O Level Mathematics 4024 Paper 22, May/June 2023

A shop buys some fruit. The table shows the bill for this fruit.

Find the value of each of $p$, $q$, $r$ and $s$.

The shop sells all this fruit for a total of $\mathrm{\$}325$. Calculate the percentage profit.

In 2022, the shop's total sales were $\mathrm{\$}34974$. A pie chart is drawn to show the item types that make up these total sales. The sales for fruit were $\mathrm{\$}9520.70$. Calculate the angle representing fruit on the pie chart.

The angle representing frozen food is ${46}^{\circ }$. Calculate the sales for frozen food.

The shop's total sales of $\mathrm{\$}34974$ in 2022 were a $4.4$ increase on the total sales in 2021. Calculate the total sales in 2021.

One chocolate bar costs $p$ cents and one packet of sweets costs $75$ cents. Tanish pays $\mathrm{\$}9.10$ for $5$ chocolate bars and $8$ packets of sweets. Form an equation and solve it to find the value of $p$. Show your working.

Factorise $6ac-27c$.

Write $\frac{3m^{2}n}{10}\times \frac{5n}{9m}$ as a single fraction in its simplest form.

Rearrange the formula $y=\frac{3x^2}{5}$ to make $x$ the subject.

A group of $k$ numbers has a mean of $56.8$. The number $52$ is added to the group. The new mean is $56.5$. Find the value of $k$.

The diagram shows the equilateral triangle $B$ and part of the regular polygon $A$ which have a common side. The interior angle of polygon $A$ is ${165}^{\circ }$. Find the value of $y$.

Calculate the number of sides of polygon $A$.

$A$, $B$ and $C$ are points on the circumference of a circle, centre $O$. Angle $ABC=x^{\circ }$. Show that angle $OAC=(90-x{)}^{\circ }$. Give reasons for your answer.

Angle $BAO={54}^{\circ }$ and angle $OCB={11}^{\circ }$. Find the value of $x$.

Complete the table of values for $y=\frac{2^x}{5}$.

Draw the graph of $y=\frac{2^x}{5}$ for $-1⩽x⩽4$.

By drawing a suitable line on the grid, solve $2^x=6$.

Complete the table of values for $4y=2x+1$.

On the grid on page 8, draw the graph of $4y=2x+1$ for $-1⩽x⩽4$.

Find the $x$-coordinates of the points where the line $4y=2x+1$ crosses the graph of $y=\frac{2^x}{5}$.

The $x$-coordinates in part (d)(iii) are the solutions of the equation $A\times 2^x+Bx+C=0$, where $A$, $B$ and $C$ are all integers. Use the equations $4y=2x+1$ and $y=\frac{2^x}{5}$ to find the exact value of each of $A$, $B$ and $C$.

The table shows the population and area of three countries in 2019.

Write down the value of the smallest population.

Find the difference in area between Sri Lanka and Pakistan. Give your answer in standard form.

The population density of a country is the number of people per square kilometre. Find the value of the largest population density from these countries.

In standard form, $A=8.6\times {10}^n$ and $B=1.5\times {10}^{n-1}$. Giving your answer in standard form, find in terms of $n$ $A-B$

Giving your answer in standard form, find in terms of $n$ $A\times B$.

Write these distances $\text{walk}:\text{cycle}:\text{swim}$ as a ratio in its simplest form.

Sophia walks at an average speed of $1.4\text{ m/s}$. She completes the walk at 11 05. Find the time she starts walking.

Sophia cycles a distance of $3000\text{ m}$ correct to the nearest 10 metres. She cycles this distance in a time of 450 seconds correct to the nearest 10 seconds. Calculate the upper bound for her average cycling speed in metres per second.

On Monday, the amount of money spent on a website by each customer was recorded. The table shows the results.

The histogram shows some of the results. Find the value of $p$.

Complete the histogram.

One of these customers is selected at random to receive a discount voucher. Calculate the probability that this customer spent more than $\mathrm{\$}50$ on Monday.

The table shows the amount of money spent on a website by each customer on Tuesday.

Calculate an estimate of the mean.

An error was made and one of the sales on Tuesday was not included in the table. That customer spent $\mathrm{\$}41$. Tristan says: If that value had been included in the table, then the estimated mean would have been higher. Without calculation, explain why he is correct.

The length of the rectangle $R$ is twice its width. Rectangle $R$ has a perimeter of $20.4\text{ cm}$. Find the length and width of the rectangle $R$.

Rectangle $S$ is mathematically similar to rectangle $R$. Rectangle $S$ has a perimeter of $30.6\text{ cm}$. Calculate the length of rectangle $S$.

A piece of card is a sector of a circle with sector angle ${75}^{\circ }$ and radius $8\text{ cm}$. Find an expression, in terms of $\pi$, for the arc length of the sector. Give your answer in its simplest form.

[Volume of a cone $=\frac{1}{3}\pi r^{2}h$] The piece of card forms the curved surface area of a cone. The cone is filled to the top with water. Calculate the volume of water in the cone.

Calculate $BC$.

Find the bearing of $A$ from $C$.

The point $D$ lies on the horizontal ground, due south of $A$ and due east of $C$. Show that $AD=86.1\text{ m}$, correct to 1 decimal place.

A point $X$ is at the top of a vertical mast at $A$. The angle of elevation of $X$ from $B$ is $7^{\circ }$. Calculate the angle of elevation of $X$ from $D$.

$F$ is the point $(6,1)$, $G$ is the point $(-2,4)$ and $\overrightarrow{GH}=\left(\begin{array}{c}-1\\ -8\end{array}\right)$. Calculate $\mathrm{\mid }\overrightarrow{FH}\mathrm{\mid }$.

$\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$ and $\overrightarrow{AC}=k\mathbf{b}$. $X$ is the point on $OC$ such that $\overrightarrow{OX}=m\overrightarrow{OC}$. Write $\overrightarrow{OX}$ in terms of $m$, $k$, $\mathbf{a}$ and $\mathbf{b}$.

$\overrightarrow{BX}=\frac{3}{5}\mathbf{a}-\frac{1}{2}\mathbf{b}$ Find the value of $k$.