Grade 12 Problems on Complex Numbers with Solutions and Answers
Grade 12 problems on complex numbers with detailed solutions are presented.

Evaluate the following expressions
a) (3 + 2i)  (8  5i)
b) (4  2i)*(1  5i)
c) ( 2  4i) / i
d) ( 3 + 2i) / (3  6i)

If (x + yi) / i = ( 7 + 9i ) , where x and y are real, what is the value of (x + yi)(x  yi)?

Determine all complex number z that satisfy the equation
z + 3 z' = 5  6i
where z' is the complex conjugate of z.

Find all complex numbers of the form z = a + bi , where a and b are real numbers such that z z' = 25 and a + b = 7
where z' is the complex conjugate of z.

The complex number 2 + 4i is one of the root to the quadratic equation x^{2} + bx + c = 0, where b and c are real numbers.
a) Find b and c
b) Write down the second root and check it.

Find all complex numbers z such that z^{2} = 1 + 2 sqrt(6) i.

Find all complex numbers z such that (4 + 2i)z + (8  2i)z' = 2 + 10i, where z' is the complex conjugate of z.

Given that the complex number z = 2 + 7i is a root to the equation:
z^{3} + 6 z^{2} + 61 z + 106 = 0
find the real root to the equation.

a) Show that the complex number 2i is a root of the equation
z^{4} + z^{3} + 2 z^{2} + 4 z  8 = 0
b) Find all the roots root of this equation.

P(z) = z^{4} + a z^{3} + b z^{2} + c z + d is a polynomial where a, b, c and d are real numbers. Find a, b, c and d if two zeros of polynomial P are the following complex numbers: 2  i and 1  i.


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