Higher Tier - Paper 2storage.googleapis.com/prudhoe/2014/12/Maths-Higher... · 2017. 10. 22. ·...

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Higher Tier - Paper 2

Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser ! You will need a calculator

Guidance

1. Read each question carefully before you begin answering it.2. Donʼt spend too long on one question.3. Attempt every question.4. Check your answers seem right.5. Always show your workings

Revision for this test

Corbettmaths © 2015

Questions Topic Video number

1, 2 Estimated Mean 55

3, 4 Currency 214a

5 Use of a Calculator 352

6, 7 Ratio 269, 270, 271

8, 9 Constructions 70, 72, 78, 79, 80

10 Loci 75, 76, 77

11, 12 Bearings 26, 27

13 Trial and Improvement 116

14, 15 The nth Term 288, 289

16 Quadratic Graphs 264

17, 18, 19 Simultaneous Equations 295

20, 21, 22, 23 Trigonometry (SOH CAH TOA) 329, 330, 331

24, 25, 26 Similar Shapes 292, 293a, 293b

27, 28, 29 Circle Theorems 64, 65

30, 31, 32 Standard Form 300 to 304

33, 34 Compound Interest 236

35, 36 Reverse Percentages 240

37 Direct Proportion 254

38 Inverse Proportion 255

39 Stratified Sampling 281

40, 41 Histograms 157, 158, 159

42, 43 Limits of Accuracy 183, 184

44 Trigonometry (The Sine Rule) 333

45 Trigonometry (The Cosine Rule) 335, 336

46 Trigonometry (1/2abSinC - Area of a Triangle) 337

47 The Quadratic Formula 267

48, 49 Simultaneous Equations (A*) 298

50 Vectors 353


1.! The table shows the heights of 50 students.

! (a) Write down the modal class interval.

........................................(1)

! (b) Work out an estimate for the mean height of the students.

..........................cm(4)


2.! Timothy asked 30 people how long it takes them to get to school.!! The table shows some information about his results.

! Work out an estimate for the mean time taken.

..........................minutes(4)

3.! Vicky buys an ice cream for €2.16 in Berlin.! The exchange rate is £1 = €1.60.

! Find the cost of the ice cream in pounds.

£.........................(2)


4.! Kevin is going on holiday to Japan.! He wants to change some money into yen.

! The bank only stocks ¥1000 notes.! James wants to change up to £300 into yen.! He wants as many ¥1000 notes as possible.

! The exchange rate is £1 = ¥168

! How many ¥1000 notes should he get?

.........................(3)

5.! Calculate the value of

! !

! (a) Write down your full calculator display.

.............................................................(1)

! (b) Give your answer to three significant figures.

.........................(1)


6. A piece of carpet is 240cm long.! ! Mr Jones cuts it into three pieces in the ratio 1 : 2 : 5

! ! Work out the length of the longest piece of carpet.

................................(3)

7.! Sarah has some chocolates.

! ! 24 are white chocolate.! ! 16 are milk chocolate.! ! 8 are dark chocolate.

! ! (a) Write down the ratio of white chocolate to milk chocolate to dark chocolate. ! ! Give your ratio in its simplest form.

............... : ............... : ...............(2)

! ! Rachel has some apples and bananas.! ! The ratio of apples to bananas is 2 : 3! ! She has 14 apples.

! ! (b) Work out how many bananas Rachel has.

.........................(2)


8.! Use ruler and compasses to construct the perpendicular bisector of AB.! You must show clearly all your construction arcs.

(2)

9.! Using ruler and compasses, construct the bisector of angle ABC.

(2)


10.! A phone box is located near three houses, A, B and C.

! The phone box is less than 500m from the railway track.! The phone box is between 300m and 500m from house A.! The phone box is closer to house C than house B.

! Shade the region on the map where the phone box could is located.

(5)


11.! The diagram shows the position of two cities C and D.

! (a) Work out the actual distance of D from C.

.........................km(2)

! (b) Find the three figure bearing of D from C.

...............................⁰(1)

! E is South-East of C.! (c) Write down the bearing of E from C.

...............................⁰(1)

! E is also on a bearing of 190⁰ from D.

! (d) Mark the position of E on the diagram.(2)


12.! The diagram shows the position of two towns, A and B.

! A rugby club, R, has bearing of 110⁰ from town A.! The rugby club, R, has bearing 245⁰ from town B.

! In the space above, show the position of the rugby club R.! Mark the position with a cross (x) and label it R.

(3)


13.! The equation

! ! ! x³ + 2x = 50

! has a solution between 3 and 4.

! Use trial and improvement to find this solution.! Give your answer correct to 1 decimal place.! You must show all your working.

x = ...............................(4)


14.! Find the nth term of the sequences

! (a) 1, 4, 9, 16, 25, ...

....................(1)

! (b) 3, 6, 11, 18, 27, ...

....................(1)

! (c) -3, 0, 5, 12, 21, ...

....................(1)

! (d) 2, 8, 18, 32, 50, ...

....................(1)

15.! The nth term of a sequence is ! 3n - 2

! (a) Write down the first two terms of this sequence.

1st term ..............., 2nd term ...............(2)

! (b) Explain why 101 is not a term in the sequence.

! ................................................................................................................................

! ................................................................................................................................

(2)


16! (a) Complete the table of values for y = x² + 2x + 1

(2)

! (b) On the grid, draw the graph of y = x² + 2x + 1 for the values of x from -3 to 3.

(2)


17.! Solve the simultaneous equations

! ! 3x − y = 23! ! 2x + 3y = 8

! Do not use trial and improvement

x = ......................... y = ..........................(3)


! ! 4x + 3y = 5! ! 2x − 5y = 9


x = ......................... y = ..........................(4)


! ! 4x − y = 17! ! y = x − 2


x = ......................... y = ..........................(3)


20.!

! Calculate the size of angle BAC.

....................⁰(3)

21.!

!! Find the length of the side RT in the triangle above.

....................cm(3)


22.! A ladder is placed against a wall.! To be safe, it must be inclined at between 70⁰ and 80⁰ to the ground.

! (a) Is the ladder safe?

.........................(3)

!! (b) Calculate the length of the ladder.

.........................m(3)


23.! The diagram shows two right-angled triangles.

! Calculate the value of x.

.........................cm(5)


24.!

! ACE and BCD are straight lines.! DE is parallel to AB.

! (a) Work out the size of CE.

........................cm(3)

! (b) Work out the size of AB.

........................cm(3)


25.! Below are two similar parallelograms.

! The area of parallelogram A is 28cm²

! Work out the area of parallelogram B.

..........................cm²(2)

26.! Two clay models of the Statue of Liberty are mathematically similar.

! The smaller model has a height of 15cm.! The larger model has a height of 20cm.

! The smaller model weighs 108g.

! Work out the weight of the larger model.

.........................g(3)


27.! P, Q, R and S are four points on the circumference of a circle.! PR meets QS at A.! Angle PRS = 57° and Angle PAQ = 78°

! Work out the value of angle y.! Include your method.

.........................°(3)

28.! RS is a tangent to the circle at R.

! Calculate the value of x.

! Give reasons for your answer.

.........................°(3)


29.! PQRS is a cyclic quadrilateral.! APB is a tangent to the circle at P.! PQ is parallel to SR.! Angle SPB = 35° and angle PSQ = 80°

! (a) Work out the size of angle QRS.

.........................°(4)

! (b) Work out the size of angle RQS.

.........................°(2)


30.! Write the following numbers in standard form.

! (a)! 0.000345

..........................................(1)

! (b)! 41200000

..........................................(1)

31.! Calculate, writing your answer in standard form

! !

..........................................(2)

32.! The distance of the moon to the Earth is 384,400 km.! The speed of light is 2.998 x 108 m/s.!! Work out how long it will take light to travel from the moon to the Earth.! Include suitable units.

..........................................(3)


33.! A radioactive substance decays over time.! Every year its mass decreases by 14%.

! How many years will it take for 500kg of the substance to decay to a mass less ! than 200kg?

.........................years(3)

34.! Fiona leaves £1600 in the bank for four years.! It earns compound interest of 4% each year.

! Calculate the total amount Fiona has in the bank at the end of the four years.

£.........................(3)

35.! A lamp is on sale at £22.05! This is a 10% reduction of the normal price.

! What was the price of the lamp before the reduction?

£.........................(3)

36. ! There are 6000 people at an ice hockey match.! The announcer says this is exactly 40% more people that the previous match.

! Explain why the announcer is incorrect.

(4)Corbettmaths © 2015

37.! The cost of a circular table is directly proportional to the square of the radius.! A circular table with a radius of 40cm cost £50.

! What is the cost of a circular table with a radius of 60cm?! !

£.......................(3)

38.! The force, F newtons, exerted by a magnet on a metal object is inversely ! proportional to the square of the distance d cm.

! When d = 2 cm, F = 50 N.

! (a) Express F in terms of d.

F = .......................(3)

! (b) Find the force when the distance between the magnet and metal object is 10cm

F = .......................N(1)

! (c) Find the distance between the magnet and metal object when the force is 8N.

d = .......................cm(1)


39.! The table shows information about the inhabitants of a village.

! Bernard is going to carry out a survey about the local library.! He wants to find out how often people have been to the library in the last year.

! Bernard decides to take a stratified sample.

! (a) Explain why it is appropriate to take a stratified sample.

! ................................................................................................................................

! ................................................................................................................................(1)

! Bernard takes a stratified sample of 100.!! (b) Calculate the number of each age group that Bernard should choose.

0 - 20 ...............................

21 - 40 ...............................

41 - 60 ...............................

Over 60 ...............................(3)


40.! The test scores from the students in a school are summarised in the table.

!

! Draw a histogram for this data.

(3)


41.! A group of students were asked to complete a puzzle.! The histogram shows the distribution of the times taken.

! (a) Work out how many students took between 50 and 70 seconds to complete ! the puzzle.

.........................(1)

! (b) Calculate an estimate of the number of students who took under 30 seconds! to complete the puzzle.

.........................(2)


42.! Declan ran a distance of 200m in a time of 26.2 seconds.

! The distance of 200m was measured to the nearest 10 metres.! The time of 26.2 was measured to the nearest tenth of a second.

! (a) Work out the upper bound for Declanʼs average speed.

.........................m/s(2)

! (b) Work out the lower bound for Declanʼs average speed.

.........................m/s(2)

43.! Sophie estimated that the distance between Bristol and Newcastle is about ! 290 miles and that her average driving speed would be 60 mph.

! She estimated the distance to the nearest 10 miles and the speed to the nearest! 10 mph.

! Calculate the lower bound of the time the journey should take.! Give your answer in hours and minutes.! Give your answer to the nearest minute.

...................hours ....................minutes(4)


44.!

! In triangle ABC the length of AC is 15cm.! Angle ABC = 112°! Angle BAC = 33°

! Work out the length of BC.

.........................cm(3)

45.!

! !

! Calculate the length of BC.

.........................cm(3)


46.

! Calculate the area of the triangle.

.........................cm²(2)

47.! Solve the equation 4x² + x − 7 = 0

! Give your answers to two decimal places.

x = ..................... or x = .....................(3)


! ! ! 2x − y = 7! ! ! xy = 15

.....................................................(4)


49.! Solve the equations

! ! ! x² + y² = 25! ! ! x + y = 7

!

.....................................................(5)

50! In the diagram OBDE and OAFG are parallelograms.! B is the midpoint of OG.! A is the midpoint of OE.

! (a) Express, in terms of a and b, the following vectors.! Give your answers in their simplest form.

! (i)

.........................(1)

! (ii)

.........................(1)


! (iii)

.........................(1)

! (b) Show are parallel.

(2)


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