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Let S=z:z¯=iz2+Re(z¯). Then zS|z|2 is equal to

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Important Questions on Complex Numbers

HARD
Let n denote the number of solutions of the equation z2+3z¯=0, where z is a complex number. Then the value of k=01nk is equal to
HARD
If the roots of the equation (z-4)3=8i are a-2i, b+i, and c+i, then abc=
MEDIUM
If x+iy=11+cosθ+isinθ, then show that 4x2-1=0.
MEDIUM

If (x-iy)1/3=a-ib, then show that:

xa-yb=4a2-b2

HARD
If S=zC:z¯=iz2, then the maximum value of |z-3-i|2  on S is
HARD
Let z¯ denote the complex conjugate of a complex number z and let i=-1. In the set of complex numbers, the number of distinct roots of the equation z¯-z2=iz¯+z2 is _______.
HARD
If the least and the largest real values of α, for which the equation z+αz-1+2i=0 zC and i=-1 has a solution, are p and q respectively; then 4p2+q2 is equal to_______.
HARD
The area of the polygon, whose vertices are the non-real roots of the equation z¯=iz2 is
HARD
The imaginary part of 3+2-5412-3-2-5412, can be
MEDIUM
Let S=z:z2+z¯=0. Then zSRez+Imz is equal to _______.
MEDIUM
Let z=a+ib, b0 be complex numbers satisfying z2=z¯·21-z. Then the least value of nN, such that zn=z+1n, is equal to _____ .
EASY
If x-13+i+y-13-i=i then the true statement among the following is
EASY
The solutions of the equation z(z-3i)¯=2(2+3i) are:
MEDIUM
If z=x+iy, z13=a-ib, then xa-yb=ka2-b2 where k is equal to
MEDIUM
For all complex numbers z of the form 1+iα, αR, if z2=x+iy, then