IGCSE Mathematics 0580 Paper 41, May/June 2024

The table shows the areas, in ${\mathrm{k}\mathrm{m}}^2$, of the four largest rainforests in the world.

Find the area of the Valdivian rainforest as a percentage of the area of the Amazon rainforest.

Write, in its simplest form, the ratio of the areas of the rainforests Valdivian : Atlantic : Congo.

The Amazon rainforest has 60% of its area in Brazil and 10% of its area in Colombia. $43\frac{1}{3}\mathrm{\%}$ of the remaining area of the rainforest is in Peru. Find the percentage of the Amazon rainforest that is in Brazil, Colombia and Peru.

The area of the Amazon rainforest represents $\frac{27}{50}$ of the total area of rainforest in the world. Calculate the total area of rainforest in the world. Give your answer correct to the nearest $100000{\mathrm{k}\mathrm{m}}^2$.

In the world, 60.7 hectares of rainforest are lost every minute. Calculate the total area, in hectares, of rainforest that is lost in 365 days. Give your answer in standard form.

The Amazon river has a length of $6440\mathrm{k}\mathrm{m}$, correct to the nearest $10\mathrm{k}\mathrm{m}$. The Congo river has a length of $4400\mathrm{k}\mathrm{m}$, correct to the nearest $100\mathrm{k}\mathrm{m}$. Calculate the upper bound of the difference between the lengths of the Amazon river and the Congo river.

On the grid, draw the image of triangle $T$ after a reflection in the $x$-axis

triangle $T$ after a translation by the vector $\left(\begin{array}{c}-5\\ -2\end{array}\right)$

triangle $T$ after an enlargement by scale factor $-\frac{1}{2}$ with centre $(-1,1)$.

A shape $P$ is enlarged by scale factor 3 to give shape $Q$. Shape $Q$ is then enlarged by scale factor $\frac{2}{5}$ to give shape $R$. The area of shape $P$ is $10{\mathrm{c}\mathrm{m}}^2$. Calculate the area of shape $R$.

$C=\frac{1}{4}x{y}^2$ Find $C$ when $x=5$ and $y=8$.

Find the positive value of $y$ when $C=15$ and $x=2.4$.

Write as a single fraction in its simplest form. $$\frac{4}{x-1}-\frac{3}{2x+5}$$

Expand and simplify. $$(2x+3)(4-x{)}^2$$

Simplify. $${\left(\frac{y^8}{16x^{16}}\right)}^{-\frac{3}{4}}$$

Jianyu records the time, in seconds, that some cars take to travel $195\mathrm{m}$. The box and whisker plot shows this information. Find the median time.

Find the interquartile range.

Find the difference between the average speed of the fastest car and the average speed of the slowest car. Give your answer in kilometres per hour.

Matilda records the distances that 80 different cars can travel with a full tank of fuel. The table shows this information.

Write down the class interval that contains the median.

Calculate an estimate of the mean.

A histogram is drawn to show the information in the table. The height of the bar for the interval $250<d⩽300$ is $2.8\mathrm{c}\mathrm{m}$. Calculate the height of the bar for each of the following intervals. $300<d⩽400$ $400<d⩽420$ $420<d⩽450$

Two of the 80 cars are chosen at random. Find the probability that, with a full tank of fuel, one of the cars can travel more than $450\mathrm{k}\mathrm{m}$ and the other car can travel not more than $300\mathrm{k}\mathrm{m}$.

Find $\overrightarrow{PQ}$.

Show that $\mathrm{\mid }\overrightarrow{OP}\mathrm{\mid }=\mathrm{\mid }\overrightarrow{OQ}\mathrm{\mid }$.

$PQ$ is a chord of a circle with centre $O$. Calculate the circumference of this circle.

$PQ$ is the diameter of a different circle with centre $R$. Find the coordinates of $R$.

Find the equation of the perpendicular bisector of $PQ$. Give your answer in the form $y=mx+c$.

The position vector of $A$ is $\mathbf{a}$. The position vector of $B$ is $\mathbf{b}$. $M$ is a point on $AB$ such that $AM:MB=2:3$. Find, in terms of $\mathbf{a}$ and $\mathbf{b}$, the position vector of $M$. Give your answer in its simplest form.

Find the bearing of the harbour from boat $C$.

Show that angle $CBH={100}^{\circ }$.

Show that $BH=13.7\mathrm{k}\mathrm{m}$, correct to 1 decimal place.

Calculate the bearing of $A$ from $B$.

At 1 pm boat $C$ sails $32\mathrm{k}\mathrm{m}$ directly to the harbour at a speed of 10 knots. Calculate the time when boat $C$ arrives at the harbour. Give this time correct to the nearest minute. $[1\text{ knot}=1.852\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}]$

Calculate the distance of boat $C$ to the harbour when boat $C$ is at the shortest distance from lighthouse $B$.

The diagram shows a box in the shape of a cuboid. The box is open at the top. Work out the surface area of the inside of the open box.

Cylinders with height $20\mathrm{c}\mathrm{m}$ and diameter $15\mathrm{c}\mathrm{m}$ are placed in the box. Work out the maximum number of these cylinders that can completely fit inside the box.

A solid bronze cone has a mass $750\mathrm{g}$. The density of the bronze is $8.9{\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}}^3$. The ratio $\text{radius of cone}:\text{height of cone}=1:3$. Show that the radius of the cone is $2.99\mathrm{c}\mathrm{m}$, correct to 3 significant figures. $[\text{Density}=\text{mass}÷\text{volume}]$ $[\text{The volume, }V\text{, of a cone with radius }r\text{ and height }h\text{ is }V=\frac{1}{3}\pi r^{2}h\mathrm{.}]$

Calculate the total surface area of the cone. $[\text{The curved surface area, }A\text{, of a cone with radius }r\text{ and slant height }l\text{ is }A=\pi rl\mathrm{.}]$

On the axes, sketch the graph of $y=x^2+7x-18$. On your sketch, write the values where the graph meets the $x$-axis and the $y$-axis.

Find the derivative of $y=x^2-3x-28$.

Find the coordinates of the turning point of $y=x^2-3x-28$.

The line $y=5-2x$ intersects the graph of $y=x^2-3x-28$ at point $P$ and point $Q$. Find the coordinates of $P$ and $Q$. You must show all your working and give your answers correct to 2 decimal places.

Find $\mathrm{f}(3)$

$\mathrm{g}\mathrm{f}(3)$.

Find ${\mathrm{g}}^{-1}(x)$.

Find $x$ when $\mathrm{f}(x)=\mathrm{g}(2x-7)$.

Find the value of $\mathrm{h}\mathrm{h}(2)$.

Find $x$ when ${\mathrm{h}}^{-1}(x)=10$.