A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.

The model is of the form

$\left(\begin{array}{c}{X}_{n+1}\\ Z_{n+1}\end{array}\right)=\mathit{M}\left(\begin{array}{c}{X}_n\\ Z_n\end{array}\right)$

where $X_n$ is the probability of the gene being in its normal state after dividing for the $n\text{th}$ time, and $Z_n$ is the probability of it being in another state after dividing for the $n\text{th}$ time, where $n\in \mathrm{\mathbb{N}}$.

Matrix $\mathit{M}$ is found to be $\left(\begin{array}{cc}0.94 & b\\ 0.06 & 0.98\end{array}\right)$.

The gene is in its normal state when $n=0$. Calculate the probability of it being in its normal state

A transformation, $T$, of a plane is represented by $\mathit{r}\prime =\mathit{P}\mathit{r}+\mathit{q}$, where $\mathit{P}$ is a $2 \times 2$ matrix, $\mathit{q}$ is a $2 \times 1$ vector, $\mathit{r}$ is the position vector of a point in the plane and $\mathit{r}\prime$ the position vector of its image under $T$.

The triangle $\text{OAB}$ has coordinates $(0, 0)$, $(0, 1)$ and $(1, 0)$. Under T, these points are transformed to $(0, 1)$, $\left(\frac{1}{4}, 1+\frac{\sqrt{3}}{4}\right)$ and $\left(\frac{\sqrt{3}}{4}, \frac{3}{4}\right)$ respectively.

$\mathit{P}$ can be written as $\mathit{P}=\mathit{R}\mathit{S}$, where $\mathit{S}$ and $\mathit{R}$ are matrices.

$\mathit{S}$ represents an enlargement with scale factor $0.5$, centre $(0, 0)$.

$\mathit{R}$ represents a rotation about $(0, 0)$.

The transformation $T$ can also be described by an enlargement scale factor $\frac{1}{2}$, centre $(a, b)$, followed by a rotation about the same centre $(a, b)$.

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

$\frac{d^{2}x}{d{t}^2}+5\frac{dx}{dt}+6x=0$

The equation for the motion of the particle is amended to

$\frac{d^{2}x}{d{t}^2}+5\frac{dx}{dt}+6x=3t+4$.

When $t=0$ the particle is stationary at $\text{O}$.

A change in grazing habits has resulted in two species of herbivore, $\text{X}$ and $\text{Y}$, competing for food on the same grasslands. At time $t=0$ environmentalists begin to record the sizes of both populations. Let the size of the population of $\text{X}$ be $x$, and the size of the population $\text{Y}$ be $y$. The following model is proposed for predicting the change in the sizes of the two populations:

$\dot{x}=0.3x-0.1y$

$\dot{y}=-0.2x+0.4y$

for $x, y>0$

For this system of coupled differential equations find

When $t=0$ $\text{X}$ has a population of $2000$.

It is known that $\text{Y}$ has an initial population of $2900$.

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

$k=A{\text{e}}^{-\frac{c}{T}}$

This equation links a variable $k$ with the temperature $T$, where $A$ and $c$ are positive constants and $T>0$.

The Arrhenius equation predicts that the graph of $\ln k$ against $\frac{1}{T}$ is a straight line.

Write down

The following data are found for a particular reaction, where $T$ is measured in Kelvin and $k$ is measured in ${\text{cm}}^3 {\text{mol}}^{-1} {\text{s}}^{-1}$:

Find an estimate of

Phil takes out a bank loan of $150 000 to buy a house, at an annual interest rate of 3.5%. The interest is calculated at the end of each year and added to the amount outstanding.

To pay off the loan, Phil makes annual deposits of $P at the end of every year in a savings account, paying an annual interest rate of 2% . He makes his first deposit at the end of the first year after taking out the loan.

David visits a different bank and makes a single deposit of $Q , the annual interest rate being 2.8%.

A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.

The displacement, $x$, of the spring is measured, in centimetres, from the equilibrium position of $x=0$. The value of $x$ can be modelled by the following second order differential equation, where $t$ is the time, measured in seconds, after the initial displacement.

$\ddot{x}+3\dot{x}+1.25x=0$

The differential equation can be expressed in the form $\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=\mathit{A}\left(\begin{array}{c}x\\ y\end{array}\right)$, where $\mathit{A}$ is a $2\times 2$ matrix.

A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin $\text{O}$ and a set of $x$-$y$-axes.

In each case, the drone moves to a new position represented by the following transformations:

• a rotation anticlockwise of $\frac{\mathrm{\pi }}{6}$ radians about $\text{O}$
• a reflection in the line $y=\frac{x}{\sqrt{3}}$
• a rotation clockwise of $\frac{\mathrm{\pi }}{3}$ radians about $\text{O}$.

All the movements are performed in the listed order.

The function $f$ is given by $f(x)=m{x}^3+n{x}^2+px+q$, where $m$, $n$, $p$, $q$ are integers.

The graph of $f$ passes through the point (0, 0).

The graph of $f$ also passes through the point (3, 18).

The graph of $f$ also passes through the points (1, 0) and (–1, –10).

Consider the equation $x^5-<!--\; -\; -->3x^4+m{x}^3+n{x}^2+px+q=0$, where $m$, $n$, $p$, $q\in <!--\; \in \; -->\mathbb{R}$.

The equation has three distinct real roots which can be written as $\text{lo}{\text{g}}_{2}a$, $\text{lo}{\text{g}}_{2}b$ and $\text{lo}{\text{g}}_{2}c$.

The equation also has two imaginary roots, one of which is $d\text{i}$ where $d\in <!--\; \in \; -->\mathbb{R}$.

The values $a$, $b$, and $c$ are consecutive terms in a geometric sequence.

Let $z=1-\text{i}$.

Let $w_1={\text{e}}^{\text{i}x}$ and $w_2={\text{e}}^{\text{i}(x-\frac{\mathrm{\pi }}{2})}$, where $x\in \mathrm{ℝ}$.

The current, $I$, in an AC circuit can be modelled by the equation $I=a \cos (bt-c)$ where $b$ is the frequency and $c$ is the phase shift.

Two AC voltage sources of the same frequency are independently connected to the same circuit. If connected to the circuit alone they generate currents $I_{\text{A}}$ and $I_{\text{B}}$. The maximum value and the phase shift of each current is shown in the following table.

When the two voltage sources are connected to the circuit at the same time, the total current $I_{\text{T}}$ can be expressed as $I_{\text{A}}+I_{\text{B}}$.

On the day of her birth, 1st January 1998, Mary’s grandparents invested $\mathrm{\$<!--\; \$\; -->}x$ in a savings account. They continued to deposit $\mathrm{\$<!--\; \$\; -->}x$ on the first day of each month thereafter.

The account paid a fixed rate of 0.4% interest per month. The interest was calculated on the last day of each month and added to the account.

Let $\mathrm{\$<!--\; \$\; -->}{A}_n$ be the amount in Mary’s account on the last day of the $n\text{th}$ month, immediately after the interest had been added.

Let A .

Let A² + $m$A + $n$I = O where $m$, $n\in <!--\; \in \; -->\mathbb{Z}$ and O = .

The 3rd term of an arithmetic sequence is 1407 and the 10th term is 1183.

Let A = .

Let B = .

In this question, give all answers to two decimal places.

Bryan decides to purchase a new car with a price of €14 000, but cannot afford the full amount. The car dealership offers two options to finance a loan.

Finance option A:

A 6 year loan at a nominal annual interest rate of 14 % compounded quarterly. No deposit required and repayments are made each quarter.

Finance option B:

A 6 year loan at a nominal annual interest rate of $r$ % compounded monthly. Terms of the loan require a 10 % deposit and monthly repayments of €250.

Matrices A, B and C are defined by

A =    B =    C = .

Let X be an unknown 2 × 2 matrix satisfying the equation

AX + B = C.

This equation may be solved for X by rewriting it in the form

X = A⁻¹ D.

where D is a 2 × 2 matrix.

Let $S$_$n$ be the sum of the first $n$ terms of the arithmetic series 2 + 4 + 6 + ….

Let M = .

It may now be assumed that M^$n$ = , for $n$ ≥ 4. The sum T_$n$ is defined by

T_$n$ = M¹ + M² + M³ + ... + M^$n$.

Let M = .

Let A =  and B = . Giving your answers in terms of $a$, $b$, $c$, $d$ and $e$,

Let $z=a+b\text{i}$, $a$, $\text{b}\in <!--\; \in \; -->{\mathbb{R}}^{+}$ and let $\text{arg}z=\mathrm{\theta <!--\; \theta \; -->}$.

Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.

Let , where $a\in <!--\; \in \; -->\mathbb{Z}$.

Let $\mathrm{\gamma <!--\; \gamma \; -->}=\frac{1+\text{i}\sqrt{3}}{2}$.

The matrix A is defined by A = .

Deduce that

Long term experience shows that if it is sunny on a particular day in Vokram, then the probability that it will be sunny the following day is $0.8$. If it is not sunny, then the probability that it will be sunny the following day is $0.3$.

The transition matrix $\mathit{T}$ is used to model this information, where $\mathit{T}=\left(\begin{array}{cc}0.8& 0.3\\ 0.2& 0.7\end{array}\right)$.

The matrix $\mathit{T}$ can be written as a product of three matrices, $\mathit{P}\mathit{D}\mathbf{ }{\mathit{P}}^{-1}$ , where $\mathit{D}$ is a diagonal matrix.

The matrix M is given by M .

Let A =  and B = .

Given that X = B – A^–1 and Y = B^–1 – A,

You are told that , for $n\in <!--\; \in \; -->{\mathbb{Z}}^{+}$.

Given that , for $n\in <!--\; \in \; -->{\mathbb{Z}}^{+}$,

An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable $x$ measures the concentration of mercury in micrograms per litre.

The situation is modelled using the second order differential equation

$\frac{{\text{d}}^{2}x}{d{t}^2}+3\frac{dx}{dt}+2x=0$

where $t\ge 0$ is the time measured in days since the leak started. It is known that when $t=0, x=0$ and $\frac{dx}{dt}=1$.

If the mercury levels are greater than $0.1$ micrograms per litre, fishing in the river is considered unsafe and is stopped.

The river authority decides to stop people from fishing in the river for $10\%$ longer than the time found from the model.

Let M² = M where M = . 

It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.

At the start of 2018, there are 2500 fish in the lake.

Consider the following system of coupled differential equations.

$\frac{dx}{dt}=-4x$

$\frac{dy}{dt}=3x-2y$

Find the value of $\frac{dy}{dx}$

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.