If a is a complex number such that `|a|=1`, then find the value of a, so that equation `az^2 + z +1=0` has one purely imaginary root.

If z is a complex number such that |z|>=2 then the minimum value of |z+1/2| is

If z is a complex number, then find the minimum value of |z|+|z-1|+|2z-3|dot

If z is a complex number, then find the minimum value of |z|+|z-1|+|2z-3|dot

If a ,b are complex numbers and one of the roots of the equation x^2+a x+b=0 is purely real, whereas the other is purely imaginary, prove that a^2- (bar a)^2=4b .

Consider the complex number z = (1 - isin theta)//(1+ icos theta) . The value of theta for which z is purely imaginary are

If (z-1)/(z+1) is a purely imaginary number (z ne - 1) , then find the value of |z| .

If the complex number z is such that |z-1|le1 and |z-2|=1 find the maximum possible value |z|^(2)

Consider a quadratic equation az^2 + bz + c =0 where a, b, c, are complex numbers. Find the condition that the equation has (i) one purely imaginary root (ii) one purely real root (iii) two purely imaginary roots (iv) two purely real roots

Consider the quadratic equation az^(2)+bz+c=0 where a,b,c are non-zero complex numbers. Now answer the following. The condition that the equation has both roots purely imaginary is

If a and b are complex and one of the roots of the equation x^(2) + ax + b =0 is purely real whereas the other is purely imaginary, then