Complex Numbers and Quadratic Equations

Some equations, such as x^2 + 1 = 0, have no solution in real numbers because the square of a real number is never negative. To solve them, mathematicians introduced the imaginary unit i, with i^2 = -1. This chapter defines a complex number as z = x + iy, where x is the real part and y is the imaginary part. It explains equality and the four operations on complex numbers. You will learn the powers of i, which repeat in a cycle of four (i, -1, -i, 1), the square roots of negative real numbers. You will study the modulus, mod(z) = sqrt(x^2 + y^2), and the conjugate x - iy of a complex number, and use them to find the multiplicative inverse. The chapter shows how complex numbers are represented as points in the Argand plane, with the real axis and the imaginary axis. Finally, it uses complex numbers to solve quadratic equations ax^2 + bx + c = 0 with real coefficients and negative discriminant, whose roots are a pair of complex conjugates.

Detailed Worksheet: Complex Numbers and Quadratic Equations Section A - Definitions (10 marks) 1. Define a complex number. Identify the real and imaginary parts of -3 + 5i. (2 marks) 2. Evaluate i^35 and i^(-39). (2 marks) 3. Find the modulus and conjugate of 3 - 4i. (2 marks) 4. Find the multiplicative inverse of 4 - 3i. (2 marks) 5. What is the Argand plane? Represent 2 + 3i and -1 - 2i in it. (2 marks) Section B - Calculations and Applications (15 marks) 6. Express (5 - 3i)^3 in the form x + iy. (3 marks) 7. Express (3 + 2i)/(1 - i) in the form x + iy and find its modulus. (3 marks) 8. Find the real numbers x and y if (3x - 7) + 2iy = -5y + (5 + x)i. (3 marks) 9. Solve the quadratic equation 2x^2 + x + 1 = 0. (3 marks) 10. Solve sqrt5 x^2 + x + sqrt5 = 0. (3 marks) Section C - Diagrams (10 marks) 11. Plot the complex numbers 3 + 2i, -2 + i, -1 - 3i and 2 - 2i on the Argand plane and name the quadrant of each. (4 marks) 12. Represent z = 3 + 4i and its conjugate on the Argand plane and show the geometrical relation between them. Find mod(z). (3 marks) 13. Show on the Argand plane the sum of 2 + i and 1 + 3i using the parallelogram law. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. If x + iy = (a + ib)/(a - ib), prove that x^2 + y^2 = 1. (5 marks) 15. If z1 = 2 - i and z2 = 1 + i, find the modulus of (z1 + z2 + 1)/(z1 - z2 + 1). (5 marks) 16. Show that (1 - ix)/(1 + ix) has modulus 1 for every real number x, and that its conjugate equals its reciprocal. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all steps. Write answers in the form x + iy.