It is given that \(\frac{2 + 3ai}{a + 2i} = \lambda(2 - i)\), where \(a\) and \(\lambda\) are real constants.
(a) Show that \(3a^2 + 4a - 4 = 0\).
(b) Hence find the possible values of \(a\) and the corresponding values of \(\lambda\).
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z + 1 - i| \leq 1\) and \(\arg(z - 1) \leq \frac{3}{4}\pi\).
(a) Solve the equation \(z^2 - 2piz - q = 0\), where \(p\) and \(q\) are real constants.
In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
(b) Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
(c) Given instead that triangle \(OAB\) is equilateral, express \(q\) in terms of \(p\).
\(The complex numbers u and v are defined by u = -4 + 2i and v = 3 + i.\)
(a) Find \(\frac{u}{v}\) in the form x + iy, where x and y are real.
(b) Hence express \(\frac{u}{v}\) in the form \(re^{i\theta}\), where r and \(\theta\) are exact.
In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and 2u + v respectively.
(c) State fully the geometrical relationship between OA and BC.
(d) Prove that angle AOB = \(\frac{3}{4}\pi\).
The complex number u is defined by
\(u = \frac{7+i}{1-i}\).
(a) Express u in the form \(x + iy\), where \(x\) and \(y\) are real.
(b) Show on a sketch of an Argand diagram the points A, B and C representing u, \(7 + i\) and \(1 - i\) respectively.
(c) By considering the arguments of \(7 + i\) and \(1 - i\), show that
\(\arctan\left(\frac{4}{3}\right) = \arctan\left(\frac{1}{7}\right) + \frac{1}{4}\pi\).
(a) Verify that \(-1 + \sqrt{5}i\) is a root of the equation \(2x^3 + x^2 + 6x - 18 = 0\).
(b) Find the other roots of this equation.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z| \geq 2\) and \(|z - 1 + i| \leq 1\).
(a) The complex numbers u and w are such that
\(u - w = 2i\) and \(uw = 6\).
Find u and w, giving your answers in the form x + iy, where x and y are real and exact.
(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities
\(|z - 2 - 2i| \leq 2\), \(0 \leq \arg z \leq \frac{\pi}{4}\) and \(\text{Re } z \leq 3\).
(a) Solve the equation \((1 + 2i)w + iw^* = 3 + 5i\). Give your answer 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 \(z\) satisfying the inequalities \(|z - 2 - 2i| \leq 1\) and \(\arg(z - 4i) \geq -\frac{1}{4}\pi\).
(ii) Find the least value of \(\text{Im } z\) for points in this region, giving your answer in an exact form.
(a) The complex number u is defined by \(u = \frac{3i}{a + 2i}\), where a is real.
(b)
(a) The complex numbers \(v\) and \(w\) satisfy the equations
\(v + iw = 5\) and \((1 + 2i)v - w = 3i\).
Solve the equations for \(v\) and \(w\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real.
(b) (i) On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(|z - 2 - 3i| = 1\).
(ii) Calculate the least value of \(\arg z\) for points on this locus.
(a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 4 - 3i| \leq 2\) and \(\text{Re} \, z \leq 3\).
(b) Find the greatest value of \(\arg z\) for points in this region.
The complex number with modulus 1 and argument \(\frac{1}{3} \pi\) is denoted by \(w\).
(i) Express \(w\) in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [1]
The complex number \(1 + 2i\) is denoted by \(u\). The complex number \(v\) is such that \(|v| = 2|u|\) and \(\arg v = \arg u + \frac{1}{3} \pi\).
(ii) Sketch an Argand diagram showing the points representing \(u\) and \(v\). [2]
(iii) Explain why \(v\) can be expressed as \(2uw\). Hence find \(v\), giving your answer in the form \(a + ib\), where \(a\) and \(b\) are real and exact. [4]
(a) Find the complex number \(z\) satisfying the equation
\(z + \frac{iz}{z^*} - 2 = 0,\)
where \(z^*\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + iy\), where \(x\) and \(y\) are real.
(b) (i) On a single Argand diagram sketch the loci given by the equations \(|z - 2i| = 2\) and \(\text{Im} \, z = 3\), where \(\text{Im} \, z\) denotes the imaginary part of \(z\).
(ii) In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).
(a) The complex number u is given by u = -3 - (2\sqrt{10})i. Showing all necessary working and without using a calculator, find the square roots of u. Give your answers in the form a + ib, where the numbers a and b are real and exact.
(b) On a sketch of an Argand diagram shade the region whose points represent complex numbers z satisfying the inequalities |z - 3 - i| \leq 3, arg z \geq \frac{1}{4}\pi and Im z \geq 2, where Im z denotes the imaginary part of the complex number z.
The complex number u is defined by
\(u = \frac{4i}{1 - (\sqrt{3})i}\).
\(It is given that the complex number -1 + (\sqrt{3})i is a root of the equation\)
\(kx^3 + 5x^2 + 10x + 4 = 0\),
where \(k\) is a real constant.
(i) Write down another root of the equation.
(ii) Find the value of \(k\) and the third root of the equation.
The complex number \((\sqrt{3}) + i\) is denoted by \(u\).
(a) Showing all working and without using a calculator, solve the equation
\((1 + i)z^2 - (4 + 3i)z + 5 + i = 0.\)
Give your answers in the form x + iy, where x and y are real.
(b) The complex number u is given by
\(u = -1 - i.\)
On a sketch of an Argand diagram show the point representing u. Shade the region whose points represent complex numbers satisfying the inequalities |z| < |z - 2i| and \(\frac{1}{4}\pi < \text{arg}(z - u) < \frac{1}{2}\pi\).
(a) (i) Without using a calculator, express the complex number \(\frac{2 + 6i}{1 - 2i}\) in the form \(x + iy\), where \(x\) and \(y\) are real.
(ii) Hence, without using a calculator, express \(\frac{2 + 6i}{1 - 2i}\) in the form \(r(\cos \theta + i \sin \theta)\), where \(r > 0\) and \(-\pi < \theta \leq \pi\), giving the exact values of \(r\) and \(\theta\).
(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - 3i| \leq 1\) and \(\text{Re } z \leq 0\), where \(\text{Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places.