Worked Solutions

Pre Public Exam Paper 1

June 2016 Higher Tier

Edexcel Style

Non-Calculator

Time 1 Hour 30 mins

Marks Available 80

Commissioned by The PiXL Club Ltd.

Name

Class

Question Mark Maximum mark

1 3

2 2

3 3

4 3

5 5

6 2

7 3

8 5

9 2

10 2

11 4

12 2

13 4

14 5

15 2

16 2

17 3

18 3

19 3

20 3

21 5

22 2

23 3

24 4

25 5

Total 80

Question 1.

Work out 2.7 × 4.63

463 = 4.63 x 100 27 = 2.7 x 10

463 x 27 answer will be 1000 times larger

x 400 60 3

20 8000 1200 60

7 2800 420 21

463 x 27 = 9260 + 3241 = 12501 Move decimal place to divide by 1000 12501 / 1000 = 12.501

12.501

(Total 3 marks)

Question 2.

Expand and simplify 𝑥 + 4 (𝑥 + 9)

𝒙𝟐 + 𝟗𝒙+ 𝟒𝒙+ 𝟑𝟔 = 𝒙𝟐 + 𝟏𝟑𝒙+ 𝟑𝟔

𝒙𝟐 + 𝟏𝟑𝒙+ 𝟑𝟔 (Total 2 marks)

Question 3.

The line L is drawn on the grid below.

Find an equation for the straight line L.

Give your answer in the form 𝑦 = 𝑚𝑥 + 𝑐.

𝒎 = 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭, 𝒄 = 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭 gradient = 𝟒𝟐= 𝟐 intercept = - 3

𝒚 = 𝟐𝒙 −3 (Total 3 marks)

L

Question 4.

Mr Adi types up a short story. It takes him 9 minutes to finish typing up the story at an average of 40 words per minute.

Mr Blade also types up the same story. He takes 12 minutes to type it.

Work out Mr Blade’s average number of words per minute.

Mr Adi: 40 words per minutes- so in 9 minutes: 9 x 40 = 360 words Mr Blade: same number of words- ie 360 but takes him 12 minutes (he is slower) 360 ÷ 12 = 30 Mr Blade: 30 words per minute

30 words per minute (Total 3 marks)

Question 5.

In a shop, the ratio of the number of male staff to the number of female staff is 2: 3.

20% of the male staff are under the age of 20. 40% of the female staff are under the age of 20.

(a) What percentage of all the people in the company are under the age of 20? 2:3 means 𝟐

𝟓= 𝟒𝟎% 𝐚𝐫𝐞 𝐦𝐚𝐥𝐞 𝐚𝐧𝐝 𝟑

𝟓= 𝟔𝟎% 𝐚𝐫𝐞 𝐟𝐞𝐦𝐚𝐥𝐞

20% × 40% male + 40% × 60% female = 8 % + 24 % = 32%

32%

(4)

(b) A new member of staff joins the shop.

Karen is a 35 year old female. Explain whether this will change your answer in part (a)

Yes, this will change the answer. It will reduce it slightly since there will be a lower proportion of staff aged under 20

(1)

(Total 5 marks)

B

Question 6.

The diagram below shows a solid made from some cubes.

(a) On the grid below, draw a side elevation from the direction of the arrow A.

(1)

(b) On the grid below, draw the solid from the direction of the arrow B.

(1)

(Total 2 marks)

A

Question 7.

There are 900 students at a school.

Joe is organising a party. He is going to order packets of crisps.

Joe takes a sample of 30 students at the school. He asks them which flavour of crisps they want.

The table below shows their results.

Flavour Number of students

Salt and vinegar 10

Ready salted 4

Barbecue 7

Cheese and Onion 9

Work out how much packets of Cheese and Onion crisps Joe should order.

Write down any assumptions you make and explain how this could affect your answer.

𝟗𝟑𝟎 =

𝟑𝟏𝟎 𝐨𝐟 𝐭𝐡𝐞 𝐬𝐭𝐮𝐝𝐞𝐧𝐭𝐬 𝐥𝐢𝐤𝐞 𝐜𝐡𝐞𝐞𝐬𝐞 𝐚𝐧𝐝 𝐨𝐧𝐢𝐨𝐧

𝟑𝟏𝟎 ×𝟗𝟎𝟎 = 𝟐𝟕𝟎

270 packets of cheese and onion.

This assumes that all the students are coming to the party and that the same proportion as the sample like cheese and onion. 30 is quite a small sample, so 270 is only an estimate.

(Total 3 marks)

Question 8.

This is an isosceles trapezium.

Work out the value of x and the value of y.

Since isosceles 𝟐𝒙 + 𝟒𝟎 = 𝟒𝒙 − 𝟑𝟎

𝟒𝟎 + 𝟑𝟎 = 𝟒𝒙 − 𝟐𝒙

𝟕𝟎 = 𝟐𝒙

𝟑𝟓 = 𝒙

𝟑𝒙 − 𝟓𝒚 + 𝟒𝒙 − 𝟑𝟎 = 𝟏𝟖𝟎 Complementary angles total 180

𝟕𝒙 − 𝟓𝒚 − 𝟑𝟎 = 𝟏𝟖𝟎 𝟕𝒙 = 𝟕 ×𝟑𝟓 = 𝟐𝟒𝟓

𝟐𝟒𝟓 − 𝟓𝒚 = 𝟐𝟏𝟎

𝟐𝟒𝟓 − 𝟐𝟏𝟎 = 𝟓𝒚

𝟑𝟓 = 𝟓𝒚

𝟕 = 𝒚

x = 350

y = 70

(Total 5 marks)

Diagram not drawn accurately

Question 9.

Write the following numbers in order of size. Start with the largest number. 0.067 x 102 0.67 x 10-1 6700 x 10-4 67

67, 0.067 x 102, 6700 x 10-4, 0.67 x 10-1 (M1 A1)

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

(Total 2 marks)

Question 10.

(a) Write down the value of 4 -2

𝟏𝟏𝟔

(B1)

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

(1)

(b) Write √63 in the form k√7 , where k is an integer.

3√7 (B1)

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

(1)

(Total 2 marks)

Question 11.

A scientist models a nut as a sphere with a radius of 2.3mm.

Volume of a sphere = !!

𝜋r3

(a) Work out an estimate for the volume of the seed.

𝟒𝟑 x 3 x 23 = 32 (P1 A1)

...................................mm3

(2)

(b) Is your answer to (a) an underestimate or an overestimate?

Give a reason for your answer.

((B1) underestimate (C1) 3< π and 2 < 2.3

(2)

(Total 4 marks)

Question 12.

A box has a volume of 5.6m3.

Change 5.6m3 into cm3.

Write your answer in standard form.

(M1) 5.6m3 = 5.6 x 1003 (cm3) (A1) = 5. 6 x 106

.................................................cm3

(Total 2 marks)

Question 13.

Two cylinders, A and B are mathematically similar.

The base of cylinder A has a radius of 3 cm.

The base of cylinder B has a radius of 6 cm.

The surface area of shape B is 400 cm2.

(a) Work out the surface area of shape A.

(P1) 400 ÷ 22

(A1) 100

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

(2)

The volume of shape A is 90 cm3

(b) Work out the volume of shape B.

(P1) 90 x 23

(A1) 720

...........................cm3

(2)

3cm

6cm

A B

(Total 4 marks)

Question 14.

Here is a table showing data about some heights in metres of some trees in a forest.

There are 160 trees in the forest.

Lowest score 5

Lower quartile 15

Median 18

Interquartile range 10

Range 25

(a) Draw a box plot for this data.

(C1 C1)

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

(2)

(b) How many trees were over 25m high?

25% of 160 = 40 (A1) .......................................

(1)

(c) Estimate how many trees had heights of less than 10m. As 25% of the trees are between 5 and 15m, about 12.5% are under 10m

½ of 25% = 12.5% = 𝟏

𝟖 160 ÷ 8 = 20 (M1 A1)

..........................20............. (2)

(Total 5 marks)

Question 15.

Ben says that to find 10% of a number you divide by 10.

David then says that to find 20% of a number you divide by 20.

Is David right?

You must give a reason for your answer.

No, 20% = 𝟏𝟓 , dividing by 20 gives 𝟏

𝟐𝟎 (C1 C1)

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

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

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

(Total 2 marks)

Question 16.

Factorise fully 7m – 28m3

7m(1 – 4m2) (M1)

7m(1 + 2m)(1 –2m) (A1)

……………………….......

(Total 2 marks)

Question 17.

Make x the subject of y + 4 = !"!!!!

xy + 4x = 50 – 4x (P1) xy + 8x = 50

x(y + 8) = 50 (P1)

x = 𝟓𝟎

𝒚!𝟖 (A1)

x= ……………………….......

(Total 3 marks)

Question 18.

Work out 4√81 x 20-1

Give your answer as a decimal.

(B1) 4√81 = 3 , 20-1 = 𝟏𝟐𝟎

(B1) 𝟑𝟐𝟎

oe

(A1) 0.15

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

(Total 3 marks)

Question 19.

(a) Solve the inequality f – 15 > 35 + 5f

-50 > 4f (M1)

f<-12.5 (A1)

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

(2)

(b) f is an integer.

Write down the largest value of f that satisfies f – 15 > 35 + 5f

-13 (B1)

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

(1)

(Total 3 marks)

Question 20.

(a) Enlarge the triangle by a scale of -1.5, centre (0,1)

Label your image B

(M1 A1)

(2)

(b) Describe fully the transformation that will map shape B onto shape A.

Enlargement

B

Scalefactor–𝟐

𝟑centre(0,1)(C1)

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

(1)

(Total 3 marks)

Question 21.

Shelley puts on her roller blades. She accelerates constantly to a velocity of 4 m/s in 20 secs.

She remains at this constant velocity for a further 40 seconds.

She then decelerates constantly to come to rest in 10 seconds.

(a) Sketch a velocity – time graph of Shelley’s journey.

(P1 P1 A1)

(3)

(b) Calculate the total distance Shelley travelled.

((20 x 4)/2) + (40 x 4) + ((10 x 4)/2) (P1)

= 220 (A1)

Velocity (m/s)

Time (s)

4

3

2

1

0

10 30 20 40 50 60 70

(alternative: find area of trapezium 0.5(70 + 40) x 4) = 220 (P1)(A1) )

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

(2)

(Total 5 marks)

Question 22.

The frequency table below gives information about items sold in a toy shop one Saturday.

(M1)

On the grid below, draw a histogram to represent the information about the toys sold that day

(A1)

(Total 2 marks)

Price (P) in pounds (£)

Frequency Frequency Density

0 < P ≤ 5 80 80/5= 16

5 < P ≤ 10 20 20/5= 4

10 < P ≤ 20 120 120/10=12

20 < P ≤ 40 200 200/20=10

FrequencyDensity

Question 23.

Write √20 - !"√!

in the form a√b where a and b are prime numbers

√20 - 𝟒𝟓√𝟓

x √𝟓√𝟓

(C1)

= 2√5 - 9√5 (C1)

= -7√5

a = -7 b = 5 (A1)

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

(Total 3 marks)

Question 24.

There are three different types of biscuits in a tin.

There are

5 chocolate biscuits

3 wafer biscuits

and 2 plain biscuits

Caira takes 2 of these biscuits at random.

Work out the probability that she takes 2 different types of biscuits.

1 – (P(C,C) + P(W,W) + P(P,P)) (P1)

= 1- ( ( 𝟓𝟏𝟎

x 𝟒𝟗) + 𝟑

𝟏𝟎x 𝟐𝟗+ ( 𝟐

𝟏𝟎 x 𝟏

𝟗)) (P1)

= 1 - 𝟐𝟖𝟗𝟎

(P1)

= 𝟔𝟐𝟗𝟎

(A1)

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

(Total 4 marks)

Question 25.

A and B are points that lie on a straight line.

The coordinates of A are (2, -4) and the coordinates of B are (3,-1).

Another point C does not lie on the line and has coordinates (6, 2)

Find the equation of the line that passes through the point C and is perpendicular to AB.

Give your answer in the form ay + bx = c where a, b and c are integers.

Gradient of line AB: !𝟏! !𝟒𝟑!𝟐

= 𝟑 (P1)

Equation of line perpendicular to AB: y = !𝟏𝟑

x + c (P1)

2 = !𝟏𝟑

(6) + c

c = 4 (P1)

y = !𝟏𝟑

x + 4 (A1)

3y + x= 12 (A1)

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

(Total 5 marks)

TOTAL FOR PAPER IS 80 MARKS