4.02k Argand diagrams: geometric interpretation

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]

Given that \(-2\) is a root of the equation \(z^3 + 6z + 20 = 0\),

1. Find the other two roots of the equation, [3]
2. show, on a single Argand diagram, the three points representing the roots of the equation, [1]
3. prove that these three points are the vertices of a right-angled triangle. [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]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z}{z-i}, \quad z \neq i.$$ The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. [8]

The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.

1. Shade the region \(R\) on an Argand diagram. [2]

A complex number \(z\) is represented by the point \(P\) in the Argand diagram.

1. Given that \(|z - 6| = |z|\), sketch the locus of \(P\). [2]
2. Find the complex numbers \(z\) which satisfy both \(|z - 6| = |z|\) and \(|z - 3 - 4i| = 5\). [3]

The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac{30}{z}\).

1. Show that \(T\) maps \(|z - 6| = |z|\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle. [5]

The point \(P\) represents the complex number \(z\) on an Argand diagram, where $$|z - i| = 2.$$ The locus of \(P\) as \(z\) varies is the curve \(C\).

1. Find a cartesian equation of \(C\). [2]
2. Sketch the curve \(C\). [2]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + i}{3 + iz}, \quad z \neq 3i.$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,

1. show that \(Q\) lies on \(C\). [5]

1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
2. Hence prove, by the method of differences, that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [6]

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6| = 2|z - 3|.$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]

where \(a\) and \(b\) are constants to be found.

1. Hence show that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [3]
2. Find the complex number for which both \(|z - 6| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 3| = 2|z|.$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, and give the coordinates of the centre and the radius of the circle.(5)

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z + 3| = |z - i\sqrt{3}|.$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.(5)
2. On your diagram shade the region which satisfies $$|z - 3| \geq 2|z| \text{ and } |z + 3| \geq |z - i\sqrt{3}|.$$ (2)

The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac{z + 1}{z + i}, \quad z \neq -i.$$

1. Show that \(T\) maps points on the half-line \(\arg(z) = \frac{\pi}{4}\) in the \(z\)-plane into points on the circle \(|w| = 1\) in the \(w\)-plane. [4]
2. Find the image under \(T\) in the \(w\)-plane of the circle \(|z| = 1\) in the \(z\)-plane. [6]
3. Sketch on separate diagrams the circle \(|z| = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane. [2]
4. Mark on your sketches the point \(P\), where \(z = i\), and its image \(Q\) under \(T\) in the \(w\)-plane. [2]

A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$|z - 3i| = 3,$$

1. sketch the locus of \(P\). [2]
2. Find the complex number \(z\) which satisfies both \(|z - 3i| = 3\) and \(\arg (z - 3i) = \frac{3}{4}\pi\). [4] The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{2i}{w}.$$
3. Show that \(T\) maps \(|z - 3i| = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line. [5]

The loci \(C_1\) and \(C_2\) are given by $$|z - 2i| = 2 \quad \text{and} \quad |z + 1| = |z + i|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

The loci \(C_1\) and \(C_2\) are given by $$|z - 2\text{i}| = 2 \quad \text{and} \quad |z + 1| = |z + \text{i}|$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]

Indicate, on separate Argand diagrams,

1. the set of points \(z\) for which \(|z-(3-\mathrm{j})| \leqslant 3\), [3]
2. the set of points \(z\) for which \(1 < |z-(3-\mathrm{j})| \leqslant 3\), [2]
3. the set of points \(z\) for which \(\arg(z-(3-\mathrm{j})) = \frac{1}{4}\pi\). [3]

Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}

Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2\mathrm{j}\) and \(\beta = -2 - \mathrm{j}\).

1. Represent \(\beta\) and its complex conjugate \(\beta^*\) on an Argand diagram. [2]
2. Express \(\alpha\beta\) in the form \(a + b\mathrm{j}\). [2]
3. Express \(\frac{\alpha + \beta}{\beta}\) in the form \(a + b\mathrm{j}\). [3]

Two loci, \(L_1\) and \(L_2\), in an Argand diagram are given by $$L_1 : |z + 6 - 5\text{i}| = 4\sqrt{2}$$ $$L_2 : \arg(z + \text{i}) = \frac{3\pi}{4}$$ The point \(P\) represents the complex number \(-2 + \text{i}\).

1. Verify that the point \(P\) is a point of intersection of \(L_1\) and \(L_2\). [2 marks]
2. Sketch \(L_1\) and \(L_2\) on one Argand diagram. [6 marks]
3. The point \(Q\) is also a point of intersection of \(L_1\) and \(L_2\). Find the complex number that is represented by \(Q\). [2 marks]