4.02k Argand diagrams: geometric interpretation

In this question you must show detailed reasoning. You are given that \(f(z) = 4z^4 - 12z^3 + 41z^2 - 128z + 185\) and that \(2 + \mathrm{i}\) is a root of the equation \(f(z) = 0\).

1. Express \(f(z)\) as the product of two quadratic factors with integer coefficients. [5]
2. Solve \(f(z) = 0\). [3] Two loci on an Argand diagram are defined by \(C_1 = \{z:|z| = r_1\}\) and \(C_2 = \{z:|z| = r_2\}\) where \(r_1 > r_2\). You are given that two of the points representing the roots of \(f(z) = 0\) are on \(C_1\) and two are on \(C_2\). \(R\) is the region on the Argand diagram between \(C_1\) and \(C_2\).
3. Find the exact area of \(R\). [4]
4. \(\omega\) is the sum of all the roots of \(f(z) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\). [2]

Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]

Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

A circle \(C\) in the complex plane has equation \(|z - 2 - 5i| = a\). The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(arg(z_1) = \frac{\pi}{4}\). Prove that \(a = \frac{3\sqrt{2}}{2}\). [6]

In an Argand diagram the loci \(C_1\) and \(C_2\) are given by $$|z| = 2 \quad \text{and} \quad \arg z = \frac{1}{4}\pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence find, in the form \(x + iy\), the complex number representing the point of intersection of \(C_1\) and \(C_2\). [2]

The complex number \(z\) satisfies the equations $$|z^* - 1 - 2i| = |z - 3|$$ and $$|z - a| = 3$$ where \(a\) is real. Show that \(a\) must lie in the interval \([1 - s\sqrt{t}, 1 + s\sqrt{t}]\), where \(s\) and \(t\) are prime numbers. [6 marks]

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

$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]

Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and Re (z) \(\geq\) 9. [6]

1. Shade on an Argand diagram the set of points $$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]

The complex number \(w\) satisfies \(|w - 4i| = 3\).

1. Find the maximum value of \(\arg w\) in the interval \((-\pi, \pi]\). Give your answer in radians correct to 2 decimal places. [2]

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation \(|z - 3| = 2\) [1]

\includegraphics{figure_9}

1. There is a unique complex number \(w\) that satisfies both \(|w - 3| = 2\) and \(\arg(w + 1) = \alpha\) where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).
   1. Find the value of \(\alpha\). [2]
   2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4]

In an Argand diagram, the points \(A\), \(B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2i\).

1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [6]

The points \(D\), \(E\) and \(F\) are the midpoints of the sides of triangle \(ABC\).

1. Find the exact area of triangle \(DEF\). [3]

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

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]

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg(z - 6) = -\frac{3\pi}{4}$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies. [4]
2. Find the complex number for which both \(|z - 6i| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]

1. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg(z + i) = \frac{\pi}{6}$$ [2 marks]

\includegraphics{figure_5}

1. \(z_1\) is a point on \(L\) such that \(|z_1|\) is a minimum. Find the exact value of \(z_1\) in the form \(a + bi\) [4 marks]

In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\). [3]
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]

You are given the complex number \(w = 2 + 2\sqrt{3}i\).

1. Express \(w\) in modulus-argument form. [3]
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]

Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and \(\text{Re}(z) \geq 9\). [6]

A locus \(C_1\) is defined by \(C_1 = \{z : |z + i| \leq |z - 2i|\}\).

1. Indicate by shading on the Argand diagram below the region representing \(C_1\). [2] \includegraphics{figure_8}
2. Find the cartesian equation of the boundary line of the region representing \(C_1\), giving your answer in the form \(ax + by + c = 0\). [2]

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_8}

1. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
   1. [(b) (i)] Find the value of \(\alpha\). [2 marks]
   2. [(b) (ii)] Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). [4 marks]

In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).

1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \quad b \in \mathbb{R}$$ [3]

The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\).

1. Find the complex numbers \(z_1\) and \(z_2\). [6]

$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{21}i\) is a root of the equation \(f(z) = 0\)

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]