4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

5 The complex number \(z\) satisfies the equation \(2 z - \mathrm { i } = \mathrm { i } z + 2\).

1. Express \(z\) in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are rational numbers.
2. Find the exact value of \(| z |\) and of \(\arg ( z )\).
3. Express \(z ^ { 2 }\) in the form \(c + \mathrm { i } d\) where \(c\) and \(d\) are rational numbers.
4. Verify that \(\tan ( 2 \arg ( z ) ) = \tan \left( \arg \left( z ^ { 2 } \right) \right)\) using an appropriate trigonometrical identity.

The complex numbers \(u\) and \(v\) are defined by \(u = -4 + 2\text{i}\) and \(v = 3 + \text{i}\).

1. Find \(\frac{u}{v}\) in the form \(x + \text{i}y\), where \(x\) and \(y\) are real. [3]
2. Hence express \(\frac{u}{v}\) in the form \(re^{\text{i}\theta}\), where \(r\) and \(\theta\) are exact. [2]

In an Argand diagram, with origin \(O\), the points \(A\), \(B\) and \(C\) represent the complex numbers \(u\), \(v\) and \(2u + v\) respectively.

1. State fully the geometrical relationship between \(OA\) and \(BC\). [2]
2. Prove that angle \(AOB = \frac{3}{4}\pi\). [2]

The complex number \(2 + \mathrm{i}\) is denoted by \(u\). Its complex conjugate is denoted by \(u^*\).

1. Show, on a sketch of an Argand diagram with origin \(O\), the points \(A\), \(B\) and \(C\) representing the complex numbers \(u\), \(u^*\) and \(u + u^*\) respectively. Describe in geometrical terms the relationship between the four points \(O\), \(A\), \(B\) and \(C\). [4]
2. Express \(\frac{u}{u^*}\) in the form \(x + \mathrm{i}y\), where \(x\) and \(y\) are real. [3]
3. By considering the argument of \(\frac{u}{u^*}\), or otherwise, prove that $$\tan^{-1}\left(\frac{4}{3}\right) = 2\tan^{-1}\left(\frac{1}{2}\right).$$ [2]

Throughout this question the use of a calculator is not permitted.

1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2v = 2i \quad \text{and} \quad iu + v = 3.$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(|z + i| = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg(w - 2) = \frac{\pi}{4}\). Find the least value of \(|z - w|\) for points on these loci. [5]

The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$

1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
2. Without using your calculator and showing all stages of your working,
   1. determine the value of \(|z_1|\) [1]
   2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]

Given that \(z_1 = 3 + 2i\) and \(z_2 = \frac{12 - 5i}{z_1}\).

1. Find \(z_2\) in the form \(a + ib\), where \(a\) and \(b\) are real. [2]
2. Show, on an Argand diagram, the point \(P\) representing \(z_1\) and the point \(Q\) representing \(z_2\). [2]
3. Given that \(O\) is the origin, show that \(\angle POQ = \frac{\pi}{2}\). [2]

The circle passing through the points \(O\), \(P\) and \(Q\) has centre \(C\). Find

1. the complex number represented by \(C\), [2]
2. the exact value of the radius of the circle. [2]

The complex numbers \(z\) and \(w\) are given by $$z = 8 + 3\text{i}, \quad w = -2\text{i}$$ Express in the form \(a + b\text{i}\), where \(a\) and \(b\) are real constants,

1. \(z - w\), [1]
2. \(zw\). [2]

The complex number \(w\) is given by $$w = 10 - 5\text{i}$$

1. Find \(|w|\). [1]
2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [2]

The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + \text{i})(z + 3\text{i}) = w$$

1. Use algebra to find \(z\), giving your answer in the form \(a + b\text{i}\), where \(a\) and \(b\) are real numbers. [4]

Given that $$\arg(\lambda + 9\text{i} + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,

1. find the value of \(\lambda\). [2]

Given that \(z = 22 + 4i\) and \(\frac{z}{w} = 6 - 8i\), find

1. \(w\) in the form \(a + bi\), where \(a\) and \(b\) are real, [3]
2. the argument of \(z\), in radians to 2 decimal places. [2]

Given that \(z = 3 - 3i\) express, in the form \(a + ib\), where \(a\) and \(b\) are real numbers,

1. \(z^2\), [2]
2. \(\frac{1}{z}\), [2]
3. Find the exact value of each of \(|z|\), \(|z^2|\) and \(\left|\frac{1}{z}\right|\). [2]

The complex numbers \(z\), \(z^2\) and \(\frac{1}{z}\) are represented by the points \(A\), \(B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.

1. Show the points \(A\), \(B\), \(C\) and \(D\) on an Argand diagram. [2]
2. Prove that \(\triangle OAB\) is similar to \(\triangle OCD\). [3]

$$z = \frac{a + 3i}{2 + ai}, \quad a \in \mathbb{R}.$$

1. Given that \(a = 4\), find \(|z|\). [3]
2. Show that there is only one value of \(a\) for which \(\arg z = \frac{\pi}{4}\), and find this value. [6]

Given that \(z = 2 - 2i\) and \(w = -\sqrt{3} + i\),

1. find the modulus and argument of \(wz^2\). [6]
2. Show on an Argand diagram the points \(A\), \(B\) and \(C\) which represent \(z\), \(w\) and \(wz^2\) respectively, and determine the size of angle \(BOC\). [4]

Given that \(z = 1 + \sqrt{3}i\) and that \(\frac{w}{z} = 2 + 2i\), find

1. \(w\) in the form \(a + ib\), where \(a, b \in \mathbb{R}\), [3]
2. the argument of \(w\), [2]
3. the exact value for the modulus of \(w\). [2]

On an Argand diagram, the point \(A\) represents \(z\) and the point \(B\) represents \(w\).

1. Draw the Argand diagram, showing the points \(A\) and \(B\). [2]
2. Find the distance \(AB\), giving your answer as a simplified surd. [2]

Given that \(z = -2\sqrt{2} + 2\sqrt{2}i\) and \(w = 1 - i\sqrt{3}\), find

1. \(\left|\frac{z}{w}\right|\), [3]
2. \(\arg \left( \frac{z}{w} \right)\). [3]

1. On an Argand diagram, plot points \(A\), \(B\), \(C\) and \(D\) representing the complex numbers \(z\), \(w\), \(\left( \frac{z}{w} \right)\) and 4, respectively. [3]
2. Show that \(\angle AOC = \angle DOB\). [2]
3. Find the area of triangle \(AOC\). [2]

$$z = -4 + 6i.$$

1. Calculate \(\arg z\), giving your answer in radians to 3 decimal places. [2]

The complex number \(w\) is given by \(w = \frac{A}{2 - i}\), where \(A\) is a positive constant. Given that \(|w| = \sqrt{20}\),

1. find \(w\) in the form \(a + ib\), where \(a\) and \(b\) are constants, [4]
2. calculate \(\arg \frac{w}{z}\). [3]

Given that \(\frac{z + 2i}{z - \lambda i} = i\), where \(\lambda\) is a positive, real constant,

1. show that \(z = \left( \frac{\lambda}{2} + 1 \right) + i \left( \frac{\lambda}{2} - 1 \right)\). [5]

Given also that \(\arg z = \arctan \frac{1}{3}\), calculate

1. the value of \(\lambda\), [3]
2. the value of \(|z|^2\). [2]

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations $$2z + iw = -1,$$ $$z - w = 3 + 3i.$$

1. Use algebra to find \(z\), giving your answers in the form \(a + ib\), where \(a\) and \(b\) are real. [4]
2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. [2]

The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 5 + 3i,$$ $$z_1 = 1 + pi,$$ where \(p\) is an integer.

1. Find \(\frac{z_2}{z_1}\), in the form \(a + ib\), where \(a\) and \(b\) are expressed in terms of \(p\). [3]

Given that \(\arg \left( \frac{z_2}{z_1} \right) = \frac{\pi}{4}\),

1. find the value of \(p\). [2]

$$z = \sqrt{3} - i.$$ \(z^*\) is the complex conjugate of \(z\).

1. Show that \(\frac{z}{z^*} = \frac{1}{2} - \frac{\sqrt{3}}{2} i\). [3]
2. Find the value of \(\left| \frac{z}{z^*} \right|\). [2]
3. Verify, for \(z = \sqrt{3} - i\), that \(\arg \frac{z}{z^*} = \arg z - \arg z^*\). [4]
4. Display \(z\), \(z^*\) and \(\frac{z}{z^*}\) on a single Argand diagram. [2]
5. Find a quadratic equation with roots \(z\) and \(z^*\) in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are real constants to be found. [2]

$$z = -2 + i.$$

1. Express in the form \(a + ib\)
   1. \(\frac{1}{z}\)
   2. \(z^2\). [4]
2. Show that \(|z^2 - z| = 5\sqrt{2}\). [2]
3. Find \(\arg (z^2 - z)\). [2]
4. Display \(z\) and \(z^2 - z\) on a single Argand diagram. [2]

The complex number \(z\) is defined by $$z = \frac{a + 2i}{a - 1}, \quad a \in \mathbb{R}, a > 0 .$$ Given that the real part of \(z\) is \(\frac{1}{2}\) , find

1. the value of \(a\), [4]
2. the argument of \(z\), giving your answer in radians to 2 decimal places. [3]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + 2i}{iz}$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line. [4]