IGCSE Mathematics 0580 Paper 43, May/June 2023

Tomas sells a computer, a bike and a phone. The amounts he receives are in the ratio computer : bike : phone = 14 : 17 : 9. Calculate the amount he receives for the phone as a percentage of the total.

The total amount he receives is $560. Calculate how much he receives for the bike.

Tomas originally bought the bike for $195. He wanted to make a profit of at least 25% when he sold it. Does Tomas make a profit of at least 25%? You must show all your working to support your decision.

Ulla invests $725 for 6 years in an account paying simple interest at a rate of 1.3% per year. Calculate the total interest earned at the end of 6 years.

In a sale, all prices are reduced by 24%. Victor pays $36.86 for a pair of shoes in the sale. Calculate the original price of the shoes.

Anna records the number of text messages she receives for 14 days. 17 15 31 38 31 22 13 18 21 27 28 21 31 29 Complete the stem-and-leaf diagram. Key: ...........................................................

Find the median.

Find the mode.

Find the range.

In a shop, there are 4 red and 8 grey phones. Anna and Pete each pick one of these phones at random. Work out the probability that they both pick a grey phone.

The scale drawing shows two sides, $AB$ and $BC$, of a field. The scale is 5 centimetres represents 200 metres. Measure angle $ABC$.

$X$ is a point on $BC$. $BX=332\text{ m}$. Mark the point $X$ on the diagram.

Find the scale in the form $1:n$.

A bronze statue is $4.5\text{ m}$ high and has a mass of $195200\text{ kg}$. The density of bronze is $8000{\text{ kg/m}}^3$. The volume of a mathematically similar model of the statue is $0.385{\text{ m}}^3$. Calculate the height of the model. [$\text{Density}=\text{Mass}÷\text{Volume}$]

$A$, $B$ and $C$ are three towns and the bearing of $C$ from $A$ is ${114}^{\circ }$. $B$ is due south of $A$ and $AC=BC$. Calculate the bearing of $B$ from $C$.

$P$, $Q$, $R$ and $S$ lie on a circle. $MPN$ is a tangent to the circle at $P$. Angle $MPS={58}^{\circ }$, angle $PSR={74}^{\circ }$ and angle $QPN={27}^{\circ }$. Find angle $PRS$.

Find angle $PQR$.

Find angle $RPQ$.

$A$, $B$ and $C$ lie on a circle, centre $O$, with diameter $AC$. $TAM$ and $TBN$ are tangents to the circle and angle $ATO={34}^{\circ }$. Using values and geometrical reasons, complete these statements to show that $CB$ is parallel to $OT$. In triangles $AOT$ and $BOT$, $OT$ is common. Angle $OAT=\text{angle }OBT={90}^{\circ }$ because .................................................................................... ..................................................................................................................................................... $AT=BT$ because ........................................................................................................................ ..................................................................................................................................................... Triangle $AOT$ is congruent to triangle $BOT$ because of congruence criterion ........................... Angle $AOT=\text{angle }BOT={56}^{\circ }$ because angles in a triangle add up to ${180}^{\circ }$. Angle $BOC=$ ...................... ${}^{\circ }$ because ...................................................................................... Angle $OBC=$ ...................... ${}^{\circ }$ because ...................................................................................... ..................................................................................................................................................... $CB$ is parallel to $OT$ because ......................................................................................................

$ABC$ is a scalene triangle on horizontal ground. $AYX$ is a straight vertical post, held in place by two straight wires $XB$ and $YC$. $AC=4.8\text{ m}$, $BC=5.6\text{ m}$ and angle $ACB={20.4}^{\circ }$. Calculate $AB$.

Angle $XBA={64}^{\circ }$. Calculate $AX$.

$AY=2.9\text{ m}$. Calculate the area of triangle $YAC$.

In triangle $PQR$, $M$ is the midpoint of $PQ$. $RM=8\text{ cm}$, angle $PRM={30}^{\circ }$ and angle $RMQ={75}^{\circ }$. Calculate $PQ$.

The cumulative frequency table shows information about the speed of each of 200 cars as they pass a speed camera.

On the grid, draw the cumulative frequency diagram.

Use your cumulative frequency diagram to find an estimate of the median

Use your cumulative frequency diagram to find an estimate of the interquartile range

Use your cumulative frequency diagram to find an estimate of the number of cars with a speed greater than $110\text{ km/h}$.

The frequency table shows information about the mass of each of 50 trucks.

Calculate an estimate for the mean mass of the trucks.

In a histogram showing this information, the height of the first block is $6\text{ cm}$. Calculate the heights of the remaining three blocks. Height of block for $2600<m⩽3500$ .......................................... cm Height of block for $3500<m⩽5000$ .......................................... cm Height of block for $5000<m⩽5700$ ........................................... cm

The diagram shows the graph of a function. Put a ring around the word which correctly identifies the type of function. reciprocal quadratic cubic exponential linear

On the diagram, sketch the graph of $y=\frac{1}{2x},x\mathrm{\ne }0$.

Solve the equation $\frac{1}{2x}=2x$.

On the diagram, sketch the graph of $y=\sin x$ for $0^{\circ }⩽x⩽{360}^{\circ }$.

Solve the equation $3\sin x+1=0$ for $0^{\circ }⩽x⩽{360}^{\circ }$.

A shop sells shirts for $\mathrm{\$}x$ and jackets for $\mathrm{\$}(x+27)$. The shop sells 4 shirts and 3 jackets for a total of $\mathrm{\$}194.75$. Write down and solve an equation to find the cost of one shirt.

Solve the simultaneous equations. You must show all your working. $x^2+4y=37$ $5x+y=-8$

A solid cylinder has radius $x$ and height $6x$. A sphere of radius $r$ has the same surface area as the total surface area of the cylinder. Show that $r^2=\frac{7}{2}{x}^2$. [The surface area, $A$, of a sphere with radius $r$ is $A=4\pi r^2$.]

The diagram shows a shape made from a square $ABCD$ and two equal sectors of a circle. The square has side $11\text{ cm}$. $MAB$ and $DCN$ are straight lines. Calculate the area of the shape.

Calculate the perimeter of the shape.

The diagram shows a cube $ABCDEFGH$ of edge $7\text{ cm}$. Calculate the angle between $AG$ and the base of the cube.

Complete the table.

On the grid, draw the graph of $y=2^x-3$ for $-2⩽x⩽2.5$.

Use your graph to solve the equation $2^x-3=2$.

By drawing a suitable straight line, solve the equation $2^x-x-1.5=0$.

Find the length of $MN$.

Find the gradient of $MN$.

Find the equation of the perpendicular bisector of $MN$.

Find the derivative, $\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)$, of $y=x^4-8x^2+5$.

Find the coordinates of the three turning points. You must show all your working.

Determine which one of these turning points is a maximum. Justify your answer.