Question1
(a) The cost of a cinema ticket is \( \$6.30\) .
(i) Work out the total cost of $4$ tickets.
(ii) The cinema sells 10 tickets for the price of $9$ tickets. A group of $10$ people go to the cinema.
Work out how much each person must pay
(b) The cinema has $650$ seats.
(i) There are $26$ seats in each row.
Work out the number of rows.
(ii) $84\%$ of the seats are occupied
Work out how many seats are not occupied
(c) Suki buys a bag of popcorn and a bottle of water.
A bag of popcorn costs twice as much as a bottle of water.
She receives \( \$3.55\) change from \( \$10\).
Work out the cost of a bag of popcorn and the cost of a bottle of water.

▶️Answer/Explanation
(a)(i): $25.20$
(a)(ii): $5.67$
(b)(i): $25$
(b)(ii): $104$
(c): $4.30$, $2.15$ (and later corrected to $303.60$)
(d)(i): $2$ hours $35$ minutes
(d)(ii): $1705$
Question2
(a) Chen asks some people if they prefer a beach, cruise, lake or mountain holiday.
The pie chart shows the results.
(i) Find the fraction of people who prefer a mountain holiday.
Give your answer in its simplest form.
(ii) Find the percentage of people who prefer a beach holiday.
(iii) Find the ratio of people who prefer each type of holiday in the form
$\text{beach:cruise: lake: mountain.}$
Give your answer in its simplest form.
(iv) One person is chosen at random
Find the probability that this person prefers a cruise or a lake holiday
(v) $675$ people prefer a beach holiday.
Show that the total number of people Chen asks is $1800$
(vi) (a) Complete the table.
(b) Complete the bar chart, including the scale on the frequency axis.
\(\begin{aligned}\mathrm{(b)}&\text{Mr Gibb pays S2208 for a holiday.}\\&\text{Mr Shah pays 2050 euros for the same holiday.}\\&\text{The exchange rate is 1 euro=S1.15.}\\&\text{Work out how much more, in curos, Mr Shah pays for the holiday than Mr Gibb.}\end{aligned}\)
▶️Answer/Explanation
(a)(i): $\frac{1}{3}$
(a)(ii): $37.5$
(a)(iii): $9:2:5:8$
(a)(iv): $\frac{105}{360}$
(a)(v): $\frac{675 \times 360}{135} = 1800$
(a)(vi)(a): $375$, $600$
(a)(vi)(b): A correct bar chart drawn with linear scale labelled.
(b): $130$
Question3
(a) Here is part of the timetable for trains from Hinton to Jarmouth.
All trains take the same time to travel from Hinton to Jarmouth.
(i) Complete the timetable.
(ii) Marge arrives at Hinton station exactly 20 minutes before the 10 47 train leaves.
Complete the clock diagram to show the time she arrives at Hinton station
(b) Each day, a bus leaves Texford to travel to Cranbrook every 45 minutes
The first bus leaves Texford at 07 10
The last bus leaves Texford at 22 10.
Work out the number of buses that travel from Texford to Cranbrook each day
(c) The cost of a bus pass increases every year.
On Ist January 2022 a bus pass costs \(\$50\).
On 1st January 2023 the cost of the bus pass increases by $10\%.$ On lst January 2024 the cost of the bus pass increases by $5\%$. Calculate the cost of the bus pass on lst January 2024.
(d) The Venn diagram shows information about the number of workers in a hotel who travel to work
by bus $(B)$ and train $(T).$
(i) Work out the number of workers in the hotel
(ii) Work out n$(B\cup T).$
(iii) Explain in words what the number 85 in the Venn diagram represents.
(iv) One of the workers is chosen at random.
Find the probability that this worker travels to work by bus and train.
(e) The hotel has single and double bedrooms in the ratio $\text{single: double=3:8}$.
There are $75$ more double rooms than single rooms
Work out the number of single rooms.
▶️Answer/Explanation
(a)(i): $12~07$
(a)(ii): 10:27 drawn correctly on clock face.
(b): $21$
(c): $57.75$ (and later corrected to $45$)
(d)(i): $232$
(d)(ii): $125$
(d)(iii): The number of workers who travel to work by train but who do not travel on a bus.
(d)(iv): $\frac{9}{232}$
Question4
4 The stem-and-leaf diagram shows the ages of the 16 workers in a shop on 1st January 2022.
(a) (i) Work out the range.
(ii) Write down the mode.
(iii) Work out the median.
(iv) Work out the percentage of workers that are older than $40$ but younger than $60.$
(b) On 1st January $2023$ the shop has the same 16 workers.
Write down the range, the mode and the median on 1st January $2023$.
(c) On 1st January $2024$ the oldest worker leaves.
His replacement is a new worker born on $12$th November $1970$.
There are no other changes to the workers.
Complete the stem-and-leaf diagram for 1st January $2024$.
(d) The shop owner is $x$ years old.
$x$ is a prime number.
$x+2$ is a square number.
$x-2$ is a multiple of $9$.
Find the value of $x$.
▶️Answer/Explanation
(a)(i): $38$
(a)(ii): $36$
(a)(iii): $40$
(a)(iv): $43.75$
(b): $38$, $37$, $41$
(c):
(d): $47$
Question5
(a)
The diagram shows a triangle $EFG$ and a straight line $HFI$
$HFI$ is parallel to $EG.$
$( \mathbf{i} )$ Angle $EFG=105^{\circ}.$
Write down the mathematical name for this type of angle
$( \mathbf{ii})$ Work out the value of $e.$
Give a geometrical reason for your answer.
$\begin{array}{ll}{(\mathrm{iii})}&{\text{Find the value of}f.}\\&{\text{Give a geometrical reason for your answer}}\end{array}$
$(\mathbf{b})$ Calculate the interior angle of a regular $7$-sided polygon
Give your answer correct to 2 decimal places.
(c) The diagram shows a regular pentagon with sides $15$cm.
Work out the values of $\text{w, x and y}$.
▶️Answer/Explanation
(a)(i): obtuse
(a)(ii): $23$, angles in a triangle add to $180$
(a)(iii): $52$, alternate
(b): $128.57$
(c): $7.5$, $-4$, $11.5$
Question6
The diagram shows a point, P, three triangles, A, B and C, and part of triangle D on a 1 cm2 grid.
$(\mathbf{a})$ On the grid, mark the image of point $P$ after a reflection in the line $y=0.$
(b) Use trigonometry to calculate angle $a.$
(c) Describe fully the single transformation that maps
(i) triangle $A$ onto triangle $B$
(ü) triangle $A$ onto triangle $C.$
$(\mathbf{d})$ Triangle $A$ has been enlarged with centre $(1,0)$ to give triangle $D.$
The grid is only large enough to show one vertex and part of two of the sides of triangle $D.$
(i) Write down the scale factor of the enlargement
(ii) Find the coordinates of the other two vertices of triangle D.
▶️Answer/Explanation
(a): $P$ plotted at $(13, 4)$
(b): $36.9$ or $36.86$ to $36.87$
(c)(i): Rotation, centre $(0, 0)$, $90^\circ$ clockwise
(c)(ii): Translation $\begin{pmatrix} 6 \\ -5 \end{pmatrix}$
(d)(i): $7$
(d)(ii): $(1, 28)$, $(29, 7)$
Question7
(a) The scale drawing shows the position of a castle, $C$.
The scale is $1$ centimetre represents $2$ kilometres.
Micah walks at 4.8 km/h for 3 hours on a bearing of 236° from $C$, to his house, $H.$
Mark the position of $H$ on the scale drawing
$\mathbf{( b) }$ A restaurant, $R$, is on a bearing of 083° from $C.$
$( \mathbf{i} )$ Work out the bearing of $C$ from $R.$
(ii) The distance, $d$ km, from $R$ to $C$ is 5 km, correct to the nearest kilometre
Complete this statement about the value of $d.$
(c) A model of the castle is made.
On the model, the length of one of the castle walls is $5$ centimetres.
The actual length of the wall is $90$ metres.
Find the scale of the model in the form $1:n.$
(d) The castle has two kitchens with rectangular floors.
Rectangle $ABCD$ is mathematically similar to rectangle $EFGH$
Calculate the value of x.
(e) The castle has a cylindrical tower.
The tower has a radius of $2.4$ m and a height of $25$ m.
Calculate the curved surface area of the tower.
Give the units of your answer.
▶️Answer/Explanation
(a): $H$ correctly marked $7.2$ cm from $C$ on bearing $236^\circ$
(b)(i): $263$
(b)(ii): $4.5$ or $5.5$ (It seems like there might be two acceptable answers here)
(c): $[1:1800]$
(d): $13.8$
(e): $377$ or $376.9$ to $377.04$ $m^2$
Question8
The table shows some values for $y=x^2-x-3.$
(a) Complete the table.
(b) On the grid, draw the graph of $y=x^2-x-3$ for$-3\leqslant x\leqslant4.$
(c) Write down the coordinates of the lowest point on the graph.
(d) Use your graph to solve the equation $x^2-x-3=7$
▶️Answer/Explanation
(a): $9, 3, -3, -3, 3$
(b): Completely correct curve
(c): $(0.5, k)$, $-3.6 \le k < -3$
(d): $3.7$ to $-2.7$