Worked solutions

Pre Public Examination GCSE Mathematics (Edexcel style) November 2016 Higher Tier Paper 2H

Name ……………………………………………………………… Class ………………………………………………………………

TIME ALLOWED

1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

• Answer all the questions. • Read each question carefully. Make sure you know what you have to

do before starting your answer. • You may use a calculator. • Do all rough work in this book.

INFORMATION FOR CANDIDATES

• The number of marks is given in brackets [ ] at the end of each question or part question on the Question Paper.

• You are reminded of the need for clear presentation in your answers.

• The total number of marks for this paper is 80. © The PiXL Club Limited 2016 This resource is strictly for the use of member schools for as long as they remain members of The PiXL Club. It may not be copied, sold nor transferred to a third party or used by the school after membership ceases. Until such time it may be freely used within the member school. All opinions and contributions are those of the authors. The contents of this resource are not connected with nor endorsed by any other company, organisation or institution.

Que

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1 6

2 2

3 3

4 3

5 3

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8 5

9 5

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11 3

12 3

13 7

14 6

15 5

16 1

17 5

18 2

19 3

20 5

Total 80

Question 1.

Maria is going on holiday to America. She changes £250 into dollars ($). The exchange rate is £1 = $1.60 (a) How many dollars should Maria get?

250 × 1.60 = £400

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

Maria wants to buy some music. She can buy the music in America or in Germany. The cost of the music in America is $29 The cost of the music in Germany is €21.50 The exchange rates are £1 = $1.60 £1 = €1.18 (b) In which country is the music cheaper? America - $29/$1.60 = £18.13

Germany - €21.50/€1.18 = £18.22

So cheaper in America

cheaper in America

……………… (4)

(Total 6 marks)

Question 2.

5 female giraffes have a mean weight of x kg. 7 male giraffes have a mean weight of y kg.

Write down an expression, in terms of x and y, for the mean weight of all 12 giraffes.

Total weight of female giraffes is 5x

Total weight of male giraffes is 7y

Mean weight of all giraffes is

5𝑥 + 7𝑦12

5𝑥 + 7𝑦12

…………… (Total 2 marks)

Question 3.

A solid cube has a volume of 64 cm3. Work out the total surface area of the cube.

One side of cube is 4 cm (4 × 4 × 4 = 64) Area of one side is 4 × 4 = 16 cm2

6 faces on a cube so surface area is 6 × 16 = 96 cm2

……96………cm2

(Total 3 marks)

Question 4.

The diagram shows a straight line, L1, drawn on a grid.

A straight line, L2, is parallel to the straight line L1 and passes through the point (0, −5). Find an equation of the straight line L2.

Gradient of line given is ½ So equation of new line is y = ½ x - 5 ...........................................................

(Total 3 marks)

Question 5.

–9 < 4n ≤ 12 n is an integer. Write down all the possible values of n.

All the integers between -9/4 and 12/4 Which are -2, -1, 0, 1, 2, 3

-2, -1, 0, 1, 2, 3

..............................................................................................................................................

(Total 3 marks)

Question 6.

Here is a garden in the shape of a quarter circle with a path of width 2m shaded as shown. Calculate the perimeter of the shaded path. Give your answer to 3 significant figures.

Inner arc length is (π×20)/4 = 5π

Outer arc length is (π×24)/4 = 6π Total Perimeter is 11π + 4 = 38.5575 m 38.6m (3sf)

........ 38.6..................m

(Total 5 marks)

Question 7.

Here are three graphs.

Here are four equations of graphs. y = x3 y = x2 + 4 y = 1⁄x y = 2x

Match each to the correct equation.

A and y = x2 + 4 B and y = x3

C and y = 2x.

(Total 3 marks)

Question 8.

ABC is a triangle. EFGH is a trapezium.

All measurements are in centimetres. The perimeters of these two shapes are the same.

Work out the area of the trapezium.

Perimeter triangle is 5x + 5x + 4x + 8 = 14x + 8 Perimeter trapezium is x – 1 + 5x + 1 + 5x + 1 + 7x – 7 = 18x – 6

So 18x – 6 = 14x + 8 4x = 14

x = 3.5 Sub in x: Area trapezium is ½ (2.5 + 17.5)×14 = 140 cm2

........................... 140................................ cm2 (Total 5 marks)

A

B C

5x

4x+8

E F

G H

x–1

5x+15x+1

7x-7

4x

Question 9.

Prove that the triangle in the figure shown is not isosceles.

5x–10+x+17+3x+20=180

9x+27=180

x=17

Threeanglesare:3×17+20,17+17,5×17–10

Whichare71,34and75.Sincenotwoanglesarethesame,thetriangleisnotisosceles.

(Total 5 marks)

Question 10.

At 9 am, Bradley began a journey on his bicycle. From 9 am to 9.36 am, he cycled at an average speed of 15 km/h. From 9.36 am to 10.45 am, he cycled a further 8 km. (a) Draw a travel graph to show Bradley's journey.

(3)

From 10.45 am to 11 am, Bradley cycled at an average speed of 18 km/h.

(b) Work out the distance Bradley cycled from 10.45 am to 11 am. Distance = Speed × Times = 0.25 × 18 = 4.5 km

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

(Total 5 marks)

Question 11.

Dionne has 70 chocolate bars. Each of these chocolate bars weighs 45 grams to the nearest gram.

Work out the greatest possible total weight of all 70 chocolate bars. Give your answer in kilograms. 70 × 45.5 (upper bound) = 3185 g = 3.185 Kg

......... 3.185.................kg

(Total 3 marks)

Question 12.

f(x) = 3x2 − 2x - 21 Express f(x + 3) in the form ax2 + bx

f(x + 3) = 3(x + 3)2 – 2(x + 3) – 21

= 3(x2 + 6x + 9) – 2x - 6 – 21

= 3x2 + 18x + 27 – 2x - 6 – 21

= 3x2 + 16x

3x2 + 16x

......................................................

(Total 3 marks)

Question 13.

A train company has to pay a fine when its trains are late. The probability that it will snow on any day in January is 0.3 If it snows, the probability that trains will be late on that day is 0.2 If it does not snow, the probability that trains will be late on that day is 0.05

(a) Complete the tree diagram (3)

(b) The train company has to pay a fine of £28000 on any day that trains are late. Work out an estimate of the total fine that the train company will have to pay for the 31 days of January.

Probability Late is 0.3 × 0.2 + 0.7 + 0.05 = 0.095 So estimate of days late is 0.095 × 31 = 2.945 days on average So fine is 3 × 28000 = £84000

£..........84000 (4)

(Total 7 marks)

0.7

NotLate

NotLate

Late

0.2

0.8

0.05

0.95

Question 14.

(a) Show that the equation x3 + 4x = 1 has a solution between x = 0 and x = 1 (0)3 + 4(0) – 1 = -1 (1)3 + 4(1) – 1 = 4. Since there is a sign change there must be a root between 0 and 1. (2)

(b) Show that the equation x3 + 4x = 1 can be arranged to give

4x = 1 – x3 x = !

!− !!

!

(1) (c) Starting with x0 = 0, use the iteration formula twice, to find an estimate for the solution of x3 + 4x = 1

x1 = !!− !!

! = !

!

x2 = !!−

(!!)!

! = !"

!"#

63256

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

(Total 6 marks)

Question 15.

In triangle ABC shown, all lengths are in cm. Work out the length BE.

Scale factor is 15/10 = 1.5

By Pythagoras AE = 10! − 8!= 6

AB = 6 × 1.5 = 9

So BE = 3 cm

.................. 3...............cm

(Total 5 marks)

Diagram not

accurately drawn

Question 16.

The graph of y = f(x) is shown on the grid below.

On the grid draw the graph of y = f(x – 3)

(Total 1 mark)

Question 17.

There is a coastguard station at point A and at point B. B is due East of A. The distance from A to B is 12 km.

There is a rowing boat at point R. R is on a bearing of 160° from A. R is on a bearing of 220° from B.

There is a speedboat at point T. T is 5 km due South of A.

Work out the shortest distance from T to R. Give your answer correct to 1 decimal place. You must show all your working. Angle ABR = 500, Angle RAB = 700, Angle ARB = 600

Sine Rule: !"!"#(!")

= !"!"#(!")

AR = 10.61 km

Angle TAR = 200

Cosine Rule: (TR)2 = 52 + 10.612 – 2(5)(10.61)Cos(20) TR = √(52 + 10.612 – 2(5)(10.61)Cos(20)) = 6.16 km = 6.2 km to 1 dp

6.2 km to 1 dp

........................................................... (Total 5 marks)

Question 18.

Jonah has a fishing boat. Jonah went fishing 123 times last year. The histogram shows information about the total weight of the fish Jonah caught each time he went fishing.

Find an estimate for the median weight of these fish.

Total frequency is 123, so median value will be 62.

This will lie in the 12.5 to 15 bar. 50% of the total area (frequency) is to the left and right of 13.5 and to the left of 13.5.

So median weight is 13.5 kg

............................... 13.5 kg...........................

(Total 2marks)

Question 19.

In the diagram, P, S and T are points on the circumference of a circle.

O is the point such that OPS is a straight line. OT is a tangent to the circle.

Prove that triangle OPT is similar to triangle OTS.

Angle POT = Angle SOT (shared angle)

Angle OTP = Angle OST (alternate segment theorem)

If two angles are the same, the other must be the same also.

Proves the two triangles are similar

(Total 3 marks)

Question 20.

A solid is made by putting a hemisphere on top of a cone.

The total height of the solid is 5x The radius of the base of the cone is x The radius of the hemisphere is x

A cylinder has the same volume as the solid. The cylinder has radius 2x and height h All measurements are in centimetres. Find a formula for h in terms of x Give your answer in its simplest form. Volume of given solid is: !

!𝜋𝑥! 4𝑥 + !

!𝜋𝑥! = !

!𝜋𝑥! + !

!𝜋𝑥! = 2𝜋𝑥!

Volume of cylinder is: 𝜋(2𝑥)!ℎ = 4𝜋𝑥!ℎ

Volumes are equal, so: 4𝜋𝑥!ℎ = 2𝜋𝑥!

So h = !!𝑥

(Total 5 marks)

TOTAL FOR PAPER IS 80 MARKS