Complex Numbers and Quadratic Equations quiz

The value of i^4 is:

The modulus of 3 + 4i is:

The conjugate of 2 - 5i is:

The value of (1 + i)^2 is:

The roots of x^2 + 4 = 0 are:

The multiplicative inverse of i is:

The point representing -3 + 2i lies in the:

If z = 1 + i, then z times its conjugate is:

The discriminant of x^2 + x + 1 = 0 is:

The value of sqrt(-16) is:

Express (1 - i)^4 in the form x + iy. (2 marks)

Express i^9 + i^19 in the form x + iy. (2 marks)

Find the conjugate of (3 - 2i)(2 + 3i)/((1 + 2i)(2 - i)). (2 marks)

Solve x^2 + 3 = 0. (2 marks)

Solve x^2 + x + 1 = 0. (2 marks)

Find the modulus of (1 + i)/(1 - i) - (1 - i)/(1 + i). (2 marks)

If z = 2 + 3i, find z + conjugate of z and z - conjugate of z. (2 marks)

Express (-5 + 3i) - (4 + 7i) in the form x + iy. (2 marks)

Show that 1 + i^2 + i^4 + i^6 = 0. (2 marks)

Find the real numbers x and y if (x - iy)(3 + 5i) is the conjugate of -6 - 24i. (2 marks)

Express (1/3 + 3i)^3 in the form x + iy. (3 marks)

Find the multiplicative inverse of sqrt5 + 3i. (3 marks)

Solve 3x^2 - 2x + 3 = 0. (3 marks)

Find the modulus of (2 + 3i)(1 - i). (3 marks)

Evaluate (1 + i)^6. (3 marks)

Read the passage and answer the questions. A student was asked to solve the equation x^2 - 4x + 13 = 0. She found the discriminant was negative and used the formula x = (-b +/- sqrt(b^2 - 4ac))/2a. (i) Find the discriminant. (ii) Find the roots. (iii) Show that the roots are conjugates of each other. (5 marks)

Read the passage and answer the questions. In an electrical circuit, impedance is represented as a complex number. Two impedances Z1 = 3 + 4i ohms and Z2 = 1 - 2i ohms are connected in series, so the total impedance is Z1 + Z2. (i) Find the total impedance. (ii) Find its modulus. (iii) Find Z1 Z2. (5 marks)

Read the passage and answer the questions. The powers of i repeat in a cycle: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. To find i^n, we divide n by 4 and use the remainder. (i) Find i^50. (ii) Find i^101. (iii) Find i^50 + i^101. (5 marks)

Read the passage and answer the questions. A point P on the Argand plane represents the complex number z = -4 + 3i. Its reflection in the real axis is Q. (i) Which complex number does Q represent? (ii) Find the distance of P from the origin. (iii) In which quadrant is Q? (5 marks)

Read the passage and answer the questions. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. A student was given (x + 2y) + i(2x - y) = 4 + 3i. (i) Write the two equations. (ii) Solve them for x and y. (iii) Find the modulus of x + iy. (5 marks)

Define a complex number. Explain the algebraic operations of addition, subtraction, multiplication and division with examples. (6 marks)

Define the modulus and conjugate of a complex number. Prove that mod(z1 z2) = mod(z1) x mod(z2) and that the conjugate of z1 z2 equals the product of the conjugates. (6 marks)

Explain how complex numbers are represented on the Argand plane. Represent four complex numbers and their conjugates. (6 marks)

Solve the quadratic equations x^2 - x + 2 = 0 and 2x^2 - 4x + 3 = 0. (6 marks)

If (x + iy)^3 = u + iv, show that u/x + v/y = 4(x^2 - y^2). (6 marks)