Core Pure Mathematics A Level Further Maths Worksheets, Questions and Revision

Solve the equation z^2 - 2z + 5 = 0, giving each root in the form a + bi, then find the modulus and argument of the root with positive imaginary part.

1. Use the quadratic formula: z = [2 +- sqrt((-2)^2 - 4(1)(5))] / (2(1)).
2. Simplify inside the root: 4 - 20 = -16, so sqrt(-16) = 4i.
3. z = (2 +- 4i)/2 = 1 +- 2i, so the roots are z = 1 + 2i and z = 1 - 2i.
4. For z = 1 + 2i, the modulus is sqrt(1^2 + 2^2) = sqrt(5).
5. The argument is arctan(2/1) = 1.107 radians, since the point (1,2) lies in the first quadrant.
6. Final answer: z = 1 + 2i or z = 1 - 2i; the root 1 + 2i has modulus sqrt(5) and argument 1.107 radians.