Complex Numbers

Complex Numbers: Level 3 Challenges


If the square of xx and the square root of xx are equal to each other, but not equal to xx, then what is the value of x2+xx^2+x?

Let z=x+iyz=x+iy be a complex number where xx and yy are integers. Find the area of the rectangle whose vertices are the roots of the equation zzˉ3+zˉz3=350.\large z \bar{z}^3+\bar{z}z^3=350.

Details: zˉ=xiy \bar{z}=x-iy and i=1i=\sqrt{-1}.

How many complex numbers are the conjugate of their own cube?

The complex numbers pp and qq satisfy p3=5+i2p^3=5+i\sqrt{2} and q3=5i2q^3=5-i\sqrt{2}.

Find the only possible integer value of p+qp+q.

eiθe2iθe3iθ...eniθ=1\large e ^{ i\theta }e^{ 2i\theta} e^{ 3i\theta }...\large e^{ ni\theta } = 1

Find the value of θ\theta satisfying the above equation for mNm\in\mathbb{N}.


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