 3 i 4 ...(1) Since we know that cube root of unity are, 1, , ^2 . Where = -1+i 3 2 and ^2= -1-i 3 2 Now = -1+i 3 2 aligned & 3 +1 2 = -1 3 3 i 2 & 3 +1 2 =z aligned (from (1)) Similary, ^2= -1-i 3 2 aligned & 3 ^2+1 2 = -1-3 3 i 4 3 ^2+1 2 =z & 3 ^2+1

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