IB Mathematics AI HL 3.2 Geometry and Trigonometry Question Bank

Practise IB Mathematics AI HL 3.2 by combining vectors, trigonometric identities, loci, 3D geometry and technology-supported modelling.

Exam points

Use vector, coordinate and 3D geometry methods to solve lines, planes, angles and distances.
Apply advanced trigonometric identities, transformations and parameter restrictions with technology checks.
Interpret dynamic geometry, loci and intersections while justifying exact results and assumptions.

Covered syllabus points

AHL 3.7 (HL)—Radian measure AHL 3.8 (HL)—Unit circle and trigonometric equations AHL 3.9 (HL)—Matrix transformations and fractals AHL 3.10 (HL)—Vectors AHL 3.11 (HL)—Vector equations of lines AHL 3.12 (HL)—Vector kinematics AHL 3.13 (HL)—Scalar and vector products AHL 3.14 (HL)—Graph theory basics AHL 3.15 (HL)—Adjacency matrices and transition matrices AHL 3.16 (HL)—Graph algorithms

[Maximum number: 9]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

Figure for Question 3.2 Geometry and trigonometry - AHL content question 1 — IB Maths AI HL

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

Find the angle through which Mars rotates on its axis each hour.

The time of sunrise on Mars depends on the angle, $\delta$ , at which it tilts towards the Sun. During a Martian year, $\delta$ varies from -0.440 to 0.440 radians.

The angle, $\omega$ , through which Mars rotates on its axis from the start of a Martian day to the moment of sunrise, at point A , is given by $\cos \omega=0.839 \tan \delta, 0 \leq \omega \leq \pi$ .

[ 3 ]

length of day $=24 \frac{2}{3}$ hours

Note: Award A1 for $\frac{2}{3}, 0.666 \ldots, 0 . \overline{6}$ or 0.667 .

\frac{2 \pi}{24 \frac{2}{3}}

Note: Accept $\left(\frac{360}{24 \frac{2}{3}}\right)$ .

=0.255 \text { radians }\left(=0.254723 \ldots, \frac{3 \pi}{37}, 14.5945 \ldots{ }^{\circ}\right)

(b)

Show that the maximum value of $\omega=1.98$ , correct to three significant figures.

[ 3 ]

substitution of either value of $\delta$ into equation

Note: Both (M1) lines may be seen in either part (c)(i) or part (c)(ii).

\cos \omega=0.839 \tan (-0.440)

\omega=1.97684 \ldots

\approx 1.98

Note: For substitution of 1.98 award MOAO.

(c)

Find the minimum value of $\omega$ .

[ 1 ]

$\delta=0.440$

A1

(d)

Hence show that a=1.6, correct to two significant figures.

[ 2 ]

$a=\frac{7.76075 \ldots-4.57258 \ldots}{2}$ $\approx 1.59408 \ldots$

Note: Award M1 for substituting their values into a correct expression.

Award A1 for a correct value of a from their expression which has at least 3 significant figures and rounds correctly to 1.6 .
$\approx 1.6$ (correct to 2 sf )

[Maximum number: 10]

François is a video game designer. He designs his games to take place in two dimensions, relative to an origin O . In one of his games, an object travels on a straight line $L_{1}$ with vector equation

r=\binom{-1}{1}+\lambda\binom{2}{-1}

(a)

Write down $L_{1}$ in the form $x=x_{0}+\lambda l$ and $y=y_{0}+\lambda m$ , where $l, m \in \mathbb{Z}$ .

[ 1 ]

$x=-1+2 \lambda, y=1-\lambda$

A1

(b)

Francois uses the matrix $\boldsymbol{T}=\begin{pmatrix}1 & 7 \\ 7 & -1\end{pmatrix}$ to transform $L_{1}$ into a new straight line $L_{2}$ . The object will then travel along $L_{2}$ .

Find the vector equation of $L_{2}$ .

[ 4 ]

\begin{pmatrix}1&7\\7&-1\end{pmatrix}\binom{-1+2\lambda}{1-\lambda}=\binom{6-5\lambda}{-8+15\lambda}

\mathbf{r}=\binom{6}{-8}+\lambda\binom{-5}{15}

(or equivalent)

Note: Award (M1) for the correct format of a vector equation of a line, A1 for the line being completely correct.

(c)

François knows that the transformation given by matrix $\boldsymbol{T}$ is made up of the following three separate transformations (in the order listed):
- A rotation of $\frac{\pi}{4}$ , anticlockwise (counter-clockwise) about the origin O
- An enlargement of scale factor $5\sqrt{2}$ , centred at O
- A reflection in the straight line y=mx, where $m=\tan \alpha, 0 \leq \alpha<\pi$

Write down the matrix that represents

[ 4 ]

(i)

the rotation.

[ 2 ]

\begin{pmatrix}\cos\left(\frac{\pi}{4}\right)&-\sin\left(\frac{\pi}{4}\right)\\\sin\left(\frac{\pi}{4}\right)&\cos\left(\frac{\pi}{4}\right)\end{pmatrix}

\begin{pmatrix}0.707&-0.707\\0.707&0.707\end{pmatrix}

\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}

(ii)

the enlargement.

[ 2 ]

\begin{pmatrix}5\sqrt{2}&0\\0&5\sqrt{2}\end{pmatrix}

(d)

The matrix R represents the reflection. Write down R in terms of $\alpha$ .

[ 1 ]

\boldsymbol{R}=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}

All question bank results loaded

Covered syllabus points

3.2 Geometry and trigonometry - AHL content question 1

Question (a)

Question (b)

Question (c)

Question (d)

3.2 Geometry and trigonometry - AHL content question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (d)