4.02i Quadratic equations: with complex roots

The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).

1. Use an algebraic method to find \(z\). [5]
2. Show that \(z^3 = 2 + 11\text{i}\). [1]

The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).

1. Find \(w\). [5]

The cubic equation \(x^3 + Ax^2 + Bx + 15 = 0\), where \(A\) and \(B\) are real numbers, has a root \(x = 1 + 2\mathrm{j}\).

1. Write down the other complex root. [1]
2. Explain why the equation must have a real root. [1]
3. Find the value of the real root and the values of \(A\) and \(B\). [9]

The cubic equation \(3z^3 + pz^2 + 17z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2\mathrm{i}\).

1. 1. Write down the value of another non-real root, \(\beta\), of this equation. [1 mark]
   2. Hence find the value of \(\alpha\beta\). [1 mark]
2. Find the value of the third root, \(\gamma\), of this equation. [3 marks]
3. Find the values of \(p\) and \(q\). [3 marks]

\(2 - 3i\) is one root of the equation $$z^3 + mz + 52 = 0$$ where \(m\) is real.

1. Find the other roots. [3 marks]
2. Determine the value of \(m\). [2 marks]

\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) is a root of the equation \(p(z) = 0\)

1. Express \(p(z)\) as a product of quadratic factors with real coefficients. [5 marks]
2. Solve the equation \(p(z) = 0\). [1 mark]

The quadratic equation \(z^2 - 5z + 8 = 0\) has roots \(\alpha\) and \(\beta\)

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\) [2 marks]
2. Without finding the value of \(\alpha\) or the value of \(\beta\), show that \(\alpha^4 + \beta^4 = -47\) [4 marks]
3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha^3 + \beta\) and \(\beta^3 + \alpha\) [5 marks]

You are given that \(z = 1 + 2i\) is a root of the equation \(z^3 - 5z^2 + qz - 15 = 0\), where \(q \in \mathbb{R}\). Find • the other roots, • the value of \(q\). [5]

In this question you must show detailed reasoning. You are given that \(f(z) = 4z^4 - 12z^3 + 41z^2 - 128z + 185\) and that \(2 + \mathrm{i}\) is a root of the equation \(f(z) = 0\).

1. Express \(f(z)\) as the product of two quadratic factors with integer coefficients. [5]
2. Solve \(f(z) = 0\). [3] Two loci on an Argand diagram are defined by \(C_1 = \{z:|z| = r_1\}\) and \(C_2 = \{z:|z| = r_2\}\) where \(r_1 > r_2\). You are given that two of the points representing the roots of \(f(z) = 0\) are on \(C_1\) and two are on \(C_2\). \(R\) is the region on the Argand diagram between \(C_1\) and \(C_2\).
3. Find the exact area of \(R\). [4]
4. \(\omega\) is the sum of all the roots of \(f(z) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\). [2]

Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]

The complex number \(z\) satisfies the equation \(z^2 - 4iz^* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

The complex number \(z\) satisfies the equation \(z^2 - 4iz + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

The complex number \(z\) satisfies the equation \(z^2 - 4iz* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]

In this question you must show detailed reasoning. The distinct numbers \(\omega_1\) and \(\omega_2\) both satisfy the quadratic equation \(4x^2 + 4x + 17 = 0\).

1. Write down the value of \(\omega_1 \omega_2\). [1]
2. \(A\), \(B\) and \(C\) are the points on an Argand diagram which represent \(\omega_1\), \(\omega_2\) and \(\omega_1 \omega_2\). Find the area of triangle \(ABC\). [6]

The complex number \(z_1\) is such that \(z_1 = a + ib\), where \(a\) and \(b\) are positive real numbers.

1. Given that \(z_1^2 = 2 + 2i\), show that \(a = \sqrt{\sqrt{2} + 1}\) and find the exact value of \(b\) in a similar form. [5]

The complex number \(z_2\) is such that \(z_2 = -a + ib\).

1. 1. Determine \(\arg z_2\) as a rational multiple of \(\pi\). [You may use the result \(\tan(\frac{1}{8}\pi) = \sqrt{2} - 1\).] [2]
   2. The point \(P_n\) in an Argand diagram represents the complex number \(z_2^n\), for positive integers \(n\). Find the least value of \(n\) for which \(P_n\) lies on the half-line with equation $$\arg(z) = \frac{1}{4}\pi.$$ [3]