Maths Higher GCSE 2018 Topics after paper 1

Number Algebra Geometry Ratio and Proportion Statistics and ProbabilityRules of indicesEstimateFraction calculationsPercentage profit

Factorising difference of two squaresProofExpanding double brackets and simplifyingAlgebraic fractionsSolving inequalitiesPerpendicular lines

Plans and elevationsSurface areaArea triangleTransformationsGeometric problems on co-ordinate axes

Sharing by ratioDirect proportionInverse proportion Speed

Box Plot

Best buysCurrencyLCM/HCFProduct of primesStandard FormError intervalsReciprocalProduct rule for counting

Conversion graphsSequences (nth term)Changing the subjectGraphing inequalitiesSimultaneous equationsQuadratic / cubic graphsIterationCompleting the squareSolve linear and quadratic equationsDerive an equation

Angles in parallel linesBearingsAngles in polygonsConstructions and lociCircles, cones, spheresPythagorasTrigonometrySimilar shapesArea of triangleVectors

Density, pressureFractions in ratio problemsConvert between standard units (including area and volume)Reverse percentagesCompound percentages

Frequency treesTwo-way tablesPie ChartsScatter GraphsHistogramsFrequency polygonsStem and LeafCumulative FrequencyEstimated meanMean subsetEstimate using probabilityProbability tree diagramsConditional probabilityCapture RecaptureVenn Diagrams

Q1. Stacey went to the theatre in Paris.

Her theatre ticket cost €96The exchange rate was £1 = €1.20

(a) Work out the cost of her theatre ticket in pounds (£). (2)

Stacey bought a handbag in Paris.

The handbag cost €64.80In Manchester, the same type of handbag costs £52.50

The exchange rate was £1 = €1.20

*(b) Compare the cost of the handbag in Paris with the cost of the handbag in Manchester.

(3)Q2. *  Soap powder is sold in three sizes of box.A 2 kg box of soap powder costs £1.89A 5 kg box of soap powder costs £4.30A 9 kg box of soap powder costs £8.46

Which size of box is the best value for money?Explain your answer.You must show all your working.

(4)Q3. (a)  Write 168 as a product of its prime factors.       You must show your working.

(3)(b)  Find the highest common factor (HCF) of 168 and 180

(2) 

Q4. Buses to Ashby leave a bus station every 24 minutes. Buses to Barford leave the same bus station every 20 minutes.

A bus to Ashby and a bus to Barford both leave the bus station at 7 30 am.

When will a bus to Ashby and a bus to Barford next leave the bus station at the same time?

(3)

 

Q5.(a)  Write 7.8 × 10−4 as an ordinary number.

(1)(b)  Write 95 600 000 as a number in standard form.

(1)Q6. Work out

Give your answer in standard form.

(2)Q7. A machine puts drinks into cups. The volume of a cup is 200 ml, correct to the nearest 0.5 ml.

(a)  Write down the lower bound for the volume of a cup.

(1)

The machine puts tea and milk into each cup. It puts into each cup

175 ml of tea measured correct to the nearest ml24 ml of milk measured correct to the nearest ml

*(b)  Is it possible that the total volume of tea and milk put in a cup is greater than the volume of the cup? You must show how you get your answer.

(3)Q8. (a)  Find the reciprocal of 2.5

(1)Q9. There are 95 girls and 87 boys in Year 13 at a school.

One girl is going to be chosen for the role of Head Girl. A different girl is going to be chosen for the role of Deputy Head Girl. One boy is going to be chosen for the role of Head Boy. A different boy is going to be chosen for the role of Deputy Head Boy.

Work out how many different ways this can be done.

(3) 

Q10.*  You can use this graph to convert between litres and gallons.

Jack buys 8 gallons of diesel. He pays £52Francoise buys 40 litres of diesel. She pays £58

Who got the better value for their money, Jack or Francoise?

You must show your working.

(3)Q11. Here are the first four terms of an arithmetic sequence.

6         10         14         18

(a)  Write an expression, in terms of n, for the nth term of this sequence.

(2)The nth term of a different arithmetic sequence is 3n + 5

(b)  Is 108 a term of this sequence? Show how you get your answer.

(2)Q12. Make t the subject of the formula    w = 3t + 11

(2)Q13. Make p the subject of the formula     y = 3p2 – 4

(3)Q14. On the grid below show, by shading, the region defined by the inequalities

Mark this region with the letter R.

(4)Q15. Solve the simultaneous equations

4x + 2y = 7 3x − 5y = −24

(4)Q16. * Paper clips are sold in small boxes and in large boxes.

There is a total of 1115 paper clips in 4 small boxes and 5 large boxes.

There is a total of 530 paper clips in 3 small boxes and 2 large boxes.

Work out the number of paper clips in each small box and in each large box.

(Total for Question is 5 marks)Q17. Here is the graph of  y = x2 – 2x – 4

(a)  Write down estimates for the roots of  x2 – 2x – 4 = 0

(2)(b)  Write down the coordinates of the turning point of  y = x2 – 2x – 4

(1)

Q18. Here are six graphs.

Write down the letter of the graph that could have the equation

(i)  y = 2x

(ii)  y = x3 − 3x

(2)Q19. Write   x2 + 6x – 7   in the form   (x + a)2 + b   where a and b are integers.

(2) Q20. (a)  Show that the equation   x3 + 5x – 4 = 0   has a solution between x = 0 and x = 1

(2)(b)  Show that the equation   x3 + 5x – 4 = 0   can be arranged to give  

(2)

(c)  Starting with   x0 = 0,   use the iteration formula   twice, to find an estimate for the solution of x3 + 5x – 4 = 0

(3)Q21. Asha and Lucy are selling pencils in a school shop. They sell boxes of pencils and single pencils.

Asha sells 7 boxes of pencils and 22 single pencils. Lucy sells 5 boxes of pencils and 2 single pencils. Asha sells twice as many pencils as Lucy.

Work out how many pencils there are in a box.

(4)Q22. (a)  Solve     x2 + 2x − 35 = 0

(3)Q23. *  ABC and DE are parallel lines. AEG and BEF are straight lines.Angle AED = 54°

Angle FEG = 70°

Work out the size of the angle marked x. Give a reason for each stage of your working.

(4)

Q26. The diagram shows the position of two churches, A and B. Church C is on a bearing of 130° from church A. Church C is on a bearing of 245° from church B.

In the space above, draw an accurate diagram to show the position of church C.

Mark the position of church C with a cross ( ). Label it C.

(3) 

Q27.The map shows the positions of two schools, Alford and Bancroft.

A new school is going to be built.The new school will be less than 3 kilometres from Alford.

It will be nearer to Bancroft than to Alford.Shade the region on the map where the new school can be built.

(3)

Q28. The diagram shows a sector of a circle of radius 4 cm.

Work out the length of the arc ABC.

Give your answer correct to 3 significant figures.

(2)

Q29. A frustum is made by removing a small cone from a large cone as shown in the diagram.

The frustum is made from glass. The glass has a density of 2.5 g / cm3

Work out the mass of the frustum.

(5)Q30. The diagram shows a quadrilateral JKLM.

Work out the size of angle KLM. Give your answer correct to 3 significant figures.

(4)

Q31. The diagram shows two vertical posts, AB and CD, on horizontal ground.

AB = 1.7 m

CD : AB = 1.5 : 1

The angle of elevation of C from A is 52°

Calculate the length of BD. Give your answer correct to 3 significant figures.

(4)

Q32. Here are two pots.Pot A and pot B are mathematically similar.

The area of the base of pot B is 160 cm2.

Work out the area of the base of pot A.

(2)Q33. ABC is a triangle.

D is a point on AB and E is a point on AC.

DE is parallel to BC.

AD = 4 cm, DB = 6 cm, DE = 5 cm, AE = 5.8 cm.

Calculate the perimeter of the trapezium DBCE.

(4)Q34. The area of triangle ABC is 42 cm2

Find the length of AB. Give your answer correct to 3 significant figures.

(5)

Q35. M is the midpoint of BC.Q is the midpoint of AM.

(a) Find in terms of a and b. (1)

(b) Find in terms of c.(4)

Q36. Liquid A has a density of 1.42 g/cm3

7 cm3 of liquid A is mixed with 125 cm3 of liquid B to make liquid C.

Liquid C has a density of 1.05 g/cm3

Find the density of liquid B. Give your answer correct to 2 decimal places.

(3) Q37. On a school trip the ratio of the number of teachers to the number of students is 1 : 15

The ratio of the number of male students to the number of female students is 7 : 5

Work out what percentage of all the people on the trip are female students. Give your answer correct to the nearest whole number.

(3)Q38. Change 2 m3 to cm3.

(2) 

Q39. Claire is making a loaf of bread. A loaf of bread loses 12% of its weight when it is baked.

Claire wants the baked loaf of bread to weigh 1.1 kg.

Work out the weight of the loaf of bread before it is baked.

(3)

Q40. The value of a van depreciates at the rate of 20% per year. Gary buys a new van for £27 500 After n years the value of the van is £11 264

Find the value of n.

(2)Q41. 200 people live in a village.

23 people do not have a garden. 10 males do not have a garden. 95 people are male.

(a)  Use this information to complete the frequency tree.

(3)One of the people who does not have a garden is chosen at random.

(b)  Write down the probability that this person is female.

(2)

Q42. 100 adults were asked how they keep fit.

Each adult goes to the gym or runs or cycles.

45 of these adults are female. 30 of the 52 adults who go to the gym are female. 35 adults run. 9 males cycle.

How many females run? 

(3)

Q43. Linda planted 400 flower bulbs. She planted daffodil bulbs, tulip bulbs and hyacinth bulbs.

The incomplete table and pie chart show some information about the bulbs.

Complete the table and the pie chart. (4)Q44. The scatter diagram shows information about 10 students.

For each student, it shows the number of hours spent revising and the mark the student achieved in the Spanish test.

One of the points is an outlier.

(a)  Write down the coordinates of the outlier. (1)

For all the other points

(b)  (i)  draw the line of best fit, (ii)  describe the correlation. (2)A different student studies for 9 hours.

(c)  Estimate the mark gained by this student.

(1)The Spanish test was marked out of 100

Lucia says, "I can see from the graph that had I revised for 18 hours I would have got full marks."

(d)  Comment on what Lucia says.

(1)Q45. The table shows some information about the weights of oranges.

Weight (w grams) Frequency

0 < w ≤ 20   20 < w ≤ 30 15 30 < w ≤ 50   50 < w≤ 60 13 60 < w≤ 75 15

75 < w ≤ 100 10

(a) Use the histogram to complete the table. (2)(b) Use the table to complete the histogram. (2)

Q46. Helen went on 35 flights in a hot air balloon last year.

The table gives some information about the length of time, t minutes, of each flight.

On the grid below, draw a frequency polygon for this information. (2)

 Q47. *  Zoe recorded the heart rates, in beats per minute, of each of 15 people.Zoe then asked the 15 people to walk up some stairs.She recorded their heart rates again.

She showed her results in a back-to-back stem and leaf diagram.

Compare the heart rates of the people before they walked up the stairswith their heart rates after they walked up the stairs.

(6)Q48. Francesco carried out a survey about the ages of the people in his

office.

The table shows information about his results.

(a)  On the grid opposite, draw a cumulative frequency graph for this information.

(2)

(b)  Use your graph to find an estimate for the median age.

(1)Francesco says,"More than 60% of the people in the office are between 35 and 55 years old."

(c)  Use your graph to determine if Francesco is correct. (3)

Q49. Bob asked each of 40 friends how many minutes they took to get to work.

The table shows some information about his results.

Time taken (m minutes) Frequency

0 < m ≤ 10 3 10 < m ≤ 20 8 20 < m ≤ 30 11 30 < m ≤ 40 9 40 < m ≤ 50 9

Work out an estimate for the mean time taken.

(4)Q50. 50 students each did a mathematics test. The mean score for these 50 students was 8.4

There were 30 boys. The mean score for these 30 boys was 8.25

Work out the mean score for the girls. 

(3)Q51. The table shows the probabilities that a biased dice will land on 2, on 3, on 4, on 5 and on 6

Neymar rolls the biased dice 200 times.

Work out an estimate for the total number of times the dice will land on 1 or on 3

(3) Q52.Sameena has a round pencil case and a square

pencil case.

There are 4 blue pens and 3 red pens in the round pencil case. There are 3 blue pens and 5 red pens in the square pencil case.

Sameena takes at random one pen out of each pencil case.

(a)  Complete the probability tree diagram.

(2)(b)  Work out the probability that the pens Sameena takes are both red.

(2) 

Q53. There are 11 girls and 8 boys in a tennis club.Jake is going to pick at random a team from the tennis club.

The team will have two players.Work out the probability that Jake will pick two boys or two girls for the team.

(4)Q54. * Toga wants to estimate the number of termites in a nest.

On Monday Toga catches 80 termites. He puts a mark on each termite. He then puts all 80 termites back in the nest.

On Tuesday Toga catches 60 termites. 12 of these termites have a mark on them.

Work out an estimate for the total number of termites in the nest. You must write down any assumptions you have made.

(4) 

Q55. Here is a Venn diagram.

(a)  Write down the numbers that are in set

(i)  A ∪ B (ii)  A ∩ B (2)

One of the numbers in the diagram is chosen at random.

(b)  Find the probability that the number is in set A'

(2)

ANSWERS1. 80, more expensive in Paris (with workings)2. 5kg box3. 23×3×7 ;124. 9.30am5. 0.00078; 9.56x107

6. 2.52x1015

7. 199.75; yes, 2008. 0.4; 0.5869. 66 814 26010. Jack with comparison11. 4n+2; No with evidence

12. t=w−113

13. p=√ y+4314. Region identified

15.−12, 92

16. Small=60, large=17517. -1.2 and 3.2; (1, -5)18. D; A19. ( x+3 )2−1620. f(0)=-4 and f(1)=2 – change of sign; rearrangement shown; 0.70921. 6 (7 x+22=2(5 x+2)22. 5 and -723. 124o with reasons24. .25. 10026. Two bearings drawn and C identified27. Perpendicular bisector, circle and region shaded28. 18.229. 136130. 33.731. 0.66432. 102.433. 32.234. 12.335. 3a + b; 3c

36. 1.0337. 39%38. 2 000 00039. 1.2540. 4

41. 23, 177, 10, 13, 85, 92; 1323

42. 1143. 130, 90, 81, 16244. (4,10) line of best fit, positive, value between 60 and 70; outside

range of data45. 10 and 18; bars with fd 1.5 and 146. Points at (5,6) (15,9) (25,8) (35,7) (45,4) and joined with line

segments47. Median before = 67, after = 78; Range before = 26, after = 37

Median after bigger so heart rate higher after; Range before small (more consistent)

48. Upper bounds with CF and joined by curve or line segments; 43; Yes (with justification)

49. 28.2550. 8.62551. 9852. round: 4/7 and 3/7 square: 3/8, 5/8, 3/8, 5/8; 15/5653. 83/17154. 400 and eg population not changed or mark now worn off or sample

random55. 10, 12, 14, 15, 16, 18; 12, 18; 7/10