A function ' f ' is defined by f(z)=(4+i) z^2+ z+ for all complex number z , where and are complex numbers if f(1) and f(i) are both real and the smallest possible values of | |+| | is p then determine p^2

Step-by-step solution

aligned & lies =a+i b, =c+i d & a^2+b^2 + c^2+d^2 =p & f(z)=(4+i) z^2+z(a+i b)+(c+i d) & f(1)=(4+a+c)+i(1+b+d) aligned hence 1+b+d=0 .....(1) aligned & f(i)=-(4+i)+i(a+i b)+(c+i d) & -4-b+c+i(a+d-1) aligned hence a+d=1 .....(2) from (1) & (2) a-b=2 hence there is no restriction of ' c ' let c=0| |+| |= a^2+b^2 + d^2 = 4+2 a b +|d| 4+2 a b 2 with equality d=0, a=1, b=-1 hence p= 2