The rate of change of the height $\left(h\right)$ of a ball above horizontal ground, measured in metres, $t$ seconds after it has been thrown and until it hits the ground, can be modelled by the equation

$\frac{dh}{dt}=11.4-9.8t$

The height of the ball when $t=0$ is $1.2 \text{m}$.

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 $2000\text{ c}{\text{m}}^2$.

The cross-sectional view of a tunnel is shown on the axes below. The line $\left[\text{AB}\right]$ represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by $y=-0.1x^3+ 0.8x^2, 2\le x\le 8$, relative to an origin $\text{O}$.

Point $\text{A}$ has coordinates $(2, 0)$, point $\text{B}$ has coordinates $(2, 2.4)$, and point $\text{C}$ has coordinates $(8, 0)$.

When $x=4$ the height of the tunnel is $6.4 \text{m}$ and when $x=6$ the height of the tunnel is $7.2 \text{m}$. These points are shown as $\text{D}$ and $\text{E}$ on the diagram, respectively.

Hyungmin designs a concrete bird bath. The bird bath is supported by a pedestal. This is shown in the diagram.

The interior of the bird bath is in the shape of a cone with radius $r$, height $h$ and a constant slant height of $50 \text{cm}$.

Let $V$ be the volume of the bird bath.

Hyungmin wants the bird bath to have maximum volume.

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 $f$.

In a company it is found that 25 % of the employees encountered traffic on their way to work. From those who encountered traffic the probability of being late for work is 80 %.

From those who did not encounter traffic, the probability of being late for work is 15 %.

The tree diagram illustrates the information.

The company investigates the different means of transport used by their employees in the past year to travel to work. It was found that the three most common means of transport used to travel to work were public transportation (P ), car (C ) and bicycle (B ).

The company finds that 20 employees travelled by car, 28 travelled by bicycle and 19 travelled by public transportation in the last year.

Some of the information is shown in the Venn diagram.

There are 54 employees in the company.

Sila High School has 110 students. They each take exactly one language class from a choice of English, Spanish or Chinese. The following table shows the number of female and male students in the three different language classes.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test was carried out at the 5 % significance level to analyse the relationship between gender and student choice of language class.

Use your graphic display calculator to write down

The critical value at the 5 % significance level for this test is 5.99.

One student is chosen at random from this school.

Another student is chosen at random from this school.

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

The following diagram shows the graph of $f(x)=a\sin <!--\; \; -->bx+c$, for $0⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->12$.

The graph of $f$ has a minimum point at $(3,5)$ and a maximum point at $(9,17)$.

The graph of $g$ is obtained from the graph of $f$ by a translation of $\left(\begin{array}{c}k\\ 0\end{array}\right)$. The maximum point on the graph of $g$ has coordinates $(11.5,17)$.

The graph of $g$ changes from concave-up to concave-down when $x=w$.

A hollow chocolate box is manufactured in the form of a right prism with a regular hexagonal base. The height of the prism is $h \text{cm}$, and the top and base of the prism have sides of length $x \text{cm}$.

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

Consider a function $f\left(x\right)$, for $x\ge 0$. The derivative of $f$ is given by $f\text{'}\left(x\right)=\frac{6x}{x^2+4}$.

The graph of $f$ is concave-down when $x>n$.

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 company performs an experiment on the efficiency of a liquid that is used to detect a nut allergy.

A group of 60 people took part in the experiment. In this group 26 are allergic to nuts. One person from the group is chosen at random.

A second person is chosen from the group.

When the liquid is added to a person’s blood sample, it is expected to turn blue if the person is allergic to nuts and to turn red if the person is not allergic to nuts.

The company claims that the probability that the test result is correct is 98% for people who are allergic to nuts and 95% for people who are not allergic to nuts.

It is known that 6 in every 1000 adults are allergic to nuts.

This information can be represented in a tree diagram.

An adult, who was not part of the original group of 60, is chosen at random and tested using this liquid.

The liquid is used in an office to identify employees who might be allergic to nuts. The liquid turned blue for 38 employees.

All lengths in this question are in metres.

Let $f(x)=-<!--\; -\; -->0.8x^2+0.5$, for $-<!--\; -\; -->0.5⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->0.5$. Mark uses $f(x)$ as a model to create a barrel. The region enclosed by the graph of $f$, the $x$-axis, the line $x=-<!--\; -\; -->0.5$ and the line $x=0.5$ is rotated 360° about the $x$-axis. This is shown in the following diagram.

A box of chocolates is to have a ribbon tied around it as shown in the diagram below.

The box is in the shape of a cuboid with a height of $3$ cm. The length and width of the box are $x$ and $y$ cm.

After going around the box an extra $10$ cm of ribbon is needed to form the bow.

The volume of the box is $450{\text{\hspace{0.05em}cm}}^3$.

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

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.

A theatre set designer is designing a piece of flat scenery in the shape of a hill. The scenery is formed by a curve between two vertical edges of unequal height. One edge is $2$ metres high and the other is $1$ metre high. The width of the scenery is $6$ metres.

A coordinate system is formed with the origin at the foot of the $2$ metres high edge. In this coordinate system the highest point of the cross‐section is at $\left(2, 3.5\right)$.

A set designer wishes to work out an approximate value for the area of the scenery $(A {\text{m}}^2 )$.

In order to obtain a more accurate measure for the area the designer decides to model the curved edge with the polynomial $h\left(x\right)=a{x}^3+b{x}^2+cx+d a, b, c, d\in \mathrm{ℝ}$ where $h$ metres is the height of the curved edge a horizontal distance $x \text{m}$ from the origin.

The Happy Straw Company manufactures drinking straws.

The straws are packaged in small closed rectangular boxes, each with length 8 cm, width 4 cm and height 3 cm. The information is shown in the diagram.

Each week, the Happy Straw Company sells $x$ boxes of straws. It is known that $\frac{\text{d}P}{\text{d}x}=-<!--\; -\; -->2x+220$, $x$ ≥ 0, where $P$ is the weekly profit, in dollars, from the sale of $x$ thousand boxes.

A cafe makes $x$ litres of coffee each morning. The cafe’s profit each morning, $C$, measured in dollars, is modelled by the following equation

$C=\frac{x}{10}\left(k^2-\frac{3}{100}{x}^2\right)$

where $k$ is a positive constant.

The cafe’s manager knows that the cafe makes a profit of $\$426$ when $20$ litres of coffee are made in a morning.

The manager of the cafe wishes to serve as many customers as possible.

A sector of a circle, centre $\text{O}$ and radius $4.5 \text{m}$, is shown in the following diagram.

A square field with side $8 \text{m}$ has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to $4.5 \text{m}$ from the post.

^{[Source: mynamepong, n.d. Goat [image online] Available at: https://thenounproject.com/term/goat/1761571/
This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)
https://creativecommons.org/licenses/by-sa/3.0/deed.en [Accessed 22 April 2010] Source adapted.]}

Let $V$ be the volume of grass eaten by the goat, in cubic metres, and $t$ be the length of time, in hours, that the goat has been in the field.

The goat eats grass at the rate of $\frac{dV}{dt}=0.3 t{\text{e}}^{-t}$.

All lengths in this question are in metres.

Consider the function $f\left(x\right)=\sqrt{\frac{4-<!--\; -\; -->x^2}{8}}$, for −2 ≤ $x$ ≤ 2. In the following diagram, the shaded region is enclosed by the graph of $f$ and the $x$-axis.

A container can be modelled by rotating this region by 360˚ about the $x$-axis.

Water can flow in and out of the container.

The volume of water in the container is given by the function $g\left(t\right)$, for 0 ≤ $t$ ≤ 4 , where $t$ is measured in hours and $g\left(t\right)$ is measured in m³. The rate of change of the volume of water in the container is given by $g^{\prime }\left(t\right)=0.9-<!--\; -\; -->2.5\text{cos}\left(0.4t^2\right)$.

The volume of water in the container is increasing only when $p$ < $t$ < $q$.

Consider the function $g(x)=x^3+k{x}^2-<!--\; -\; -->15x+5$.

The tangent to the graph of $y=g(x)$ at $x=2$ is parallel to the line $y=21x+7$.

Let $f(x)=-<!--\; -\; -->0.5x^4+3x^2+2x$. The following diagram shows part of the graph of $f$.

There are $x$-intercepts at $x=0$ and at $x=p$. There is a maximum at A where $x=a$, and a point of inflexion at B where $x=b$.

The Maxwell Ohm Company is designing a portable Bluetooth speaker. The speaker is in the shape of a cylinder with a hemisphere at each end of the cylinder.

The dimensions of the speaker, in centimetres, are illustrated in the following diagram where $r$ is the radius of the hemisphere, and $l$ is the length of the cylinder, with $r>0$ and $l\ge 0$.

The Maxwell Ohm Company has decided that the speaker will have a surface area of $300 {\text{cm}}^2$.

The quality of sound from the speaker will improve as $V$ increases.

The graph of the quadratic function $f\left(x\right)=\frac{1}{2}\left(x-2\right)\left(x+8\right)$ intersects the $y$-axis at $\left(0, c\right)$.

The vertex of the function is $\left(-3, -12.5\right)$.

The equation $f\left(x\right)=12$ has two solutions. The first solution is $x=-10$.

Let $T$ be the tangent at $x=-3$.

Let $f\left(x\right)=\frac{16}{x}$. The line $L$ is tangent to the graph of $f$ at $x=8$.

$L$ can be expressed in the form r $=\left(\begin{array}{c}8\\ 2\end{array}\right)+t$u.

The direction vector of $y=x$ is $\left(\begin{array}{c}1\\ 1\end{array}\right)$.

Let $f\left(x\right)=4-<!--\; -\; -->2{\text{e}}^x$. The following diagram shows part of the graph of $f$.

Let $f\left(x\right)=\text{sin}\left(e^x\right)$ for 0 ≤ $x$ ≤ 1.5. The following diagram shows the graph of $f$.

Let g(x) = −(x − 1)² + 5.

Let f(x) = x². The following diagram shows part of the graph of f.

The graph of g intersects the graph of f at x = −1 and x = 2.

Haruka has an eco-friendly bag in the shape of a cuboid with width 12 cm, length 36 cm and height of 9 cm. The bag is made from five rectangular pieces of cloth and is open at the top.

Nanako decides to make her own eco-friendly bag in the shape of a cuboid such that the surface area is minimized.

The width of Nanako’s bag is x cm, its length is three times its width and its height is y cm.

The volume of Nanako’s bag is 3888 cm³.

Consider the function $f\left(x\right)=x^2{\text{e}}^{3x}$,  $x\in <!--\; \in \; -->\mathbb{R}$.

Let f(x) = ln x − 5x , for x > 0 .

In this question distance is in centimetres and time is in seconds.

Particle A is moving along a straight line such that its displacement from a point P, after $t$ seconds, is given by $s_{\text{A}}=15-<!--\; -\; -->t-<!--\; -\; -->6t^3{\text{e}}^{-<!--\; -\; -->0.8t}$, 0 ≤ $t$ ≤ 25. This is shown in the following diagram.

Another particle, B, moves along the same line, starting at the same time as particle A. The velocity of particle B is given by $v_{\text{B}}=8-<!--\; -\; -->2t$, 0 ≤ $t$ ≤ 25.

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a=3t^2-<!--\; -\; -->14t+8$, for $0⩽<!--\; ⩽\; -->t⩽<!--\; ⩽\; -->5$.

When $t=0$, the velocity of P is $3\text{ m}{\text{s}}^{-<!--\; -\; -->1}$.

A particle P moves along a straight line. Its velocity $v_{\text{P}}\text{ m}{\text{s}}^{-<!--\; -\; -->1}$ after $t$ seconds is given by $v_{\text{P}}=\sqrt{t}\sin <!--\; \; -->\left(\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}t\right)$, for $0⩽<!--\; ⩽\; -->t⩽<!--\; ⩽\; -->8$. The following diagram shows the graph of $v_{\text{P}}$.

A particle P starts from a point A and moves along a horizontal straight line. Its velocity $v\text{ cm}{\text{s}}^{-<!--\; -\; -->1}$ after $t$ seconds is given by

The following diagram shows the graph of $v$.

P is at rest when $t=1$ and $t=p$.

When $t=q$, the acceleration of P is zero.

A particle moves along a straight line so that its velocity, $v$ m s⁻¹, after $t$ seconds is given by $v\left(t\right)={1.4}^t-<!--\; -\; -->2.7$, for 0 ≤ $t$ ≤ 5.

The function $f\left(x\right)=\frac{1}{3}{x}^3+\frac{1}{2}{x}^2+kx+5$ has a local maximum and a local minimum. The local maximum is at $x=-<!--\; -\; -->3$.

Let $f^{″}\left(x\right)=\left(\text{cos}2x\right)\left(\text{sin}6x\right)$, for 0 ≤ $x$ ≤ 1.

The population of fish in a lake is modelled by the function

$f\left(t\right)=\frac{1000}{1+24{\text{e}}^{-<!--\; -\; -->0.2t}}$, 0 ≤ $t$ ≤ 30 , where $t$ is measured in months.

A particle P moves along a straight line. The velocity v m s⁻¹ of P after t seconds is given by v (t) = 7 cos t − 5t ^{cos t}, for 0 ≤ t ≤ 7.

The following diagram shows the graph of v.

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 $p>0$.