IGCSE Mathematics 0580 Paper 42, May/June 2023

The diagram shows an isosceles triangle with the base extended. Find the value of $x$.

The diagram shows three lines meeting at a point. The ratio $a:b:c=3:4:5$. Find the value of $c$.

A regular pentagon has an exterior angle, $d$. A regular hexagon has an interior angle, $h$. Find the fraction $\frac{d}{h}$. Give your answer in its simplest form.

Show that $PQRS$ is a cyclic quadrilateral.

The diagram shows a circle of radius $9\text{ cm}$, centre $O$. The minor sector $AOB$, with sector angle ${50}^{\circ }$, is removed from the circle. Calculate the length of the major arc $AB$.

Anil changes $\mathrm{\$}830$ into euros when the exchange rate is $1euro=\mathrm{\$}1.16$. He spends $500\text{ euros}$. He then changes the remaining money back into dollars at the same exchange rate. Work out how much, in dollars, Anil receives.

In 2021, Anil earns $\mathrm{\$}37000$. He spends $\mathrm{\$}12400$ on bills in 2021. Calculate the percentage of his earnings he spends on bills.

His earnings of $\mathrm{\$}37000$ increase by $3.2$ in 2022. Calculate his earnings in 2022.

Anil invests $\mathrm{\$}3500$ in an account that pays a rate of $2.4$ per year compound interest. Calculate the total interest earned at the end of 5 years.

Find the number of complete years before Anil has at least $\mathrm{\$}5000$ in this account.

The area of the triangle is $60{\text{ cm}}^2$. Show that $2x^2+11x-105=0$.

Solve by factorisation. $$2x^2+11x-105=0$$

Calculate angle $ACB$.

Triangle $ABC$ is similar to triangle $DEF$. Triangle $DEF$ has an area of $93.75{\text{ cm}}^2$. Find the size of the smallest angle of triangle $DEF$.

Find the length of the shortest side of triangle $DEF$.

Write down the interval containing the median.

Calculate an estimate of the mean height.

One of these children is chosen at random. Calculate the probability that they have a height of $1.4\text{ m}$ or less.

Two of these children are chosen at random. Calculate the probability that both children are taller than $1.5\text{ m}$ but only one of them is taller than $1.8\text{ m}$.

Complete the cumulative frequency table for the heights.

On the grid, draw the cumulative frequency diagram.

Use your diagram to find an estimate of the interquartile range

the 60th percentile.

A cone has base diameter $8\text{ cm}$ and perpendicular height $15\text{ cm}$. Calculate the volume of the cone. [The volume, $V$, of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\pi r^{2}h$.]

A label completely covers the curved surface area of the cone. Calculate the area of the label as a percentage of the total surface area of the cone. [The curved surface area, $A$, of a cone with radius $r$ and slant height $l$ is $A=\pi rl$.]

An empty cylindrical container has radius $0.45\text{ m}$. $300\text{ litres}$ of water is poured into the container at a rate of $375\text{ ml}$ per second. Find the time taken, in minutes and seconds, for all the water to be poured into the container.

Calculate the height of the water in the container.

A sequence has $n\text{th}$ term $\frac{n}{2n+3}$. Find the first three terms of this sequence. Give your answers as fractions.

The $k\text{th}$ term of this sequence is $\frac{12}{25}$. Find the value of $k$.

Find the $n\text{th}$ term of each sequence. $6,13,32,69,130,\dots$

$100,50,25,12.5,6.25,\dots$

Show that angle $ACB={95.1}^{\circ }$, correct to 1 decimal place.

Without stopping, a car travels from town $A$ to town $C$ then to town $B$, before returning directly to town $A$. The total time taken for the journey is 3 hours 20 minutes. Calculate the average speed of the car for this journey. Give your answer in kilometres per hour.

Show that the equation $y=(x-4)(x+1)(x-2)$ can be written as $y=x^3-5x^2+2x+8$.

On the diagram, sketch the graph of $y=x^3-5x^2+2x+8$, indicating the values where the graph crosses the axes.

The graph of $y=x^3-5x^2+2x+8$ has two tangents with a gradient of 10. Find the equations of these two tangents. You must show all your working and give your answers in the form $y=mx+c$.

Simplify. $(3x^2{y}^4{)}^3$

$${\left(\frac{16}{x^{16}{y}^8}\right)}^{-\frac{3}{2}}$$

Factorise. $$x^2-9$$

Simplify. $$\frac{x^2-9}{2xy-6y+5x-15}$$

Solve the simultaneous equations. You must show all your working and give your answers correct to 2 decimal places. $$2x+y=7$$ $$y=5x^2+2x-13$$

$ABCDEFGH$ is a cuboid with a square base of side $x\text{ cm}$. $CG=20\text{ cm}$ and $AG=28\text{ cm}$. Calculate the value of $x$.

The diagram shows a different cuboid $JKLMNPQR$. $MR=30\text{ cm}$ correct to the nearest centimetre. $KR=37\text{ cm}$ correct to the nearest centimetre. Calculate the lower bound of the angle between $KR$ and the base $JKLM$ of the cuboid.