IGCSE Mathematics 0580 Paper 41, May/June 2023

An orchard has 1250 trees. The trees are in the ratio $\text{apple}:\text{pear}:\text{cherry}=12:9:4$. Calculate the number of apple trees.

Last year in the orchard, the mean mass of fruit produced was $64\text{ kg}$ per tree. Calculate the total mass of fruit produced last year. Give your answer in tonnes. [$1\text{ tonne}=1000\text{ kg}$]

Last year, the mean mass of pears produced was $54\text{ kg}$ per tree. This was a decrease of $10\mathrm{\%}$ on the mean mass of pears produced per tree from the year before. Calculate the mean mass of pears produced by each pear tree the year before.

The orchard loses $\frac{1}{5}$ of its total number of trees in a storm. Calculate the number of trees that remain.

Paulo buys some pears from a market. Pears cost $\mathrm{\$}0.54$ each or $0.51euros$ each. Paulo pays in dollars for 12 pears. Calculate the change he receives from $\mathrm{\$}10$.

The exchange rate is $\mathrm{\$}1=0.826euros$. Calculate how much more Paulo pays for each pear when he pays in euros. Give your answer in dollars, correct to the nearest cent.

Show that angle $EBA={32}^{\circ }$, correct to the nearest degree.

Find the length of $EA$.

Calculate the shortest distance from $E$ to $AB$.

Calculate the angle $BF$ makes with the base, $ABCD$, of the prism.

The prism is made of plastic with density $938{\text{ kg/m}}^3$. Calculate the mass of the prism in grams. [$\text{Density}=\text{mass}÷\text{volume}$]

The table shows information about the mass of each of 1000 eggs.

Calculate an estimate of the mean.

An egg is picked at random from the 1000 eggs. Find the probability that this egg has a mass greater than $56\text{ g}$. Give your answer as a fraction in its simplest form.

One year, a farmer makes a profit of $\mathrm{\$}24730$ selling eggs. Write this profit correct to 2 significant figures

Write this profit in standard form.

On a farm, there are 500 hens, correct to the nearest 10. In one year, the mean number of eggs laid per hen was 320 eggs, correct to the nearest 20. Calculate the upper bound for the total number of eggs all the hens lay in that year.

Another farm has 800 hens, correct to the nearest 20. Calculate the lower bound for the difference between the number of hens on the two farms.

Calculate the area of triangle $PQR$.

Calculate angle $PRQ$.

The diagram shows a half-cylinder of radius $6\text{ cm}$ and length $11\text{ cm}$. Calculate the volume of the half-cylinder.

$ABCD$ is a rectangle with $AB=20\text{ cm}$ and $BC=15\text{ cm}$. $S$, $X$ and $T$ are points on a circle centre $O$, such that $DSA$ and $DTC$ are tangents to the circle. The radius of the circle is $4\text{ cm}$ and $TX$ is a diameter of the circle. The shape $DSXT$ is removed from the corner of the rectangle, leaving the shaded shape shown in the second diagram. Calculate the area of the shaded shape.

Calculate the perimeter of the shaded shape.

There are 160 people in a village. The cumulative frequency diagram shows information about their ages. Find an estimate for the median age

Find an estimate for the lower quartile

Find an estimate for the number of people who are 50 or more years of age

Find an estimate for the 65th percentile.

The youngest person in the village is 1 year old and the oldest is 70 years old. Draw a box-and-whisker plot to show the distribution of ages in the village.

Write down an estimate of the percentage of people in the village that are younger than the median age.

The frequency table shows information about the age of each person in another village.

On the grid, complete the histogram to show this information. The first block has been drawn for you.

In the square $ABCD$, $A$ has coordinates $(-2,1)$ and $B$ has coordinates $(1,5)$. $C$ has coordinates $(a,b)$, where $a$ and $b$ are both positive integers. Find the coordinates of $C$ and the coordinates of $D$. You may use the grid to help you.

$P$ has coordinates $(-1,3)$ and $Q$ has coordinates $(6,4)$. Find the coordinates of the midpoint of $PQ$.

Find the length $PQ$.

Find the gradient of $PQ$.

Find the equation of the line parallel to $PQ$ that crosses the $x$-axis at $x=2$.

Factorise fully. $27y^2-3$

Factorise fully. $2m-pk+2k-pm$

Solve $\frac{x-1}{x+1}-\frac{6}{x-1}=1$.

Solve $4x^2-3x-2=0$. You must show all your working and give your answers correct to 2 decimal places.

Make $k$ the subject of the formula. $$\frac{k}{m}=4+kp$$

Write down the other three inequalities in $x$ and/or $y$.

On the grid, draw 4 straight lines and shade the unwanted regions to show these inequalities. Label the region $R$ that satisfies the 4 inequalities.

Use your diagram to find the smallest number of dresses and the smallest number of shirts the tailor makes in one week.

The profit the tailor makes on one dress is $\mathrm{\$}10$ and the profit on one shirt is $\mathrm{\$}6$. Use your diagram to find the largest profit the tailor can make in one week.

The Venn diagram shows set $X$ and set $Y$. List the elements of $X$.

Find $\mathrm{n}(Y^{\mathrm{\prime }})$.

In each Venn diagram, shade the required region.

$\mathcal{E}=\{\text{positive integers }<13\}$ $A=\{x:x<9\}$ $B=\{x:x\text{ is even}\}$ $C=\{x:x\text{ is a multiple of }3\}$ Complete the Venn diagram.

Find $\mathrm{n}(A^{\mathrm{\prime }}\cup (B\cap C))$.

Find $\mathrm{f}(-3)$

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

Find $\mathrm{f}(x)\times \mathrm{g}(x)\times \mathrm{f}(x)$.

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