An arithmetic sequence has first term $60$ and common difference $-2.5$.

The temperature $T °\mathrm{C}$ of water $t$ minutes after being poured into a cup can be modelled by $T=T_0{\mathrm{e}}^{-kt}$ where $t\ge 0$ and $T_0, k$ are positive constants.

The water is initially boiling at $100 °\text{C}$. When $t=10$, the temperature of the water is $70 °\text{C}$.

Consider the curves $y=x^2 \sin x$ and $y=-1-\sqrt{1+4{\left(x+2\right)}^2}$ for $-\mathrm{\pi }\le x\le 0$.

A function $f$ is given by $f(x)=(2x+2)(5-<!--\; -\; -->x^2)$.

The graph of the function $g(x)=5^x+6x-<!--\; -\; -->6$ intersects the graph of $f$.

Consider the function $f\left(x\right)=\frac{1}{3}{x}^3+\frac{3}{4}{x}^2-<!--\; -\; -->x-<!--\; -\; -->1$.

The function has one local maximum at $x=p$ and one local minimum at $x=q$.

The following diagram shows a circle with centre $\text{O}$ and radius $3$.

Points $\text{A}$, $\text{P}$ and $\text{B}$ lie on the circumference of the circle.

Chord $\left[\text{AB}\right]$ has length $L$ and $\text{A}\hat{\text{O}}\text{B}=\theta$ radians.

Points $\text{A}$ and $\text{P}$ lie on opposite banks of a river, such that $\text{AP}$ is the shortest distance across the river. Point $\text{B}$ represents the centre of a city which is located on the riverbank. $\text{PB}=215 \text{km}$, $\text{AP}=65 \text{km}$ and $\text{A}\hat{\text{P}}\text{B}=90°$.

The following diagram shows this information.

A boat travels at an average speed of $42 {\text{km\hspace{0.05em}h}}^{-1}$. A bus travels along the straight road between $\text{P}$ and $\text{B}$ at an average speed of $84 {\text{km\hspace{0.05em}h}}^{-1}$.

Find the travel time, in hours, from $\text{A}$ to $\text{B}$ given that

There is a point $\text{D}$, which lies on the road from $\text{P}$ to $\text{B}$, such that $\text{BD}=x \text{km}$. The boat travels from $\text{A}$ to $\text{D}$, and the bus travels from $\text{D}$ to $\text{B}$.

An excursion involves renting the boat and the bus. The cost to rent the boat is $\$ 200$ per hour, and the cost to rent the bus is $\$ 150$ per hour.

Let $f\left(x\right)=4-x^3$ and $g\left(x\right)=\ln x$, for $x>0$.

Consider the curve y = 2x³ − 9x² + 12x + 2, for −1 < x < 3

Tommaso plans to compete in a regional bicycle race after he graduates, however he needs to buy a racing bicycle. He finds a bicycle that costs 1100 euro (EUR). Tommaso has 950 EUR and invests this money in an account that pays 5 % interest per year, compounded monthly.

The cost of the bicycle, $C$, can be modelled by $C=20x+1100$, where $x$ is the number of years since Tommaso invested his money.

The following table shows the probability distribution of a discrete random variable $X$, where $a⩾<!--\; ⩾\; -->0$ and $b⩾<!--\; ⩾\; -->0$.

Let $f(x)=6-<!--\; -\; -->\ln <!--\; \; -->(x^2+2)$, for $x\in <!--\; \in \; -->\mathbb{R}$. The graph of $f$ passes through the point $(p,4)$, where $p>0$.

Consider the function $f\left(x\right)=\frac{27}{x^2}-<!--\; -\; -->16x,x\ne <!--\; \ne \; -->0$.

Consider a function $f$, such that $f\left(x\right)=5.8\text{sin}\left(\frac{\pi }{6}(x+1)\right)+b$, 0 ≤ $x$ ≤ 10, $b\in <!--\; \in \; -->\mathbb{R}$.

The function $f$ has a local maximum at the point (2, 21.8) , and a local minimum at (8, 10.2).

A second function $g$ is given by $g(x)=p\text{sin}\left(\frac{2\mathrm{\pi <!--\; \pi \; -->}}{9}\left(x-<!--\; -\; -->3.75\right)\right)+q$,  0 ≤ $x$ ≤ 10;  $p$, $q\in <!--\; \in \; -->\mathbb{R}$.

The function $g$ passes through the points (3, 2.5) and (6, 15.1).

The function $f$ is defined by $f\left(x\right)=\frac{4x+1}{x+4}$, where $x\in \mathrm{ℝ}, x\ne -4$.

For the graph of $f$

The graphs of $f$ and $f^{-1}$ intersect at $x=p$ and $x=q$, where $p<q$.

The following diagram shows a semicircle with centre $\text{O}$ and radius $r$. Points $\text{P, Q}$ and $\text{R}$ lie on the circumference of the circle, such that $\text{PQ}=2r$ and $\text{R}\hat{\text{O}}\text{Q}=\theta$, where $0<\theta <\mathrm{\pi }$.

The height of water, in metres, in Dungeness harbour is modelled by the function $H\left(t\right)=a \sin \left(b\right(t-c\left)\right)+d$, where $t$ is the number of hours after midnight, and $a, b, c$ and $d$ are constants, where $a>0, b>0$ and $c>0$.

The following graph shows the height of the water for $13$ hours, starting at midnight.

The first high tide occurs at $04:30$ and the next high tide occurs $12$ hours later. Throughout the day, the height of the water fluctuates between $2.2 \text{m}$ and $6.8 \text{m}$.

All heights are given correct to one decimal place.

The following table shows a probability distribution for the random variable $X$, where $\text{E}(X)=1.2$.

A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable $X$.

A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.

Consider the function $f(x)=-<!--\; -\; -->x^4+a{x}^2+5$, where $a$ is a constant. Part of the graph of $y=f(x)$ is shown below.

It is known that at the point where $x=2$ the tangent to the graph of $y=f(x)$ is horizontal.

There are two other points on the graph of $y=f(x)$ at which the tangent is horizontal.

All living plants contain an isotope of carbon called carbon-14. When a plant dies, the isotope decays so that the amount of carbon-14 present in the remains of the plant decreases. The time since the death of a plant can be determined by measuring the amount of carbon-14 still present in the remains.

The amount, $A$, of carbon-14 present in a plant $t$ years after its death can be modelled by $A=A_0{\text{e}}^{-kt}$ where $t\ge 0$ and $A_0, k$ are positive constants.

At the time of death, a plant is defined to have $100$ units of carbon-14.

The time taken for half the original amount of carbon-14 to decay is known to be $5730$ years.

Consider the function $f\left(x\right)=x^2+x+\frac{50}{x}, x\ne 0.$

Consider the function $f\left(x\right)=\frac{48}{x}+k{x}^2-<!--\; -\; -->58$, where x > 0 and k is a constant.

The graph of the function passes through the point with coordinates (4 , 2).

P is the minimum point of the graph of f (x).

A water container is made in the shape of a cylinder with internal height $h$ cm and internal base radius $r$ cm.

The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.

The volume of the water container is $0.5{\text{m}}^3$.

The water container is designed so that the area to be coated is minimized.

One can of water-resistant material coats a surface area of $2000\text{ c}{\text{m}}^2$.

Consider the lines $L_1$ and $L_2$ with respective equations

$L_1\text{:}y=-<!--\; -\; -->\frac{2}{3}x+9$  and  $L_2\text{:}y=\frac{2}{5}x-<!--\; -\; -->\frac{19}{5}$.

A third line, $L_3$, has gradient $-<!--\; -\; -->\frac{3}{4}$.

Let $f\left(x\right)=3x-4^{0.15x^2}$ for $0\le x\le 3$.

A scientist conducted a nine-week experiment on two plants, $A$ and $B$, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant $A$ was given fertilizer regularly, while Plant $B$ was not.

The scientist found that the height of Plant $A, h_A \text{cm}$, at time $t$ weeks can be modelled by the function $h_A\left(t\right)=\sin (2t+6)+9t+27$, where $0\le t\le 9$.

The scientist found that the height of Plant $B, h_B \text{cm}$, at time $t$ weeks can be modelled by the function $h_B\left(t\right)=8t+32$, where $0\le t\le 9$.

Use the scientist’s models to find the initial height of

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=6x^2-12x+1$ and $g\left(x\right)=-x+c$, where $c\in \mathrm{ℝ}$.

Consider the function $f\left(x\right)={\mathrm{e}}^{-x^2}-0.5$, for $-2\le x\le 2$.

SpeedWay airline flies from city $\text{A}$ to city $\text{B}$. The flight time is normally distributed with a mean of $260$ minutes and a standard deviation of $15$ minutes.

A flight is considered late if it takes longer than $275$ minutes.

The flight is considered to be on time if it takes between $m$ and $275$ minutes. The probability that a flight is on time is $0.830$.

During a week, SpeedWay has $12$ flights from city $\text{A}$ to city $\text{B}$. The time taken for any flight is independent of the time taken by any other flight.

A particle moves in a straight line such that its velocity, $v \text{m\hspace{0.05em}s}{}^{-1}$, at time $t$ seconds is given by $v=\frac{\left(t^2+1\right)\cos t}{4}, 0\le t\le 3$.

Consider the function $f\left(x\right)=x^3-<!--\; -\; -->5x^2+6x-<!--\; -\; -->3+\frac{1}{x}$, $x>0$

The function $f\left(x\right)=x^3-<!--\; -\; -->5x^2+6x-<!--\; -\; -->3+\frac{1}{x}$, $x>0$, models the path of a river, as shown on the following map, where both axes represent distance and are measured in kilometres. On the same map, the location of a highway is defined by the function $g\left(x\right)=0.5{\left(3\right)}^{-<!--\; -\; -->x}+1$.

The origin, O(0, 0) , is the location of the centre of a town called Orangeton.

A straight footpath, $P$, is built to connect the centre of Orangeton to the river at the point where $x=\frac{1}{2}$.

Bridges are located where the highway crosses the river.

A straight road is built from the centre of Orangeton, due north, to connect the town to the highway.

Let $f\left(x\right)={\text{e}}^{2\text{sin}\left(\frac{\mathrm{\pi <!--\; \pi \; -->}x}{2}\right)}$, for x > 0.

The k th maximum point on the graph of f has x-coordinate x_k where $k\in <!--\; \in \; -->{\mathbb{Z}}^{+}$.

OAB is a sector of the circle with centre O and radius $r$, as shown in the following diagram.

The angle AOB is $\mathrm{\theta <!--\; \theta \; -->}$ radians, where .

The point C lies on OA and OA is perpendicular to BC.

Let $f(x)=x{\text{e}}^{-<!--\; -\; -->x}$ and $g(x)=-<!--\; -\; -->3f(x)+1$.

The graphs of $f$ and $g$ intersect at $x=p$ and $x=q$, where .

Let $f\left(x\right)=x-<!--\; -\; -->8$,  $g\left(x\right)=x^4-<!--\; -\; -->3$  and  $h\left(x\right)=f\left(g\left(x\right)\right)$.

Consider the function $f$ defined by $f\left(x\right)=90{\text{e}}^{-0.5x}$ for $x\in {\mathrm{ℝ}}^{+}$.

The graph of $f$ and the line $y=x$ intersect at point $\text{P}$.

The line $L$ has a gradient of $-1$ and is a tangent to the graph of $f$ at the point $\text{Q}$.

The shaded region $A$ is enclosed by the graph of $f$ and the lines $y=x$ and $L$.

Let $f(x)=\ln <!--\; \; -->x$ and $g(x)=3+\ln <!--\; \; -->\left(\frac{x}{2}\right)$, for $x>0$.

The graph of $g$ can be obtained from the graph of $f$ by two transformations:

$$\begin{array}{l}\text{a horizontal stretch of scale factor }q\text{ followed by}\\ \text{a translation of }\left(\begin{array}{c}h\\ k\end{array}\right).\end{array}$$

Let $h(x)=g(x)\times <!--\; \times \; -->\cos <!--\; \; -->(0.1x)$, for . The following diagram shows the graph of $h$ and the line $y=x$.

The graph of $h$ intersects the graph of $h^{-<!--\; -\; -->1}$ at two points. These points have $x$ coordinates 0.111 and 3.31 correct to three significant figures.

Gemma and Kaia started working for different companies on January 1st 2011.

Gemma’s starting annual salary was $\$45 000$, and her annual salary increases $2\%$ on January 1st each year after 2011.

Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.

A rocket is travelling in a straight line, with an initial velocity of $140$ m s⁻¹. It accelerates to a new velocity of $500$ m s⁻¹ in two stages.

During the first stage its acceleration, $a$ m s⁻², after $t$ seconds is given by $a\left(t\right)=240\text{sin}\left(2t\right)$, where $0⩽<!--\; ⩽\; -->t⩽<!--\; ⩽\; -->k$.

The first stage continues for $k$ seconds until the velocity of the rocket reaches $375$ m s⁻¹.

Let $f(x)=x^2-<!--\; -\; -->1$ and $g(x)=x^2-<!--\; -\; -->2$, for $x\in <!--\; \in \; -->\mathbb{R}$.

Let $f\left(x\right)=\frac{6x-<!--\; -\; -->1}{2x+3}$, for $x\ne <!--\; \ne \; -->-<!--\; -\; -->\frac{3}{2}$.

The following diagram shows the graph of a function $y=f(x)$, for $-<!--\; -\; -->6⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->-<!--\; -\; -->2$.

The points $(-<!--\; -\; -->6,6)$ and $(-<!--\; -\; -->2,6)$ lie on the graph of $f$. There is a minimum point at $(-<!--\; -\; -->4,0)$.

Let $g(x)=f(x-<!--\; -\; -->5)$.

A discrete random variable, $X$, has the following probability distribution:

Let $f\left(x\right)=\frac{8x-<!--\; -\; -->5}{cx+6}$ for $x\ne <!--\; \ne \; -->-<!--\; -\; -->\frac{6}{c},c\ne <!--\; \ne \; -->0$.

Let f(x) = ln x − 5x , for x > 0 .