Let z=9+ai, where `i=sqrt(-1)` and a be non-zero real.
If `Im(z^(2))=Im(z^(3))`, sum of the digits of `a^(2)` is

If z=(i) ^((i) ^(i)) where i= sqrt (−1) ​ , then z is equal to

Let z=(-1+sqrt(3)i)/(2) , where i=sqrt(-1) , and r, s in {1, 2, 3} . Let P=[((-z)^(r),z^(2s)),(z^(2s),z^(r))] and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P^(2)=-I is ______.

If z=(3+4i)^(6)+(3-4i)^(6),"where" i=sqrt(-1), then Find the value of Im(z) .

Consider four complex numbers z_(1)=2+2i, , z_(2)=2-2i,z_(3)=-2-2iandz_(4)=-2+2i),where i=sqrt(-1), Statement - 1 z_(1),z_(2),z_(3)andz_(4) constitute the vertices of a square on the complex plane because Statement - 2 The non-zero complex numbers z,barz, -z,-barz always constitute the vertices of a square.

If z=x+iy, where i=sqrt(-1) , then the equation abs(((2z-i)/(z+1)))=m represents a circle, then m can be

Let P point denoting a complex number z on the complex plane. i.e. z=Re(z)+i Im(z)," where "i=sqrt(-1) if" "Re(z)=x and Im(z)=y,then z=x+iy .The area of the circle inscribed in the region denoted by |Re(z)|+|Im(z)|=10 equal to

If |z-2i|lesqrt(2), where i=sqrt(-1), then the maximum value of |3-i(z-1)|, is

If z=ilog_(e)(2-sqrt(3)),"where"i=sqrt(-1) then the cos z is equal to

Let P point denoting a complex number z on the complex plane. i.e. z=Re(z)+i Im(z)," where "i=sqrt(-1) if Re(z)=xand Im (z)=y,then z=x+iy Number of integral solutions satisfying the eniquality |Re(z)|+|Im(z)|lt21,.is

If the equation z^(3)+(3+i)z^(2)-3z-(m+i)=0, " where " i=sqrt(-1) " and " m in R , has atleast one real root, value of m is