The complex number \(1 + i \sqrt{3}\) is denoted by \(u\).
(i) Express \(u\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). Hence, or otherwise, find the modulus and argument of \(u^2\) and \(u^3\).
(ii) Show that \(u\) is a root of the equation \(z^2 - 2z + 4 = 0\), and state the other root of this equation.
(iii) Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities \(|z-i| \leq 1\) and \(\arg z \geq \arg u\).
Solve the equation \(\frac{5z}{1 + 2i} - zz^* + 30 + 10i = 0\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real.
(a) On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(-\frac{1}{3}\pi \leq \arg(z - 1 - 2i) \leq \frac{1}{3}\pi\) and \(\text{Re} \, z \leq 3\).
(b) Calculate the least value of \(\arg z\) for points in the region from (a). Give your answer in radians correct to 3 decimal places.
Solve the quadratic equation \((1 - 3i)z^2 - (2 + i)z + i = 0\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real.
(a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z + 2| \leq 2\) and \(\text{Im} \, z \geq 1\).
(b) Find the greatest value of \(\arg z\) for points in the shaded region.
(a) Solve the equation \(z^2 - 6iz - 12 = 0\), giving the answers in the form \(x + iy\), where \(x\) and \(y\) are real and exact.
(b) On a sketch of an Argand diagram with origin \(O\), show points \(A\) and \(B\) representing the roots of the equation in part (a).
(c) Find the exact modulus and argument of each root.
(d) Hence show that the triangle \(OAB\) is equilateral.
The complex numbers u and w are defined by u = 2e\frac{1}{4} \pi i and w = 3e\frac{1}{3} \pi i.
(a) Find \(\frac{u^2}{w}\), giving your answer in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). Give the exact values of \(r\) and \(\theta\).
(b) State the least positive integer \(n\) such that both \(\text{Im} \ w^n = 0\) and \(\text{Re} \ w^n > 0\).
On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z| \leq 3\), \(\text{Re} \, z \geq -2\) and \(\frac{1}{4}\pi \leq \arg z \leq \pi\).
The complex number 3 - i is denoted by u.
(a) Show, on an Argand diagram with origin O, the points A, B and C representing the complex numbers u, u^* and u^* - u respectively. State the type of quadrilateral formed by the points O, A, B and C.
(b) Express \(\frac{u^*}{u}\) in the form \(x + iy\), where \(x\) and \(y\) are real.
(c) By considering the argument of \(\frac{u^*}{u}\), or otherwise, prove that \(\arctan\left(\frac{3}{4}\right) = 2 \arctan\left(\frac{1}{3}\right)\).
On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 1 + 2i| \leq |z|\) and \(|z - 2| \leq 1\).
The complex number \(-1 + \sqrt{7}i\) is denoted by \(u\). It is given that \(u\) is a root of the equation
\(2x^3 + 3x^2 + 14x + k = 0,\)
where \(k\) is a real constant.
(a) Find the value of \(k\). [3]
(b) Find the other two roots of the equation. [4]
(c) On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(|z - u| = 2\). [2]
(d) Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians. [2]
The complex number \(u\) is defined by \(u = \frac{\sqrt{2} - a\sqrt{2}i}{1 + 2i}\), where \(a\) is a positive integer.
(a) Express \(u\) in terms of \(a\), in the form \(x + iy\), where \(x\) and \(y\) are real and exact.
It is now given that \(a = 3\).
(b) Express \(u\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\), giving the exact values of \(r\) and \(\theta\).
(c) Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(-\pi < \theta \leq \pi\), giving the exact values of \(r\) and \(\theta\).
Find the complex numbers \(w\) which satisfy the equation \(w^2 + 2iw^* = 1\) and are such that \(\text{Re} \, w \leq 0\). Give your answers in the form \(x + iy\), where \(x\) and \(y\) are real.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z + 2 - 3i| \leq 2\) and \(\text{arg} \, z \leq \frac{3}{4}\pi\).
\(The complex number -\sqrt{3} + i is denoted by u.\)
\((a) Express u in the form re^{i\theta}, where r > 0 and -\pi < \theta \leq \pi, giving the exact values of r and \theta.\)
(b) Hence show that u^6 is real and state its value.
(c) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities 0 \leq \arg(z - u) \leq \frac{1}{4}\pi and \text{Re } z \leq 2.
(ii) Find the greatest value of |z| for points in the shaded region. Give your answer correct to 3 significant figures.
(a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 3 - 2i| \leq 1\) and \(\text{Im} \, z \geq 2\).
(b) Find the greatest value of \(\arg z\) for points in the shaded region, giving your answer in degrees.
(a) Given the complex numbers \(u = a + ib\) and \(w = c + id\), where \(a, b, c\) and \(d\) are real, prove that \((u + w)^* = u^* + w^*\).
(b) Solve the equation \((z + 2 + i)^* + (2 + i)z = 0\), giving your answer in the form \(x + iy\) where \(x\) and \(y\) are real.
The complex number 1 + 2i is denoted by u. The polynomial 2x^3 + ax^2 + 4x + b, where a and b are real constants, is denoted by p(x). It is given that u is a root of the equation p(x) = 0.
(a) Find the values of a and b.
(b) State a second complex root of this equation.
(c) Find the real factors of p(x).
(d) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z - u| โค โ5 and arg z โค 1/4 ฯ.
(ii) Find the least value of Im z for points in the shaded region. Give your answer in an exact form.
(a) Verify that \(-1 + \sqrt{2}i\) is a root of the equation \(z^4 + 3z^2 + 2z + 12 = 0\).
(b) Find the other roots of this equation.
\(The complex number u is given by u = 10 - 4โ6i.\)
Find the two square roots of u, giving your answers in the form a + ib, where a and b are real and exact.