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Date	November 2016	Marks available	4	Reference code	16N.1.AHL.TZ0.H_12
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Show that	Question number	H_12	Adapted from	N/A

Question

Let $\omega$ be one of the non-real solutions of the equation ${z^3} = 1$ .

Consider the complex numbers $p = 1 - 3{\text{i}}$ and $q = x + (2x + 1){\text{i}}$ , where $x \in \mathbb{R}$ .

Determine the value of

(i) $1 + \omega + {\omega ^2}$ ;

(ii) $1 + \omega {\text{*}} + {(\omega {\text{*}})^2}$ .

[4]

Show that $(\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13$ .

[4]

Find the values of $x$ that satisfy the equation $\left| p \right| = \left| q \right|$ .

[5]

Solve the inequality $\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}$ .

[6]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

(i) METHOD 1

$1 + \omega + {\omega ^2} = \frac{{1 - {\omega ^3}}}{{1 - \omega }} = 0$ A1

as $\omega \ne 1$ R1

METHOD 2

solutions of $1 - {\omega ^3} = 0$ are $\omega = 1,{\text{ }}\omega {\text{ = }}\frac{{ - 1 \pm \sqrt 3 {\text{i}}}}{2}$ A1

verification that the sum of these roots is 0 R1

(ii) $1 + \omega {\text{*}} + {(\omega {\text{*}})^2} = 0$ A2

[4 marks]

$(\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = - 3{\omega ^4} + 10{\omega ^3} - 3{\omega ^2}$ M1A1

EITHER

$= - 3{\omega ^2}({\omega ^2} + \omega + 1) + 13{\omega ^3}$ M1

$= - 3{\omega ^2} \times 0 + 13 \times 1$ A1

$= - 3\omega + 10 - 3{\omega ^2} = - 3({\omega ^2} + \omega + 1) + 13$ M1

$= - 3 \times 0 + 13$ A1

substitution by $\omega = \frac{{ - 1 \pm \sqrt 3 {\text{i}}}}{2}$ in any form M1

numerical values of each term seen A1

THEN

$= 13$ AG

[4 marks]

$\left| p \right| = \left| q \right| \Rightarrow \sqrt {{1^2} + {3^2}} = \sqrt {{x^2} + {{(2x + 1)}^2}}$ (M1)(A1)

$5{x^2} + 4x - 9 = 0$ A1

$(5x + 9)(x - 1) = 0$ (M1)

$x = 1,{\text{ }}x = - \frac{9}{5}$ A1

[5 marks]

$pq = (1 - 3{\text{i}})\left( {x + (2x + 1){\text{i}}} \right) = (7x + 3) + (1 - x){\text{i}}$ M1A1

$\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2} \Rightarrow (7x + 3) + 8 < {(1 - x)^2}$ M1

$\Rightarrow {x^2} - 9x - 10 > 0$ A1

$\Rightarrow (x + 1)(x - 10) > 0$ M1

$x < - 1,{\text{ }}x > 10$ A1

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » AHL 1.12—Complex numbers – Cartesian form and Argand diag

Show 43 related questions

22M.3.AHL.TZ1.2f.i:
Use this diagram to determine the roots of the corresponding equation of the form $(z - r) (z^{2} - 2 a z + a^{2} + 16) = 0$ for $z \in ℂ$ .
22M.1.AHL.TZ2.12c.i:
Express $z_{1}$ in terms of $z_{2}$ .
22M.1.AHL.TZ2.12d:
Use the result from part (c)(ii) to show that $a^{2} - 3 b = 0$ .
21N.3.AHL.TZ0.1f:
Sketch the graph of $x^{2} - y^{2} = 1$ , stating the coordinates of any axis intercepts and the equation of each asymptote.
16N.1.AHL.TZ0.H_12c:
Find the values of $x$ that satisfy the equation $\left| p \right| = \left| q \right|$ .
19N.1.AHL.TZ0.H_5b:
The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.
21M.3.AHL.TZ2.2g.ii:
Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result to part (e).
22M.3.AHL.TZ1.2e:
Deduce from part (d)(i) that the complex roots of the equation $(z - r) (z^{2} - 2 a z + a^{2} + b^{2}) = 0$ can be expressed as $a \pm i \sqrt{g' (a)}$ .
22M.3.AHL.TZ1.2f.ii:
State the coordinates of $C_{2}$ .
22M.1.AHL.TZ2.12b:
Given that $Re (z_{1} {z_{2}}^{*}) = 0$ , show that $Z_{1} {OZ}_{2}$ is a right-angled triangle.
22M.1.AHL.TZ2.12c.ii:
Hence show that ${z_{1}}^{2} + {z_{2}}^{2} = z_{1} z_{2}$ .
22M.1.AHL.TZ2.12e:
Consider the equation $z^{2} + a z + 12 = 0$ , where $z \in ℂ$ and $a \in ℝ$ .

Given that $0 < α - θ < π$ , deduce that only one equilateral triangle $Z_{1} {OZ}_{2}$ can be formed from the point $O$ and the roots of this equation.
SPM.2.AHL.TZ0.8:
The complex numbers $w$ and $z$ satisfy the equations

$\frac{w}{z} = 2{\text{i}}$

${{\text{z}}^ * } - 3w = 5 + 5{\text{i}}$ .

Find $w$ and $z$ in the form $a + b{\text{i}}$ where $a$ , ${\text{b}} \in \mathbb{Z}$ .
21M.3.AHL.TZ2.2g.i:
Express $z^{n - 1} + z^{n - 2} + \dots + z + 1$ as a product of linear factors over the set $ℂ$ .
17M.1.AHL.TZ2.H_5:
In the following Argand diagram the point A represents the complex number $- 1 + 4{\text{i}}$ and the point B represents the complex number $- 3 + 0{\text{i}}$ . The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.
21M.3.AHL.TZ2.2a.i:
Show that $(ω - 1) (ω^{2} + ω + 1) = ω^{3} - 1$ .
21M.3.AHL.TZ2.2b:
Show that $P_{0} P_{1} \times P_{0} P_{2} = 3$ .
21M.3.AHL.TZ2.2a.ii:
Hence, deduce that $ω^{2} + ω + 1 = 0$ .
21M.3.AHL.TZ2.2e:
Suggest a value for $P_{0} P_{1} \times P_{0} P_{2} \times \dots \times P_{0} P_{n - 1}$ .
21M.3.AHL.TZ2.2d:
Show that $P_{0} P_{1} \times P_{0} P_{2} \times P_{0} P_{3} = 4$ .
21M.3.AHL.TZ2.2f.ii:
Hence, write down an expression for $P_{0} P_{n - 1}$ in terms of $n$ and $ω$ .
21N.1.AHL.TZ0.12b:
Plot the points $A$ , $B$ and $C$ on an Argand diagram.
21N.1.AHL.TZ0.12d:
By using de Moivre’s theorem, show that $α = \frac{1}{1 - e^{i \frac{π}{6}}}$ is a root of this equation.
21N.3.AHL.TZ0.1b:
Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .
21N.3.AHL.TZ0.1c.i:
$f (i u)$ .
21N.3.AHL.TZ0.1g:
The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .
22M.1.AHL.TZ1.9a:
Find an expression for $z_{1} z_{2}$ in terms of $b$ .
22M.1.AHL.TZ2.12a:
Show that $z_{1} {z_{2}}^{*} = r_{1} r_{2} e^{i (α - θ)}$ where ${z_{2}}^{*}$ is the complex conjugate of $z_{2}$ .
21N.1.AHL.TZ0.12e:
Determine the value of $Re (α)$ .
21N.3.AHL.TZ0.1a:
Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .
21N.3.AHL.TZ0.1e:
Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .
16N.1.AHL.TZ0.H_12d:
Solve the inequality $\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}$ .
16N.1.AHL.TZ0.H_12a:
Determine the value of

(i) $1 + \omega + {\omega ^2}$ ;

(ii) $1 + \omega {\text{*}} + {(\omega {\text{*}})^2}$ .
21M.3.AHL.TZ2.2f.i:
Write down expressions for $P_{0} P_{2}$ and $P_{0} P_{3}$ in terms of $ω$ .
20N.1.AHL.TZ0.H_4:
Consider the equation $\frac{2 z}{3 - z *} = i$ , where $z = x + i y$ and $x, y \in ℝ$ .

Find the value of $x$ and the value of $y$ .
EXN.1.AHL.TZ0.12a:
Use the binomial theorem to expand ${(\cos θ + i \sin θ)}^{4}$ . Give your answer in the form $a + b i$ where $a$ and $b$ are expressed in terms of $\sin θ$ and $\cos θ$ .
21M.2.AHL.TZ2.8:
Consider $z = \cos θ + i \sin θ$ where $z \in ℂ, z \neq 1$ .

Show that $Re (\frac{1 + z}{1 - z}) = 0$ .
21M.3.AHL.TZ2.2c:
By factorizing $z^{4} - 1$ , or otherwise, deduce that $ω^{3} + ω^{2} + ω + 1 = 0$ .
21N.1.AHL.TZ0.12a.i:
Verify that $ω_{1} = 1 + e^{i \frac{π}{6}}$ is a root of this equation.
21N.1.AHL.TZ0.12a.ii:
Find $ω_{2}$ and $ω_{3}$ , expressing these in the form $a + e^{i θ}$ , where $a \in ℝ$ and $θ > 0$ .
21N.1.AHL.TZ0.12c:
Find $AC$ .
21N.3.AHL.TZ0.1c.ii:
$g (i u)$ .
21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .

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Topic 1—Number and algebra » AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers

Topic 1—Number and algebra

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