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[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.

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 ]

$\delta=0.440$

(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 ]

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

(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}

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IB Maths AI HL 3.2 Geometry and Trigonometry Ahl Content Topic Practice

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Question (d)

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Question (b)

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Question (d)