The complex number 3 - i is denoted by u. Its complex conjugate is denoted by u*.
The complex number 1 - i is denoted by u.
(i) Showing your working and without using a calculator, express \(\frac{i}{u}\) in the form \(x + iy\), where \(x\) and \(y\) are real.
(ii) On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(|z - u| = |z|\) and \(|z - i| = 2\).
(iii) Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).
\(The complex number u is given by u = -1 + (4\sqrt{3})i.\)
The complex number w is defined by \(w = \frac{22 + 4i}{(2 - i)^2}\).
The complex number \(z\) is defined by \(z = \frac{5a - 2i}{3 + ai}\), where \(a\) is an integer. It is given that \(\arg z = -\frac{1}{4}\pi\).
(a) Find the value of \(a\) and hence express \(z\) in the form \(x + iy\), where \(x\) and \(y\) are real. [6]
(b) Express \(z^3\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). Give the simplified exact values of \(r\) and \(\theta\). [3]
\(The complex numbers w and z are defined by w = 5 + 3i and z = 4 + i.\)
(i) Express \(\frac{i w}{z}\) in the form x + iy, showing all your working and giving the exact values of x and y. [3]
(ii) Find wz and hence, by considering arguments, show that \(\arctan \left( \frac{3}{5} \right) + \arctan \left( \frac{1}{4} \right) = \frac{1}{4} \pi\). [4]
The complex numbers w and z satisfy the relation
\(w = \frac{z + i}{iz + 2}\).
(i) Given that \(z = 1 + i\), find \(w\), giving your answer in the form \(x + iy\), where \(x\) and \(y\) are real.
(ii) Given instead that \(w = z\) and the real part of \(z\) is negative, find \(z\), giving your answer in the form \(x + iy\), where \(x\) and \(y\) are real.
(a) The complex number \(\frac{3 - 5i}{1 + 4i}\) is denoted by \(u\). Showing your working, express \(u\) in the form \(x + iy\), where \(x\) and \(y\) are real.
(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(|z - 2 - i| \leq 1\) and \(|z - i| \leq |z - 2|\).
(ii) Calculate the maximum value of \(\arg z\) for points lying in the shaded region.
(a) It is given that \(-1 + (\sqrt{5})i\) is a root of the equation \(z^3 + 2z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation.
(b) The complex number \(w\) has modulus 1 and argument \(2\theta\) radians. Show that \(\frac{w-1}{w+1} = i \tan \theta\).
The complex number \(z\) is defined by \(z = \frac{9\sqrt{3} + 9i}{\sqrt{3} - i}\). Find, showing all your working,
(i) an expression for \(z\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\),
(ii) the two square roots of \(z\), giving your answers in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\).
(a) Without using a calculator, use the formula for the solution of a quadratic equation to solve \((2 - i)z^2 + 2z + 2 + i = 0\). Give your answers in the form \(a + bi\).
(b) The complex number \(w\) is defined by \(w = 2e^{\frac{1}{4}\pi i}\). In an Argand diagram, the points \(A, B\) and \(C\) represent the complex numbers \(w, w^3\) and \(w^*\) respectively (where \(w^*\) denotes the complex conjugate of \(w\)). Draw the Argand diagram showing the points \(A, B\) and \(C\), and calculate the area of triangle \(ABC\).
(a) The complex numbers u and v satisfy the equations
\(u + 2v = 2i\) and \(iu + v = 3\).
Solve the equations for u and v, giving both answers in the form x + iy, where x and y are real.
(b) On an Argand diagram, sketch the locus representing complex numbers z satisfying \(|z + i| = 1\) and the locus representing complex numbers w satisfying \(\text{arg}(w - 2) = \frac{3}{4}\pi\). Find the least value of \(|z - w|\) for points on these loci.
The complex number z is defined by z = a + ib, where a and b are real. The complex conjugate of z is denoted by z*.
\(In an Argand diagram a set of points representing complex numbers z is defined by the equation |z - 10i| = 2|z - 4i|.\)
(a) The complex number \(w\) is such that \(\text{Re} \, w > 0\) and \(w + 3w^* = iw^2\), where \(w^*\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + iy\), where \(x\) and \(y\) are real.
(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(|z - 2i| \leq 2\) and \(0 \leq \arg(z + 2) \leq \frac{1}{4}\pi\). Calculate the greatest value of \(|z|\) for points in this region, giving your answer correct to 2 decimal places.
(a) Without using a calculator, solve the equation
\(3w + 2iw^* = 17 + 8i\),
where \(w^*\) denotes the complex conjugate of \(w\). Give your answer in the form \(a + bi\).
(b) In an Argand diagram, the loci
\(\arg(z - 2i) = \frac{1}{6}\pi\) and \(|z - 3| = |z - 3i|\)
intersect at the point \(P\). Express the complex number represented by \(P\) in the form \(re^{i\theta}\), giving the exact value of \(\theta\) and the value of \(r\) correct to 3 significant figures.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 3 - i| \leq 3\) and \(|z| \geq |z - 4i|\).
(a) Without using a calculator, solve the equation \(iw^2 = (2 - 2i)^2\).
(b) (i) Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where \(|z - 4 - 4i| \leq 2\).
(ii) For the complex numbers represented by points in the region \(R\), it is given that \(p \leq |z| \leq q\) and \(\alpha \leq \arg z \leq \beta\). Find the values of \(p, q, \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.
The complex number \(1 + (\sqrt{2})i\) is denoted by \(u\). The polynomial \(x^4 + x^2 + 2x + 6\) is denoted by \(p(x)\).
(a) The complex numbers u and w satisfy the equations
\(u - w = 4i\) and \(uw = 5\).
Solve the equations for u and w, giving all answers in the form x + iy, where x and y are real.
(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(|z - 2 + 2i| \leq 2\), \(\text{arg } z \leq -\frac{1}{4}\pi\) and \(\text{Re } z \geq 1\), where \(\text{Re } z\) denotes the real part of z.
(ii) Calculate the greatest possible value of \(\text{Re } z\) for points lying in the shaded region.
The complex number u is defined by
\(u = \frac{1 + 2i}{1 - 3i}\).