In this question, all lengths are in metres and time is in seconds.

Consider two particles, $P_1$ and $P_2$, which start to move at the same time.

Particle $P_1$ moves in a straight line such that its displacement from a fixed-point is given by $s\left(t\right)=10-\frac{7}{4}{t}^2$, for $t\ge 0$.

Sieun hits golf balls into the air. Each time she hits a ball she records $\theta$, the angle at which the ball is launched into the air, and $l$, the horizontal distance, in metres, which the ball travels from the point of contact to the first time it lands. The diagram below represents this information.

Sieun analyses her results and concludes:

$\frac{dl}{d\theta }=-0.2\theta +9, 35°\le \theta \le 75°$.

The following diagram shows part of the graph of $f\left(x\right)=\frac{k}{x}$, for $x>0, k>0$.

Let $\text{P}\left(p, \frac{k}{p}\right)$ be any point on the graph of $f$. Line $L_1$ is the tangent to the graph of $f$ at $\text{P}$.

Line $L_1$ intersects the $x$-axis at point $\text{A}\left(2p, 0\right)$ and the $y$-axis at point B.

Irina uses a set of coordinate axes to draw her design of a window. The base of the window is on the $x$-axis, the upper part of the window is in the form of a quadratic curve and the sides are vertical lines, as shown on the diagram. The curve has end points $(0, 10)$ and $(8, 10)$ and its vertex is $(4, 12)$. Distances are measured in centimetres.

The quadratic curve can be expressed in the form $y=a{x}^2+bx+c$ for $0\le x\le 8$.

Inspectors are investigating the carbon dioxide emissions of a power plant. Let $R$ be the rate, in tonnes per hour, at which carbon dioxide is being emitted and $t$ be the time in hours since the inspection began.

When $R$ is plotted against $t$, the total amount of carbon dioxide produced is represented by the area between the graph and the horizontal $t$-axis.

The rate, $R$, is measured over the course of two hours. The results are shown in the following table.

In an international competition, participants can answer questions in only one of the three following languages: Portuguese, Mandarin or Hindi. 80 participants took part in the competition. The number of participants answering in Portuguese, Mandarin or Hindi is shown in the table.

A boy is chosen at random.

Maria owns a cheese factory. The amount of cheese, in kilograms, Maria sells in one week, $Q$, is given by

$Q=882-<!--\; -\; -->45p$,

where $p$ is the price of a kilogram of cheese in euros (EUR).

Maria earns $(p-<!--\; -\; -->6.80)\text{ EUR}$ for each kilogram of cheese sold.

To calculate her weekly profit $W$, in EUR, Maria multiplies the amount of cheese she sells by the amount she earns per kilogram.

Consider the curve y = 5x³ − 3x.

The curve has a tangent at the point P(−1, −2).

A potter sells $x$ vases per month.

His monthly profit in Australian dollars (AUD) can be modelled by

$$P\left(x\right)=-<!--\; -\; -->\frac{1}{5}{x}^3+7x^2-<!--\; -\; -->120\text{,}x⩾<!--\; ⩾\; -->0.$$

Consider the function $f\left(x\right)=\frac{x^4}{4}$.

The diagram shows a circular horizontal board divided into six equal sectors. The sectors are labelled white (W), yellow (Y) and blue (B).

A pointer is pinned to the centre of the board. The pointer is to be spun and when it stops the colour of the sector on which the pointer stops is recorded. The pointer is equally likely to stop on any of the six sectors.

Eva will spin the pointer twice. The following tree diagram shows all the possible outcomes.

Let $f\left(x\right)=\sqrt{12-2x}, x\le a$. The following diagram shows part of the graph of $f$.

The shaded region is enclosed by the graph of $f$, the $x$-axis and the $y$-axis.

The graph of $f$ intersects the $x$-axis at the point $\left(a, 0\right)$.

A factory produces shirts. The cost, C, in Fijian dollars (FJD), of producing x shirts can be modelled by

C(x) = (x − 75)² + 100.

The cost of production should not exceed 500 FJD. To do this the factory needs to produce at least 55 shirts and at most s shirts.

Consider the curve $y=x^2-4x+2$.

Consider the graph of the function $f\left(x\right)=x^2-\frac{k}{x}$.

The equation of the tangent to the graph of $y=f\left(x\right)$ at $x=-2$ is $2y=4-5x$.

A function $f$ is given by $f(x)=4x^3+\frac{3}{x^2}-<!--\; -\; -->3,x\ne <!--\; \ne \; -->0$.

Let $f\left(x\right)=x^3-<!--\; -\; -->2x^2+ax+6$. Part of the graph of $f$ is shown in the following diagram.

The graph of $f$ crosses the $y$-axis at the point P. The line L is tangent to the graph of $f$ at P.

The surface area of an open box with a volume of $32 {\text{cm}}^3$ and a square base with sides of length $x \text{cm}$ is given by $S\left(x\right)=x^2+\frac{128}{x}$ where $x>0$.

A company’s profit per year was found to be changing at a rate of

$\frac{dP}{dt}=3t^2-8t$

where $P$ is the company’s profit in thousands of dollars and $t$ is the time since the company was founded, measured in years.

A modern art painting is contained in a square frame. The painting has a shaded region bounded by a smooth curve and a horizontal line.

When the painting is placed on a coordinate axes such that the bottom left corner of the painting has coordinates $(-1, -1)$ and the top right corner has coordinates $(2, 2)$, the curve can be modelled by $y=f\left(x\right)$ and the horizontal line can be modelled by the $x$-axis. Distances are measured in metres.

The artist used the equation $y=\frac{-x^3-3x^2+4x+12}{10}$ to draw the curve.

Let $f(x)=1+{\text{e}}^{-<!--\; -\; -->x}$ and $g(x)=2x+b$, for $x\in <!--\; \in \; -->\mathbb{R}$, where $b$ is a constant.

The graphs of $y=6-x$ and $y=1.5x^2-2.5x+3$ intersect at $(2, 4)$ and $(-1, 7)$, as shown in the following diagrams.

In diagram 1, the region enclosed by the lines $y=6-x$, $x=-1$, $x=2$ and the $x$-axis has been shaded.

In diagram 2, the region enclosed by the curve $y=1.5x^2-2.5x+3$, and the lines $x=-1$, $x=2$ and the $x$-axis has been shaded.

The coordinates of point A are $(6,-<!--\; -\; -->7)$ and the coordinates of point B are $(-<!--\; -\; -->6,2)$. Point M is the midpoint of AB.

$L_1$ is the line through A and B.

The line $L_2$ is perpendicular to $L_1$ and passes through M.

A quadratic function $f$ is given by $f(x)=a{x}^2+bx+c$. The points $(0,5)$ and $(-<!--\; -\; -->4,5)$ lie on the graph of $y=f(x)$.

The $y$-coordinate of the minimum of the graph is 3.

Ellis designs a gift box. The top of the gift box is in the shape of a right-angled triangle $\text{GIK}$.

A rectangular section $\text{HIJL}$ is inscribed inside this triangle. The lengths of $\text{GH, JK, HL}$, and $\text{LJ}$ are $p \text{cm}, q \text{cm}, 8 \text{cm}$ and $6 \text{cm}$ respectively.

The area of the top of the gift box is $A {\text{cm}}^2$.

Ellis wishes to find the value of $q$ that will minimize the area of the top of the gift box.

A cylinder with radius $r$ and height $h$ is shown in the following diagram.

The sum of $r$ and $h$ for this cylinder is 12 cm.

A company produces and sells electric cars. The company’s profit, $P$, in thousands of dollars, changes based on the number of cars, $x$, they produce per month.

The rate of change of their profit from producing $x$ electric cars is modelled by

$\frac{dP}{dx}=-1.6x+48, x\ge 0$.

The company makes a profit of $260$ (thousand dollars) when they produce $15$ electric cars.

The diagram shows part of the graph of a function $y=f(x)$. The graph passes through point $\text{A}(1,3)$.

The tangent to the graph of $y=f(x)$ at A has equation $y=-<!--\; -\; -->2x+5$. Let $N$ be the normal to the graph of $y=f(x)$ at A.

The following diagram shows part of the graph of $f(x)=\left(6-<!--\; -\; -->3x\right)\left(4+x\right)$, $x\in <!--\; \in \; -->\mathbb{R}$. The shaded region R is bounded by the $x$-axis, $y$-axis and the graph of $f$.

The following diagram shows the graph of $f^{\prime }$, the derivative of $f$.

The graph of $f^{\prime }$ has a local minimum at A, a local maximum at B and passes through $(4,-<!--\; -\; -->2)$.

The point $\text{P}(4,3)$ lies on the graph of the function, $f$.

The equation of a curve is $y=\frac{1}{2}{x}^4-<!--\; -\; -->\frac{3}{2}{x}^2+7$.

The gradient of the tangent to the curve at a point P is $-<!--\; -\; -->10$.

The point A has coordinates (4 , −8) and the point B has coordinates (−2 , 4).

The point D has coordinates (−3 , 1).

Let $\mathrm{\theta <!--\; \theta \; -->}$ be an obtuse angle such that $\text{sin}\mathrm{\theta <!--\; \theta \; -->}=\frac{3}{5}$.

Let $f\left(x\right)={\text{e}}^x\text{sin}x-<!--\; -\; -->\frac{3x}{4}$.

Consider the function $f\left(x\right)=x^2-\frac{3}{x}, x\ne 0$.

Line $L$ is a tangent to $f\left(x\right)$ at the point $(1, -2)$.

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

The function $f$ is defined by $f\left(x\right)=\frac{2}{x}+3x^2-3, x\ne 0$.

The diagram shows the curve $y=\frac{x^2}{2}+\frac{2a}{x}, x\ne 0$.

The equation of the vertical asymptote of the curve is $x=k$.

Let $f(x)=9-<!--\; -\; -->x^2$, $x\in <!--\; \in \; -->\mathbb{R}$.

The following diagram shows part of the graph of $f$.

Rectangle PQRS is drawn with P and Q on the $x$-axis and R and S on the graph of $f$.

Let OP = $b$.

Consider another function $g(x)={\left(x-<!--\; -\; -->3\right)}^2+k$,  $x\in <!--\; \in \; -->\mathbb{R}$.

Let $y={\left(x^3+x\right)}^{\frac{3}{2}}$.

Consider the functions $f\left(x\right)=\sqrt{x^3+x}$ and $g\left(x\right)=6-<!--\; -\; -->3x^2\sqrt{x^3+x}$, for $x$ ≥ 0.

The graphs of $f$ and $g$ are shown in the following diagram.

The shaded region $R$ is enclosed by the graphs of $f$, $g$, the $y$-axis and $x=1$.

Let $f\left(x\right)=6x^2-<!--\; -\; -->3x$. The graph of $f$ is shown in the following diagram.

The values of the functions $f$ and $g$ and their derivatives for $x=1$ and $x=8$ are shown in the following table.

Let $h(x)=f(x)g(x)$.

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

Let $g(x)=x^k$, where $k\in <!--\; \in \; -->{\mathbb{Z}}^{+}$.

Let $k=21$ and $h(x)=\left(f^{(19)}(x)\times <!--\; \times \; -->g^{(19)}(x)\right)$.

A particle P starts from point O and moves along a straight line. The graph of its velocity, $v$ ms⁻¹ after $t$ seconds, for 0 ≤ $t$ ≤ 6 , is shown in the following diagram.

The graph of $v$ has $t$-intercepts when $t$ = 0, 2 and 4.

The function $s\left(t\right)$ represents the displacement of P from O after $t$ seconds.

It is known that P travels a distance of 15 metres in the first 2 seconds. It is also known that $s\left(2\right)=s\left(5\right)$ and ${\int <!--\; \int \; -->}_2^{4}v\text{d}t=9$.

Consider a function $f$. The line L₁ with equation $y=3x+1$ is a tangent to the graph of $f$ when $x=2$

Let $g\left(x\right)=f\left(x^2+1\right)$ and P be the point on the graph of $g$ where $x=1$.

A closed cylindrical can with radius r centimetres and height h centimetres has a volume of 20$\mathrm{\pi <!--\; \pi \; -->}$ cm³.

The material for the base and top of the can costs 10 cents per cm² and the material for the curved side costs 8 cents per cm². The total cost of the material, in cents, is C.