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Complex Numbers (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · JEE

FoundationsStandardAdvanced

Complex Numbers: core idea

Roots of unity.

Example problem: hint and answer

Solve z4=16z^4=16: positive real root?

Hint: Fourth roots. Answer: 2.

Practice (9 problems — type your answers)

  • 1. Sum of non-real cube roots of unity?

  • 2. Locus ∣z−1∣=∣z+1∣|z-1|=|z+1|: the line Re(z)=(z)=?

  • 3. Principal Arg of −1-1? Enter degrees in (−180,180](-180,180].

  • 4. Product of roots of z2−z+1=0z^2-z+1=0?

  • 5. If z=eiπ/3z=e^{i\pi/3}, Re(z)(z)?

  • 6. Maximum of ∣z−i∣|z-i| for ∣z∣=1|z|=1?

  • 7. Factor z2+1=0z^2+1=0 roots product?

  • 8. (1+i)10(1+i)^{10}: magnitude?

  • 9. Number of solutions to z6=1z^6=1?

Complex Numbers Practice Problems — Advanced