Let f(x) = ln x − 5x , for x > 0 .

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 $x=b$.

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.

Contestants in a TV gameshow try to get through three walls by passing through doors without falling into a trap. Contestants choose doors at random.
If they avoid a trap they progress to the next wall.
If a contestant falls into a trap they exit the game before the next contestant plays.
Contestants are not allowed to watch each other attempt the game.

The first wall has four doors with a trap behind one door.

Ayako is a contestant.

Natsuko is the second contestant.

The second wall has five doors with a trap behind two of the doors.

The third wall has six doors with a trap behind three of the doors.

The following diagram shows the branches of a probability tree diagram for a contestant in the game.

All lengths in this question are in centimetres.

A solid metal ornament is in the shape of a right pyramid, with vertex $\text{V}$ and square base $\text{ABCD}$. The centre of the base is $\text{X}$. Point $\text{V}$ has coordinates $(1, 5, 0)$ and point $\text{A}$ has coordinates $(-1, 1, 6)$.

The volume of the pyramid is $57.2 {\text{cm}}^3$, correct to three significant figures.

160 students attend a dual language school in which the students are taught only in Spanish or taught only in English.

A survey was conducted in order to analyse the number of students studying Biology or Mathematics. The results are shown in the Venn diagram.

Set S represents those students who are taught in Spanish.

Set B represents those students who study Biology.

Set M represents those students who study Mathematics.

A student from the school is chosen at random.

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 $p>0$.

On one day 180 flights arrived at a particular airport. The distance travelled and the arrival status for each incoming flight was recorded. The flight was then classified as on time, slightly delayed, or heavily delayed.

The results are shown in the following table.

A χ² test is carried out at the 10 % significance level to determine whether the arrival status of incoming flights is independent of the distance travelled.

The critical value for this test is 7.779.

A flight is chosen at random from the 180 recorded flights.

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

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.

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.

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

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

Let $f\left(x\right)=12\text{cos}x-<!--\; -\; -->5\text{sin}x,-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->2\mathrm{\pi <!--\; \pi \; -->}$, be a periodic function with $f\left(x\right)=f\left(x+2\mathrm{\pi <!--\; \pi \; -->}\right)$

The following diagram shows the graph of $f$.

There is a maximum point at A. The minimum value of $f$ is −13 .

A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.

The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by

$d\left(t\right)=f\left(t\right)+17,0⩽<!--\; ⩽\; -->t⩽<!--\; ⩽\; -->5.$

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 $x=q$.

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.

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.

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

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 $k\in <!--\; \in \; -->{\mathbb{Z}}^{+}$.

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.

Let $f\left(x\right)=x^4-<!--\; -\; -->54x^2+60x$, for $-<!--\; -\; -->1⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->6$. The following diagram shows the graph of $f$.

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

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.

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 $v=\frac{\left(t^2+1\right)\cos t}{4}, 0\le t\le 3$.

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

A manufacturer makes trash cans in the form of a cylinder with a hemispherical top. The trash can has a height of 70 cm. The base radius of both the cylinder and the hemispherical top is 20 cm.

A designer is asked to produce a new trash can.

The new trash can will also be in the form of a cylinder with a hemispherical top.

This trash can will have a height of H cm and a base radius of r cm.

There is a design constraint such that H + 2r = 110 cm.

The designer has to maximize the volume of the trash can.

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.

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 $v_{\text{P}}$.

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

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 $p$ < $t$ < $q$.

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.

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 .

The displacement, in centimetres, of a particle from an origin, O, at time t seconds, is given by s(t) = t ² cos t + 2t sin t, 0 ≤ t ≤ 5.

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 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.

A particle moves along a straight line so that its velocity, $v {\text{m\hspace{0.05em}s}}^{-1}$, after $t$ seconds is given by $v\left(t\right)={\text{e}}^{\sin t}+4 \sin t$ for $0\le t\le 6$.

A particle moves in a straight line such that its velocity, $v {\mathrm{ms}}^{-1}$, at time $t$ seconds is given by

$v=4t^2-6t+9-2 \sin \left(4t\right), 0\le t\le 1$.

The particle’s acceleration is zero at $t=T$.

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 $3\text{ m}{\text{s}}^{-<!--\; -\; -->1}$.

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 $x>n$.

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.

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 $L$.

A particle moves in a straight line. The velocity, $v {\text{ms}}^{-1}$, of the particle at time $t$ seconds is given by $v\left(t\right)=t \sin t-3$, for $0\le t\le 10$.

The following diagram shows the graph of $v$.

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.

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.

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.

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

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

A group of 66 people went on holiday to Hawaii. During their stay, three trips were arranged: a boat trip ($B$), a coach trip ($C$) and a helicopter trip ($H$).

From this group of people:

One person in the group is selected at random.

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

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

Violeta plans to grow flowers in a rectangular plot. She places a fence to mark out the perimeter of the plot and uses 200 metres of fence. The length of the plot is $x$ metres.

Violeta places the fence so that the area of the plot is maximized.

By selling her flowers, Violeta earns 2 Bulgarian Levs (BGN) per square metre of the plot.

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).

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 $p<q$.

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.