The number of distinct complex number(s) z𝑧, such that |z|=1𝑧=1 and z3𝑧3 is purely imaginary, is/are equal to

The given conditions are |z| = 1 and z^3 is purely imaginary.

1. The condition |z| = 1 means that z lies on the unit circle in the complex plane.

2. The condition z^3 is purely imaginary means that z^3 can be written in the form 0 + yi, where y is a real number.

Let's represent z in its polar form, z = cosθ + isinθ, where θ is the argument of z.

Then z^3 = (cosθ + isinθ)^3 = cos^3θ + 3icos^2θsinθ - 3cosθsin^2θ - isin^3θ.

For z^3 to be purely imaginary, the real part of z^3 must be zero. So we have:

cos^3θ + 3cos^2θsinθ - 3cosθsin^2θ - sin^3θ = 0.

This simplifies to:

cos^3θ - sin^3θ = 0.

This equation holds true for θ = π/4, 5π/4, -3π/4, -7π/4.

So, there are 4 distinct complex numbers z that satisfy the given conditions.