IB Maths AI HL – Topic 1: Number and Algebra

Let $z_1 = 3 + 4i$ and $z_2 = 1 – 2i$.(a) Find $z_1 z_2$ in the form $a + bi$.(b) Find $\dfrac{z_1}{z_2}$ in the form $a + bi$.(c) Find $|z_1|$.

Let $z = -1 + \sqrt{3}\,i$.(a) Write $z$ in the form $r\,\text{cis}\,\theta$ with $-\pi < \theta \le \pi$.(b) Write $z$ in Euler form $re^{i\theta}$.(c) Hence find $z^6$.

Solve $z^3 = 8i$, giving the roots in the form $a + bi$ with exact values.

Let $\mathbf{A} = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & 0 \\ -2 & 5 \end{pmatrix}$.(a) Find $\mathbf{AB}$.(b) Find $\det \mathbf{A}$.(c) Find $\mathbf{A}^{-1}$.

Use the inverse matrix from question 4 to solve the simultaneous equations $2x + y = 7$ and $3x + 4y = 18$. Show the matrix method clearly.

Let $\mathbf{M} = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$.(a) Show that the characteristic equation of $\mathbf{M}$ is $\lambda^2 – 7\lambda + 10 = 0$.(b) Find the eigenvalues of $\mathbf{M}$.(c) Find a corresponding eigenvector for each eigenvalue.

A triangle has vertices $P(1, 1)$, $Q(3, 1)$ and $R(1, 4)$. It is rotated $90^\circ$ anticlockwise about the origin and the image is then reflected in the line $y = x$.(a) Write down the matrix for each transformation.(b) Find the single matrix representing the combined transformation, and describe it geometrically.(c) Find the coordinates of the image of the triangle.

The transformation $\mathbf{T} = \begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix}$ is applied to a square of area $4$ square units.(a) Find $\det \mathbf{T}$.(b) State the area of the image.(c) The image is a parallelogram. Explain why it cannot be a rectangle by considering the images of the unit vectors $\mathbf{i}$ and $\mathbf{j}$.

Using the eigenvalues and eigenvectors of $\mathbf{M} = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ from question 6:(a) Write $\mathbf{M}$ in the form $\mathbf{PDP}^{-1}$, stating $\mathbf{P}$, $\mathbf{D}$ and $\mathbf{P}^{-1}$.(b) Hence find an expression for $\mathbf{M}^n$.(c) Find $\mathbf{M}^5$.

Each year $20$ percent of the residents of town A move to town B, and $30$ percent of the residents of town B move to town A. Initially town A has $1000$ residents and town B has $0$. Let $\mathbf{x}_n = \begin{pmatrix} a_n \\ b_n \end{pmatrix}$ be the populations after $n$ years.(a) Write down the transition matrix $\mathbf{T}$ such that $\mathbf{x}_{n+1} = \mathbf{T}\mathbf{x}_n$.(b) Find the populations after $1$ year and after $2$ years.(c) Find the eigenvalues of $\mathbf{T}$.(d) Hence find the long-term populations of the two towns.

Two alternating voltages are $v_1 = 5\sin(\omega t)$ and $v_2 = 12\sin\left(\omega t + \frac{\pi}{2}\right)$, represented by the phasors (complex numbers) $5$ and $12i$ respectively.(a) Write down the phasor representing $v_1 + v_2$.(b) Find the modulus and argument of this phasor, giving the argument in radians to 3 significant figures.(c) Hence write $v_1 + v_2$ in the form $R\sin(\omega t + \alpha)$.

A transformation consists of a stretch with scale factor $3$ parallel to the $x$-axis, followed by a rotation of $45^\circ$ anticlockwise about the origin.(a) Find the single $2 \times 2$ matrix representing the combined transformation, giving entries in exact form.(b) Find the exact image of the point $(2, 0)$.(c) A shape of area $A$ is transformed. Find the area of its image in terms of $A$.