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FAKULTET ZA SAOBRAĆAJ I KOMUNIKACIJE UNSA, INŽ. MATEMATIKA I, PRVI PARCIJALNI, 30.11.2023. Odrediti kompleksan broj z_(1) ako je arg(z_(1)-1)=0,|z_(1)-1|=1, zatim riješiti jednačinu z^(4)+3=z_(1) i rješenja predstaviti grafički.

FAKULTET ZA SAOBRAĆAJ I KOMUNIKACIJE UNSA, INŽ. MATEMATIKA I, PRVI PARCIJALNI, 3030.1111.20232023.\newline11. Odrediti kompleksan broj z1 z_{1} ako je arg⁡(z1−1)=0,∣z1−1∣=1 \arg \left(z_{1}-1\right)=0,\left|z_{1}-1\right|=1 , zatim riješiti jednačinu z4+3=z1 z^{4}+3=z_{1} i rješenja predstaviti grafički.

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Q. FAKULTET ZA SAOBRAĆAJ I KOMUNIKACIJE UNSA, INŽ. MATEMATIKA I, PRVI PARCIJALNI, 3030.1111.20232023.\newline11. Odrediti kompleksan broj z1 z_{1} ako je arg⁡(z1−1)=0,∣z1−1∣=1 \arg \left(z_{1}-1\right)=0,\left|z_{1}-1\right|=1 , zatim riješiti jednačinu z4+3=z1 z^{4}+3=z_{1} i rješenja predstaviti grafički.
  1. Identify Real Axis: Since arg⁡(z1−1)=0\arg(z_{1}-1)=0, z1−1z_{1}-1 is on the positive real axis.
  2. Calculate Distance to 11: ∣z1−1∣=1|z_{1}-1|=1 means the distance from z1z_{1} to 11 on the complex plane is 11.
  3. Determine z1z_{1} Value: Combining both conditions, z1z_{1} is 11 unit to the right of 11 on the real axis, so z1=1+1=2z_{1}=1+1=2.
  4. Substitute z1z_{1} into Equation: Now solve z4+3=z1z^{4}+3=z_{1}. Substitute z1=2z_{1}=2 into the equation: z4+3=2z^{4}+3=2.
  5. Find Solutions for z4=−1z^{4}=-1: Subtract 33 from both sides: z4=−1z^{4}=-1.
  6. Identify 44th Roots of −1-1: The solutions to z4=−1z^{4}=-1 are the 44th roots of −1-1 on the complex plane.
  7. Select Valid Angles: The 44th roots of −1-1 are at angles π/2\pi/2, 3π/23\pi/2, 5π/25\pi/2, and 7π/27\pi/2 radians from the positive real axis.
  8. Select Valid Angles: The 44th roots of −1-1 are at angles π/2\pi/2, 3π/23\pi/2, 5π/25\pi/2, and 7π/27\pi/2 radians from the positive real axis.However, since angles on the complex plane are typically between 00 and 2π2\pi, we use π/2\pi/2 and 3π/23\pi/2 for the solutions.

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