applications of mathematics -...

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P45898A©2015 Pearson Education Ltd.

6/6/6/6/6/3/2/6/

*P45898A0128*

Applications of MathematicsUnit 2: Applications 2For Approved Pilot Centres ONLY

Higher Tier

Friday 6 November 2015 – MorningTime: 1 hour 45 minutes 5AM2H/01

You must have: Ruler graduated in centimetres and millimetres, protractor, pair of compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.

Instructions

• Use black ink or ball-point pen.• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answer all questions.• Answer the questions in the spaces provided

– there may be more space than you need.• Calculators may be used.

• If your calculator does not have a button, take the value of to be 3.142 unless the question instructs otherwise.

Information

• The total mark for this paper is 100• The marks for each question are shown in brackets

– use this as a guide as to how much time to spend on each question.• Questions labelled with an asterisk (*) are ones where the quality of your

written communication will be assessed.

Advice

• Read each question carefully before you start to answer it.• Keep an eye on the time.• Try to answer every question.• Check your answers if you have time at the end.

Pearson

Edexcel GCSE

2

*P45898A0228*

GCSE Mathematics 2AM01

Formulae: Higher Tier

You must not write on this formulae page.Anything you write on this formulae page will gain NO credit.

Volume of prism = area of cross section × length Area of trapezium = 12

(a + b)h

Volume of sphere = 43

3 Volume of cone = 13

2h

Surface area of sphere = 4 2 Curved surface area of cone =

In any triangle ABC The Quadratic Equation The solutions of ax2 + bx + c = 0

where 0, are given by

xb b ac

a=

− ± −( )2 42

Sine Rule aA

bB

cCsin sin sin

= =

Cosine Rule a2 = b2 + c2 – 2bc cos A

Area of triangle = 12

ab sin C

length

sectioncross

b

a

h

rl

r

h

C

ab

c BA

3

*P45898A0328* Turn over

Answer ALL questions.

Write your answers in the spaces provided.

You must write down all stages in your working.

1 Here is a map showing two ports.

(a) Find the bearing of Deerham from Seaburn.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °

(1)

There is a lighthouse at Deerham. The lighthouse can be seen from ships up to 20km away.

(b) On the scale drawing, show by shading the region of the sea from where the lighthouse can be seen.

(3)

(Total for Question 1 is 4 marks)

Land

Scale 1cm represents 5km

Deerham

N

Seaburn

Land

Land

4

*P45898A0428*

2 Stefan has x pence.

Michael has 20 pence more than Stefan. Alice has twice as much money as Stefan.

The total amount of money they have is T pence.

(a) Write a formula for T in terms of x.

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

(3)

Stefan, Michael and Alice want to buy a pizza. The pizzas will cost more than £5

(b) Find the smallest value of x, so that the total amount of money they have is more than £5

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

(2)


5


3

Emily fills a carton with boxes.

The carton is a cuboid with

length 100 cm width 40 cm height 30 cm

Each box is a cube of side 5cm.

Work out the number of boxes Emily needs to fill one carton completely.

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


Carton

100 cm

40cm

30cm

5cm

5cm5cm

Diagram NOT accurately drawn

Box

6

*P45898A0628*

4 The diagram shows two prisms.

A B

All the measurements are in cm.

The volume of prism A is 55 cm3 greater than the volume of prism B.

(a) Show that x3 – 12x = 55

(2)

(b) The equation

x3 – 12x = 55

has a solution between x = 4 and x = 5

Use a trial and improvement method to find this solution. Give your answer correct to one decimal place. You must show ALL your working.

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

(4)



2x

xx x

8

3

7


*5 The table gives information about two types of silver alloys.

silver:copper (by weight)

Sterling silver alloy 37:3

Britannia silver alloy 24:1

Hanif has two dishes.

One dish is made of Sterling silver alloy. The other dish is made of Britannia silver alloy.

Each dish has a weight of 3kg.

Which dish has the greater weight of silver? You must show your working.


8

*P45898A0828*

6 Three students test the same coin for bias. Each student throws the coin a number of times.

The table gives information about the results.

Ben Sammi Ellie

Number of heads 32 73 102

Number of throws 50 100 150

The coin is going to be thrown once more.

(a) Using Ben’s results, work out an estimate for the probability that the coin will land on heads.

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

(1)

(b) The results of which student should give the most reliable estimate for the probability that the coin will land on heads?

You must give a reason for your answer.

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(1)

(c) Use all the results to get the best estimate for the probability that the coin will land on heads.

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

(2)


9


7 Megan is planning a game to raise money for charity. She is going to use a fair spinner.

4

2

31

Megan spins the spinner twice.

The score is the sum of the numbers the spinner lands on.

(a) Complete the table to show the possible scores.

2nd spin1st spin

1 2 3 4

1

2 3 5

3 5

4 8

(1)

Here are the rules for Megan’s game.

Pay 50p to spin the spinner twice. When the score is 7 or more, get £1.50

*(b) If this game is played 100 times should Megan expect to make a profit?

(5)


10

*P45898A01028*

8 Here is a design for a plate.

A

D

C

B Diagram NOT accurately drawn

12cm

The plate is in the shape of a circle. The diameter of the plate is 12cm.

A, B, C and D are points on the circumference of the plate. ABCD is a square.

(a) Work out the area of the circular plate. Give your answer correct to 1 decimal place.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2

(2)

(b) Work out the length of BC. Give your answer correct to 2 decimal places.

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


11


*9 Sachin is selling his gold items.

He sells 13.5g of gold for £145.60

The table shows information about the items Sachin is going to sell.

Item Weight of gold (g)

Ring 9.5

Tie pin 5

Medal 8

Bracelet 15

Will Sachin get at least £410 if he sells the four items? You must show your working.


12

*P45898A01228*

10 Here is a sketch for a design for the cross section of the roof space of a building.

O

y x x= −126 2( )

y

x

The cross section of the roof space is the shaded region between the curve and the x-axis.

The curve has equation y x x= −126 2( ) where x and y are in metres.

(a) Complete the table for y x x= −126 2( )

x 0 1 2 3 4 5 6

y 0 2.5

(2)

(b) On the grid, draw the graph of y x x= −126 2( ) to show the cross section of the

roof space.

(2)

O 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

x

y8

13


(c) Calculate an estimate for the area of the cross section of the roof space.

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

(3)


14

*P45898A01428*

11 Here is a garden in the shape of a quarter circle.

(a) Calculate the perimeter of the garden. Give your answer correct to 3 significant figures.

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

10m

10m


15


A path of width 2m is made. The path is shown shaded in the diagram.

(b) Calculate the perimeter of the shaded path. Give your answer correct to 3 significant figures.

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


2m

10m

10m2m


16

*P45898A01628*

12 Here is a solid shape used as the base of a small statue.

b

a

h


All the measurements are in centimetres.

You can work out the volume, Vcm3 of the base, by using the formula

V h a ab b= + +3

2 2( )

When

a = 6.4 b = 3.2 h = 4.8

(a) Work out the exact value of V.

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

(2)

For a different base.

a = 5.3 correct to 1 decimal place. b = 3.6 correct to 1 decimal place. h = 4.0 correct to 1 decimal place.

(b) (i) Work out the upper bound for V

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

17


For this base, the lower bound of V = 77.4232916.

(ii) Use this lower bound and your answer to (b)(i) to give the value of V correct to an appropriate degree of accuracy.

Give a reason for your answer.

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

(4)

Here is a diagram of a geometrically similar solid shape.

All the measurements are in centimetres.

The volume of this solid shape is P cm3

(c) Find a formula for P in terms of a, b and h. Give your answer in its simplest form.

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

(2)


2b

2a

2h


18

*P45898A01828*

13 When you know the adult dose, D m , of medicine, you can work out the child dose, d m , of medicine by using the formula

d aa

D=+ 12

where a years is the age of the child.

The adult dose of medicine is 20 m .

(a) Work out the dose of medicine for a 4 year-old child.

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

A child is 2 years old. The dose of medicine for this child is 4.5 m

(b) Work out the dose of medicine for an adult.

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

19


Here is the graph of yx

x=

+ 12 for values of x from 0 to 12

y

0.4

0.3

0.2

0.1

O 2 4 6 8 10 12 x

For one child, the ratio of the child dose to the adult dose is 2:5

(c) Use the graph to estimate the age of the child.

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

The gradient of the curve gives information about how the dose increases with the age of the child.

(d) Work out an estimate for the gradient of the graph when x = 6

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .m / year(3)


0.5

20

*P45898A02028*

14 The diagram shows the central section of a bridge.

AD and BC represent two vertical towers. The height of each tower is 100 m. XM and YM represent cables. AX = BY M is the midpoint of AB XM = YM = 250m Angle XMY = 137°

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .m


D C

100m 100m250m 250m

137°

X Y

A BM


21


15 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 decision tree diagram.(3)

The train company has to pay a fine of £28000 on any day that trains are late.

(b) Work out an estimate of the total fine that the train company will have to pay for the 31 days of January.

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


Weather Outcome

snow

no snow

late

0.3

22

*P45898A02228*

16 The mass of substance A decreases at the rate of 20% every 6 hours.

500 grams of substance A is put into a dish.

(a) Work out the mass of substance A in the dish at the end of 12 hours.

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

500 grams of a different substance B is placed in a flask. The mass of this substance decreases at a rate of 30% every 6 hours.

At the end of 6 hours, 500 grams more of substance B is added to the flask. At the end of a further 6 hours, another 500 grams more of substance B is added to the

flask.

(b) Work out the mass of substance B in the flask at the end of 18 hours.

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


23


17 Here is a speed-time graph.

The diagram gives information about the speed of a car along a road for 60 seconds. The speed limit for the road is 110 kilometres per hour.

*(a) Does the speed of the car exceed the speed limit?

(3)

(b) Work out the acceleration of the car during the first 10 seconds.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s2

(2)


10 20 30 40 50 60 70 80

10

20

30

40

50

00

Speed(m/s)

Time (seconds)

24

*P45898A02428*

18 The diagram shows a sketch of the pyramid in front of the Louvre.

The base of the pyramid is a square ABCD with sides of length 35 m. The vertex V of the pyramid is vertically above the centre of the square. The sloping edges of the pyramid each have length 32 m.

Calculate the height of the pyramid. Give your answer correct to 3 significant figures.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m



32m

35m

V

C

B

D

A

25

*P45898A02528*

19 A research vessel is used to study underwater life.

The diagram shows a cross-section of the research vessel.

The shell of the research vessel is made from two spheres with the same centre.

The radius of the inner sphere is 0.75m. The thickness of the shell is 2.5cm.

The shell is made of cast iron. The density of the cast iron is 7850 kg per m3

Calculate the mass of the shell. Give your answer correct to 3 significant figures.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .kg


TOTAL FOR PAPER IS 100 MARKS


0.75m

2.5cm

26

*P45898A02628*

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*P45898A02728*

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*P45898A02828*

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