Prove that `z=i^i,` where `i=sqrt-1,` is purely real.

If z=x+iyandw=(1-iz)/(z-i) such that |w|=1 , then show that z is purely real.

A particle P starts from the point z_0=1+2i , where i=sqrt(-1) . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z_1dot From z_1 the particle moves sqrt(2) units in the direction of the vector hat i+ hat j and then it moves through an angle pi/2 in anticlockwise direction on a circle with centre at origin, to reach a point z_2dot The point z_2 is given by (a) 6+7i (b) -7+6i (c) 7+6i (d) -6+7i

Prove that 1+i^2+i^4+i^6 =0

If z_(1)=3i and z_(2)=-1-i , where i=sqrt(-1) find the value of arg ((z_(1))/(z_(2)))

If z=i^(i^(i)) where i=sqrt-1 then find the value of |z|

Find real theta such that (3+2i sin theta)/(1-2i sin theta) is purely real.

Let z= (-1+sqrt(3i))/2, where i=sqrt(-1) and r,s epsilon P1,2,3}. Let P= [((-z)^r, z^(2s)),(z^(2s), z^r)] and I be the idenfity matrix or order 2. Then the total number of ordered pairs (r,s) for which P^2=-I is

The reflection of the complex number (2-i)/(3+i) where (i=sqrt(-1) in the straight line z(1+i)=barz(i-1) is

If z=(3+7i)(a+ib) , where a , b in Z-{0} , is purely imaginery, then minimum value of |z| is

If z=(i)^((i)^(i)) w h e r e i=sqrt(-1),t h e n|z| is equal to a. 1 b. e^(-pi//2) c. e^(-pi) d. none of these