[Maximum number: 3]

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)

(i)

Write down $z_{1}$ and $z_{2}$ in exponential form, with a constant modulus.

[ 3 ]

[Maximum number: 3]

The growth of a particular type of seashell is being studied by Manon. At the end of each month Manon records the increase in the width of a seashell since the end of the previous month.
She models the monthly increase in the width of the seashell by a geometric sequence with common ratio 0.8 . In the first month, the width of the seashell increases by 4 mm .

(a)

Find the maximum possible width of the seashell, predicted by Manon's model.

[ 3 ]

[Maximum number: 1]

The following question explores a possible method of drawing phase portraits for non-linear coupled systems, taking a predator-prey model as a particular example.
Gander Green wildlife park contains a population of Czech geese ( x, measured in hundreds), and a population of gray foxes ( y, measured in hundreds).
Research indicates that the population growth of both geese and foxes can be modelled by the following differential equations, in which t is measured in years.

\left.\begin{array}{l} \frac{\mathrm{d} x}{\mathrm{~d} t}=2 x-\frac{x y}{2} \\ \frac{\mathrm{~d} y}{\mathrm{~d} t}=-3 y+x y \end{array}\right\} \text { for } x, y \geq 0

(a)

Given that $\binom{\dot{X}}{\dot{Y}}=\boldsymbol{M}\binom{X}{Y}$ , where M is a square matrix, write down M.

[ 1 ]

[Maximum number: 6]

In a small village there are two doctors' clinics, one owned by Doctor Black and the other owned by Doctor Green. It was noted after each year that 3.5 % of Doctor Black's patients moved to Doctor Green's clinic and 5\% of Doctor Green's patients moved to Doctor Black's clinic. All additional losses and gains of patients by the clinics may be ignored.
At the start of a particular year, it was noted that Doctor Black had 2100 patients on their register, compared to Doctor Green's 3500 patients.

(a)

Find a matrix P, with integer elements, such that $\boldsymbol{T}=\boldsymbol{P} \boldsymbol{D} \boldsymbol{P}^{-1}$ , where D is a diagonal matrix.

[ 6 ]

[Maximum number: 5]

Two complex numbers are z=2+a i and w=b+4 i, where $a, b \in \mathbb{R}$ .

(a)

Find an expression for the real part of $z^{2}$ , in terms of a.

[ 2 ]

(b)

Find the value of a and of b, given that $z^{2}+2 w=3+6 \mathrm{i}$ .

[ 3 ]

[Maximum number: 3]

Ayaka is creating a design made from a sequence of rectangles. The diagram shows part of her design, using 5 rectangles.

The first rectangle has the following dimensions: height 4.5 cm and width 3 cm .
The dimensions of each subsequent rectangle are $\frac{2}{3}$ of the dimensions of the
previous rectangle. previous rectangle.

(a)

Find the smallest total width that her design cannot exceed.

The width of Ayaka's final design must be at least 8.5 cm and use the least number of rectangles.

[ 3 ]

[Maximum number: 8]

The drivers of a delivery company can park their vans overnight either at its headquarters or at home.
Urvashi is a driver for the company. If Urvashi has parked her van overnight at headquarters on a given day, the probability that she parks her van at headquarters on the following day is 0.88 . If Urvashi has parked her van overnight at her home on a given day, the probability that she parks her van at home on the following day is 0.92 .

(a)

Write down the characteristic polynomial for the matrix T. Give your answer in the form $\lambda^{2}+b \lambda+c$ .

[ 2 ]

(b)

Calculate eigenvectors for the matrix T.

[ 4 ]

(c)

Write down matrices P and D such that $\boldsymbol{T}=\boldsymbol{P} \boldsymbol{D} \boldsymbol{P}^{-1}$ , where D is a diagonal matrix.

[ 2 ]

[Maximum number: 8]

The transformation T is represented by the matrix $\boldsymbol{M}=\left(\begin{array}{cc}2 & -4 \\ 3 & 1\end{array}\right)$ .
A pentagon with an area of $12 \mathrm{~cm}^{2}$ is transformed by T.

(a)

Find the area of the image of the pentagon.

Under the transformation T, the image of point X has coordinates (2 t-3,6-5 t), where $t \in \mathbb{R}$ .

[ 2 ]

(b)

Find, in terms of t, the coordinates of X .

[ 6 ]

[Maximum number: 6]

Given $z=\sqrt{3}-\mathrm{i}$ .

(a)

Write z in the form $z=r \mathrm{e}^{\theta \mathrm{i}}$ , where $r \in \mathbb{R}^{+},-\pi<\theta \leq \pi$ .

Let $z_{1}=\mathrm{e}^{2 \mathrm{i} \mathrm{i}}$ and $z_{2}=2 \mathrm{e}^{\left(2 t-\frac{\pi}{6}\right) \mathrm{i}}$ .

[ 2 ]

(b)

Find $\operatorname{Im}\left(z_{1}+z_{2}\right)$ in the form $p \sin (2 t+q)$ , where $p>0, t \in \mathbb{R}$ and $-\pi \leq q \leq \pi$ .

[ 4 ]

[Maximum number: 5]

A particle moves such that its displacement, x metres, from a point O at time t seconds is given by the differential equation

\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+5 \frac{\mathrm{~d} x}{\mathrm{~d} t}+6 x=0

(a)

(i)

Find the eigenvalues for the matrix $\left(\begin{array}{cc}0 & 1 \\ -6 & -5\end{array}\right)$ .

[ 5 ]