O Level Mathematics 4024 Paper 21, May/June 2023

Show that the equation for the volume of the cuboid, $y{\text{ cm}}^3$, is $y=75x-15x^2$.

Complete the table of values for $y=75x-15x^2$.

Draw the graph of $y=75x-15x^2$ for $1⩽x⩽4$.

[$\text{Volume of pyramid}=\frac{1}{3}\times \text{base area}\times \text{height}$] The diagram shows a pyramid with a square base of side length $9\text{ cm}$. The pyramid has height $x\text{ cm}$ and volume $y{\text{ cm}}^3$. Show that the equation for the volume of the pyramid is $y=27x$.

By drawing a suitable straight line on the grid on page 2, find the height of the pyramid when the pyramid and the cuboid have the same volume.

Filomena starts work at 10.45 am on Monday. She finishes work 2 hours 50 minutes later. Find the time she finishes work on Monday.

Xavier works for 5 days each week. He works for $4\frac{1}{2}$ hours on each of the 5 days. Each week he earns $261. Calculate the hourly rate he is paid.

One day, the length of time Xavier works decreases by 20%. Calculate the length of time he works that day. Give your answer in hours and minutes.

In 2021, Miguel’s income was $32\,000. In 2022, his income increased to $33\,408. Calculate the percentage increase in his income from 2021 to 2022.

Miguel invests $\mathrm{\$}x$ in an account paying simple interest at a rate of 1.2% per year. At the end of 3 years, he has $890.96 in the account. Calculate the value of $x$.

$PQ$ is parallel to $RS$. $ABCD$ is a straight line. $BE=CE$ and $A\widehat{B}E={110}^{\circ }$. Calculate $E\widehat{C}Q$, giving a reason for each step of your working.

$U$, $V$, $W$, $X$ and $Y$ are points on the circumference of a circle, centre $O$. $UY$ is a diameter of the circle and $ZX$ is a tangent to the circle at $X$. $V\widehat{U}X={35}^{\circ }$, $X\widehat{Z}Y=a^{\circ }$ and $V\widehat{W}Y=b^{\circ }$. Find an expression for $b$ in terms of $a$. Give your answer in its simplest form.

Use the diagram to estimate the median

Use the diagram to estimate the interquartile range.

That day the price of a litre of fuel at the service station was $1.75 . Use the diagram to find the fraction of customers who spent more than $91.00 on fuel.

Complete the frequency table for the amount of fuel bought by these 100 customers.

$\mathcal{E}=\{1,2,3,4,5,6,7,8,9,10,11,12\}$ $P=\{x:x\text{ is a multiple of }3\}$ $Q=\{x:x\text{ is an odd number}\}$ $R=\{x:x\text{ is a factor of }24\}$ Complete the Venn diagram.

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

List the elements of $(P\cup R)\cap Q$.

Describe, in words, the type of number represented by $P\cap R\cap Q^{\mathrm{\prime }}$.

A number, $m$, is chosen at random from the elements of $R$. Find the probability that $m$ is a multiple of 3.

$M=2^{2x}\times 3^4\times 5\times 7$ $N=2^3\times 3^{x-y}\times 5^2$ The lowest common multiple (LCM) of $M$ and $N$ is $2^8\times 3^6\times 5^2\times 7$. Find the value of $x$ and the value of $y$.

Find the largest square number that is a factor of $M$.

Find the highest common factor (HCF) of $M$ and $N$. Give your answer as a product of its prime factors.

Simplify $3u-6w-5u+9w$.

Emilio buys $m$ pencils at 40 cents each and 12 pens at 85 cents each. He pays $20 and receives $2.20 change. Form an equation in $m$ and solve it to find the number of pencils Emilio buys. Show your working.

$y$ is directly proportional to the cube of $(x-2)$. When $y=12$, $x=4$. Find $y$ when $x=5$.

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

$ABC$ is a triangle with $AC=8.3\text{ cm}$ and $B\widehat{A}C={105}^{\circ }$. Construct triangle $ABC$. Line $AB$ has been drawn for you.

By taking suitable measurements from your triangle, calculate the perimeter of triangle $ABC$.

The diagram shows quadrilateral $PQRS$. $SQ=15\text{ cm}$, $S\widehat{P}Q={67}^{\circ }$ and $P\widehat{Q}S={74}^{\circ }$. Calculate $PS$.

$P\widehat{S}R={96}^{\circ }$ and the area of triangle $QRS$ is $63{\text{ cm}}^2$. Show that $SR=10.0\text{ cm}$, correct to 1 decimal place.

Calculate $QR$.

Maya leaves the office for a factory visit. The distance–time graph shows her journey from the office to the factory. Maya stays at the factory for $1\frac{1}{2}$ hours. She then returns to the office at an average speed of $35\text{ km/h}$. Complete the distance–time graph to show this information.

Use your graph to find the time Maya arrives back at the office.

The diagram shows the distance–time graph for the journey of a cyclist. The cyclist travels $d$ metres from home to a lake and then returns home. At 08 36 the cyclist is $(d-7200)$ metres from home. The average speed of the cyclist between 08 00 and 08 36 is $\frac{4}{5}$ of the average speed of the cyclist between 09 10 and 09 55. Calculate the value of $d$.

Complete matrix $\mathbf{M}$ to represent this information.

The cost of a flight for each adult is $\mathrm{\$}x$ and the cost of a flight for each child is $\mathrm{\$}y$. The matrix $\mathbf{N}=\left(\begin{array}{c}x\\ y\end{array}\right)$ shows this information. The matrix $\mathbf{P}=\mathbf{M}\mathbf{N}=\left(\begin{array}{c}525\\ 575\end{array}\right)$. Using an algebraic method, find the value of $x$ and the value of $y$. Show your working.

Explain what each element in $\mathbf{P}$ represents.

A box is opened and the number of sweets in each bag is counted. The results are shown in the table.

Explain why the total number of bags in the box cannot be 87.

The mean number of sweets per bag in this box is 12.8 . Find the value of $p$.

Another box is opened and the number of sweets in each bag is counted. The results are shown in the table.

A bag of sweets is chosen at random from this box and not replaced. A second bag of sweets is then chosen at random from the same box. The probability that both bags contain 15 sweets is $\frac{4}{539}$. Show that $r^2+175r-2046=0$.

Solve $r^2+175r-2046=0$ to find the value of $r$. Show your working.

Calculate angle $DCB$.

Khalil pours water into the empty container at a rate of $4000{\text{ cm}}^3\text{/minute}$ for 2 minutes. He says that the container is now more than two thirds full. Is he correct? Show your working.

Calculate angle $DFH$.