IB DP Math AA HL Prediction Paper 3 for 2025 Exams
IB DP Math AA HL Prediction Paper 3 – April/May 2025 Exam
IB DP Math AA HL Prediction Paper 3: Prepare for the IB exams with subject-specific Prediction questions, model answers. All topics covered.
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Question 1
This question asks you to investigate and prove a geometric property involving the roots of the equation $z^n = 1$ where $z \in \mathbb{C}$ for integers $n$, where $n \geq 2$.
The roots of the equation $z^n = 1$ where $z \in \mathbb{C}$ are $1$, $\omega$, $\omega^2$, …, $\omega^{n-1}$, where $\omega = e^{\frac{2\pi i}{n}}$. Each root can be represented by a point $P_0$, $P_1$, $P_2$, …, $P_{n-1}$, respectively, on an Argand diagram.
Consider the case where $n = 3$.
The roots of the equation $z^3 = 1$ where $z \in \mathbb{C}$ are $1$, $\omega$, and $\omega^2$. On the following Argand diagram, the points $P_0$, $P_1$, and $P_2$ lie on a circle of radius $1$ unit with center $O(0, 0)$.
(i) Show that $(\omega – 1)(\omega^2 + \omega + 1) = \omega^3 – 1$.
(ii) Hence, deduce that $\omega^2 + \omega + 1 = 0$.
Working space:
Line segments $[P_0P_1]$ and $[P_0P_2]$ are added to the Argand diagram in part (a) and are shown on the following Argand diagram.
$P_0P_1$ is the length of $[P_0P_1]$ and $P_0P_2$ is the length of $[P_0P_2]$.
Show that $P_0P_1 \times P_0P_2 = 3$.
Working space:
Consider the case where $n = 4$.
The roots of the equation $z^4 = 1$ where $z \in \mathbb{C}$ are $1$, $\omega$, $\omega^2$, and $\omega^3$.
By factorizing $z^4 – 1$, or otherwise, deduce that $\omega^3 + \omega^2 + \omega + 1 = 0$.
Working space:
On the following Argand diagram, the points $P_0$, $P_1$, $P_2$, and $P_3$ lie on a circle of radius 1 unit with center $O(0, 0)$. $[P_0P_1]$, $[P_0P_2]$, and $[P_0P_3]$ are line segments.
Show that $P_0P_1 \times P_0P_2 \times P_0P_3 = 4$.
Working space:
For the case where $n = 5$, the equation $z^5 = 1$ where $z \in \mathbb{C}$ has roots $1$, $\omega$, $\omega^2$, $\omega^3$, and $\omega^4$.
It can be shown that $P_0P_1 \times P_0P_2 \times P_0P_3 \times P_0P_4 = 5$.
Now consider the general case for integer values of $n$, where $n \geq 2$.
Suggest a value for $P_0P_1 \times P_0P_2 \times … \times P_0P_{n-1}$.
Working space:
$P_0P_1$ can be expressed as $|1 – \omega|$.
(i) Write down expressions for $P_0P_2$ and $P_0P_3$ in terms of $\omega$.
(ii) Hence, write down an expression for $P_0P_{n-1}$ in terms of $n$ and $\omega$.
Working space:
Consider $z^n – 1 = (z – 1)(z^{n-1} + z^{n-2} + … + z + 1)$ where $z \in \mathbb{C}$.
(i) Express $z^{n-1} + z^{n-2} + … + z + 1$ as a product of linear factors over the set $\mathbb{C}$.
(ii) Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result for part (e).
Working space: