A student rolls two fair four-sided dice each numbered 1, 2, 3 and 4. They add the two scores together.
Complete the sample space diagram to show the possible outcomes from the dice.
[2]
Find the probability that the student gets an even total.
[1]
Find the probability that the student gets the same score on each dice.
[1]
The circumference of a circle is 23 cm.
Show that the area of the circle is 42.1 cm2, correct to 3 significant figures.
[4]
Light from the Sun travels 1 kilometre in 3.3 × 10-6 seconds. The distance from the Sun to the Earth is 1.5 × 108kilometres.
How long does it take light to travel from the Sun to the Earth? Give your answer in minutes and seconds.
........ minutes........ seconds
[4]
You are given that
where k and m are integers.
Find the value of k and the value of m.
[4]
In each row of the table there are two triangles.
State whether the two triangles are congruent or not. If they are congruent state a reason from SSS, SAS, ASA or RHS.
[3]
The mass of a stone is 680 g. The density of the stone is 1.6 g / cm3.
Multiply out and simplify.
[2]
Factorise.
x2 - 25
[1]
1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.
The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.
Complete the table. Round your answers to the nearest integer.
| Years after release | 0 | 1 | 2 | 3 | 4 |
| Expected number of fish | 1600 | 1680 | 1764 |
[3]
Use the table to draw a suitable graph to show the expected number of fish in the lake.
[3]
A maximum of 2000 fish can live in the lake.
What effect would you expect this to have on the shape of your graph after 4 years?
[2]
A garage is trying to sell a car. The price of the car is normally £18 000. In a sale, the price of the car is reduced by 30%. As a special offer, the sale price is then reduced by r %. The special offer price is £9450.
Find the value of r. You must show your working.
[5]
The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.
The volume of the prism is equal to the volume of the cube.
Find the surface area of the cube. Y ou must show your working.
[6]
Amir, Beth and Charlie work in a cafe. Customers give spare change as tips. At the end of each week, Amir, Beth and Charlie share the total amount of tips between them in the ratio matching the number of hours they worked that week.
This week:
• Amir’s share of the tips was £25.40.
• Beth worked twice as many hours as Amir.
• Charlie worked 5 more hours than Amir.
• The total hours worked by Amir, Beth and Charlie was 85 hours.
Calculate the total amount of tips received this week. You must show your working.
[6]
A student has a spinner with sectors numbered 1, 2, 3 and 4.
The table shows the probability of each score.
| Score | 1 | 2 | 3 | 4 |
| Probability | 0.4 | 0.3 | 0.2 | 0.1 |
The student spins the spinner twice.
Calculate the probability that the student gets the same score on each spin.
[4]
A car registration plate has two letters, a number from 10–99 and three letters. For example:
The letters I and O are not used, leaving 24 possible letters.
Show that there are approximately 720 million possible car registration plates of this form.
[4]
Sketch the graph of y = cos x - 1 for 0 ° ⩽ x ⩽ 720 °.
[3]
80 cyclists take part in a race. A summary of their times is shown in the table.
| Time (t minutes) | Frequency |
| 20 < t ⩽ 25 | 5 |
| 25 < t ⩽ 30 | 15 |
| 30 < t ⩽ 35 | 24 |
| 35 < t ⩽ 40 | 25 |
| 40 < t ⩽ 45 | 7 |
| 45 < t ⩽ 50 | 4 |
Draw the cumulative frequency graph to show the information.
[3]
Reece makes two comments about the times taken to complete the race.
For each comment, decide if Reece is right or wrong and give a reason for your answer.
of the 80 cyclists took more than 30 minutes to complete the race.
Reece is........ because........
[2]
The longest time that any of the 80 cyclists took to complete the race must have been 50 minutes.
Reece is........ because........
[1]
Frankie sketches this quadratic graph.
Frankie says
The y-intercept is 15.
Show that what Frankie says could be correct.
[3]
Explain why what Frankie says may not be correct.
[2]
Blake is asked to write 15 552 000 000 as a product of prime factors in index form. Blake writes
15552000000 = 27× 56× 65.
Write 15 552 000 000 as a product of prime factors in index form.
[2]
You are given that 140000 = 25 × 54 × 7.
Find the highest common factor (HCF) of 15 552 000 000 and 140 000.
[2]
The graph of y = 2 x + 1 is drawn on this one centimetre grid.
The region R satisfies these inequalities.
y ⩽ 2 x + 1 y ⩾ 5 x + y ⩽ 13
Show that the area of region R is 12 cm2.
[6]
Write x2 - 8x + 9 in the form 
x - a
2 - b.
[3]
Use your answer from part (a) to solve.
x2 - 8x + 9 = 0
Give your answers in exact form. You must show your working.
[2]
Two pyramids, A and B, are mathematically similar.
Pyramid A has surface area 12 cm2 and volume 8 cm3. Pyramid B has surface area 75 cm2.
Work out the volume of pyramid B. You must show your working.
[4]
The diagram shows triangle ABC. X lies on BC such that angle AXC = 90 °.
BC = 7.5 cm, angle ABC = 32 ° and angle ACB = 43 °.
Work out length AX. You must show your working.
[6]
The paper as printed
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