Geometry and trigonometry: 17 IB Maths AI HL practice questions

A particle P moves with velocity $\mathbf{v} = \begin{pmatrix} -15 \\ 2 \\ 4 \end{pmatrix}$ in a magnetic field, $\mathbf{B} = \begin{pmatrix} 0 \\ d \\ 1 \end{pmatrix}$, $d \in \mathbb{R}$.

(a) Given that $\mathbf{v}$ is perpendicular to $\mathbf{B}$, find the value of $d$.[2]

The force, $F$, produced by P moving in the magnetic field is given by the vector equation $\mathbf{F} = a\mathbf{v} \times \mathbf{B}$, $a \in \mathbb{R}^+$.

(b) Given that $|\mathbf{F}| = 14$, find the value of $a$.[4]

Let $R(\alpha)$ be the matrix representing a rotation, counter-clockwise (anticlockwise) about the origin, through an angle of $\alpha$.

(a) Write down $R(2\alpha)$ as a $2 \times 2$ matrix.[2]

(b) Calculate $R(\alpha) \times R(\alpha)$.[2]

(c) Use your answers from part (a) and part (b) to

(i) explain why $\sin(2\alpha) = 2\sin(\alpha) \cos(\alpha)$.[4]

(ii) show that $\cos(2\alpha) = 1 - 2\sin^2(\alpha)$.

A garden has a triangular sunshade suspended from three points $A(2, 0, 2)$, $B(8, 0, 2)$ and $C(5, 4, 3)$, relative to an origin in the corner of the garden. All distances are measured in metres.

(a) (i) Find $\overrightarrow{CA}$.[2]

(ii) Find $\overrightarrow{CB}$.[2]

(b) Find $\overrightarrow{CA} \times \overrightarrow{CB}$.[2]

(c) Hence find the area of the triangle $ABC$.[2]

Line $L_1$ has a vector equation $\mathbf{r} = \begin{pmatrix} 3p+4 \\ 2p-1 \\ p+9 \end{pmatrix}$, where $p \in \mathbb{R}$.

Line $L_2$ has a vector equation $\mathbf{r} = \begin{pmatrix} q-2 \\ 1-q \\ 2q+1 \end{pmatrix}$, where $q \in \mathbb{R}$.

The two lines intersect at point M.

(a) Find the coordinates of M.[3]

(b) Find the acute angle between the two lines.[4]

The equation of the line $y = mx + c$ can be expressed in vector form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$.

(a) Find the vectors $\mathbf{a}$ and $\mathbf{b}$ in terms of $m$ and/or $c$.[2]

The matrix $\mathbf{M}$ is defined by $\begin{pmatrix} 6 & 3 \\ 4 & 2 \end{pmatrix}$.

(b) Find the value of $\det \mathbf{M}$.[1]

The line $y = mx + c$ (where $m \neq -2$) is transformed into a new line using the transformation described by matrix $\mathbf{M}$.

(c) Show that the equation of the resulting line does not depend on $m$ or $c$.[4]

A submarine is located in a sea at coordinates $(0.8, 1.3, -0.3)$ relative to a ship positioned at the origin $O$. The $x$ direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards.

All distances are measured in kilometres.

The submarine travels with direction vector $\begin{pmatrix} -2 \\ -3 \\ 1 \end{pmatrix}$.

(a) Assuming the submarine travels in a straight line, write down an equation for the line along which it travels.[2]

The submarine reaches the surface of the sea at the point P.

(b) (i) Find the coordinates of P.[3]

(ii) Find OP.[2]

A geometric transformation $T: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} x' \\ y' \end{pmatrix}$ is defined by

$$T: \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 7 & -10 \\ 2 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -5 \\ 4 \end{pmatrix}$$

(a) Find the coordinates of the image of the point $(6, -2)$.[2]

(b) Given that $T: \begin{pmatrix} p \\ q \end{pmatrix} \mapsto 2\begin{pmatrix} p \\ q \end{pmatrix}$, find the value of $p$ and the value of $q$.[3]

(c) A triangle $L$ with vertices lying on the $xy$ plane is transformed by $T$. Explain why both $L$ and its image will have exactly the same area.[2]

The point $A$ has coordinates $(1, 2, 1)$ and the point $B$ has coordinates $(3, 5, 2)$.

(a) Find $AB$.[2]

Triangle $ABC$ is right-angled with its right angle at $B$. The point $C$ has coordinates $(2, 8, k)$.

(b) Find the value of $k$.[4]

(c) Calculate the size of $\angle B\overline{A}C$.[3]

Ollie has installed security lights on the side of his house that are activated by a sensor. The sensor is located at point C directly above point D. The area covered by the sensor is shown by the shaded region enclosed by triangle ABC. The distance from A to B is 4.5 m and the distance from B to C is 6 m. Angle ACB is 15°.

Point B on the ground is 5 m from point E at the entrance to Ollie's house. He is 1.8 m tall and is standing at point D, below the sensor. He walks towards point B.

(a) Find $\angle CAB$.[3]

(b) Find the distance Ollie is from the entrance to his house when he first activates the sensor.[5]

A straight line $L$ has vector equation $\mathbf{r}=\begin{pmatrix}1\\3\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\2\end{pmatrix}$ and point Q has coordinates $(11,-1,3)$.

Point P is the point on $L$ closest to Q.

(a) Find the coordinates of P.[4]

(b) Find a vector that is perpendicular to both $L$ and the line passing through points P and Q.[3]

The quadrilateral $ABCD$ has coordinates $A(1, 3, 5)$, $B(4, 7, 5)$, $C(5, 8, 7)$ and $D(2, 4, 7)$.

(a) Write down $\overrightarrow{AD}$.[1]

(b) Calculate

(i) the size of $\angle B\hat{A}D$.[3]

(ii) the area of triangle $BAD$.[4]

(c) Show that $ABCD$ is a parallelogram.[2]

Two lines $L_1$ and $L_2$ are given by the following equations, where $p \in \mathbb{R}$.

$$L_1: \mathbf{r} = \begin{pmatrix} 2 \\ p+9 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} p \\ 2p \\ 4 \end{pmatrix}$$

$$L_2: \mathbf{r} = \begin{pmatrix} 14 \\ 7 \\ p+12 \end{pmatrix} + \mu \begin{pmatrix} p+4 \\ 4 \\ -7 \end{pmatrix}$$

It is known that $L_1$ and $L_2$ are perpendicular.

(a) Find the possible value(s) for $p$.[3]

(b) In the case that $p < 0$, determine whether the lines intersect.[4]

At 1:00 pm a ship is 1 km east and 4 km north of a harbour. A coordinate system is defined with the harbour at the origin. The position vector of the ship at 1:00 pm is given by $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$.

The ship has a constant velocity of $\begin{pmatrix} 1.2 \\ -0.6 \end{pmatrix}$ kilometres per hour (km h$^{-1}$).

(a) Write down an expression for the position vector $\mathbf{r}$ of the ship, $t$ hours after 1:00 pm.[1]

(b) Find the time at which the bearing of the ship from the harbour is $045°$.[4]

In this question, $i$ denotes a unit vector due east, and $j$ denotes a unit vector due north.

Two ships, A and B, are each moving with constant velocities.

The position vector of ship A, at time $t$ hours, is given as $\mathbf{r}_A = (1 + 2t)\mathbf{i} + (3 - 3t)\mathbf{j}$.

The position vector of ship B, at time $t$ hours, is given as $\mathbf{r}_B = (-2 + 4t)\mathbf{i} + (-4 + t)\mathbf{j}$.

(a) Find the bearing on which ship A is sailing.[3]

(b) Find the value of $t$ when ship B is directly south of ship A.[2]

(c) Find the value of $t$ when ship B is directly south-east of ship A.[3]

Two AC (alternating current) electrical sources with the same frequencies are combined. The voltages from these sources can be expressed as $V_1 = 6\sin(at + 30°)$ and $V_2 = 6\sin(at + 90°)$.

The combined total voltage can be expressed in the form $V_1 + V_2 = V\sin(at + \theta°)$.

Determine the value of $V$ and the value of $\theta$.[5]