Name
Class
Worked Solutions
Pre Public Exam
Paper 2
June 2016
Higher Tier
Edexcel Style
Calculator
Time
1 Hour 30 mins
Marks Available 80
Commissioned by The PiXL Club Ltd.
Question
Mark
Maximum
mark
1
3
2
2
3
4
4
3
5
4
6
4
7
3
8
2
9
5
10
6
11
4
12
5
13
3
14
3
15
2
16
3
17
2
18
5
19
2
20
5
21
5
22
5
Total
80
Question 1.
Richard, Allan, and Fliss share some money in the ratio 4: 8: 5.
Allan gets £9 more than Fliss.
Work out the amount of money that Richard gets.
Allan has £9 more than Fliss, with three more shares
Each worth £3
Richard: 4 x £3
£12
(Total 3 marks)
Question 2.
ABC is a right-angled triangle.
xcm
11cm
260
Work out the value of x.
Give your answer correct to 1 decimal place.
11
cos 26 ×𝑥
!!
x = !"# !" = 12. 238…
12.2 cm
(Total 2 marks)
Question 3.
The first four terms of an arithmetic sequence are
5
9
13
17
(a) Write an expression, in terms of n, for the nth term of this sequence.
n
1x4
4
+1
=5
2x4
8
+1
=9
3x4
12
+1
= 13
4x4
16
+1
= 17
4n + 1
(2)
The nth term of a different sequence is 4n -5
(b) Is 103 a term of this sequence?
Show how you get your answer.
4n - 5 = 103
4n = 103 + 5
4n = 108
n=
!"#
!
= 27
Yes, it is the 27th term
(2)
(Total 4 marks)
Question 4.
Sian is driving to visit her friend Annabelle.
She has to drive 28 miles to get to Annabelle’s house, and needs to get there within 45 minutes.
The speed limit on the road is 40 mph.
Sian thinks ‘I will need to drive faster than the speed limit to travel 28 miles in 45 minutes.’
Is Sian right?
You must show how you get your answer.
Speed=
!"#$%&'(
!"#$
!" !"#$%
Speed = !" !"#$%&' = 0.62222 miles/minute
0.62222 x 60 = 37.3333 mph
No- she will not have to break the speed limit- she can keep below 40mph and travel at just over 37mph.
(Total 3 marks)
Question 5.
The table shows some information about the number of letters delivered to 50 houses.
Number of letters delivered
Number of houses
0-2
25
3-4
12
5-7
8
8 - 10
5
(a) Write down the modal class interval.
Modal = most
0- 2
(1)
(b) Calculate an estimate for the mean number of letters delivered.
Mid-interval value
Number of houses
Mid-interval value x number of
houses
1
25
25
3.5
12
42
6
8
48
9
5
45
25 + 42 + 48 + 45 160
=
50
50
3.2 letters
(3)
(Total 4 marks)
Question 6.
Triangle ABC is a right-angled triangle.
4cm
Work out the perimeter of triangle ABC.
Give your answer correct to 2 decimal places.
BC = 10! − 7!
= 100 − 49
=
51
=7.14.14 ….
Perimeter = 10 + 7 + 7.1414 …
24.14 cm
(Total 4 marks)
Question 7.
x
(a) Write down the coordinates of the turning point of the graph.
(-1.5, -4.3) (B1)
(-1.5, -4.3)
(1)
(b) Write down the roots of f(x) = 1
0.8, -3.8 (B1)
0.8, -3.8
(1)
(c) Write down the value of f (-3)
-2 (B1)
-2
(1)
(Total 3 marks)
Question 8.
Tom is twice as old as Peggy and Peggy is five times older then Sam.
Write down the ratio of the ages of Tom, Peggy and Sam.
Tom:Peggy:Sam
10 :
5
: 1 (M1) (A1)
10 : 5 : 1
(Total 2 marks)
Question 9.
A
2x + 1
B
E
x
3x – 1
D
C
5x -2
G
2x-1
F
ABCD is a parallelogram.
EFG is a right angled triangle.
The perimeters of the two shapes are the same.
Work out the value of the area of the triangle.
Rectangle = 2x + 1 + x + 2x + 1 + x = 6x + 2
Triangle = 3x – 1 + 5x + 2 + 2x – 1 = 10x -4
6x + 2
= 10x – 4
(P1)
(P1)
6 = 4x
1.5 = x
(A1)
Area of triangle = 0.5 x 2 x 3.5
(P1)
3.5
(A1)
units squared
3.5cm2
(Total 5 marks)
Question 10.
Bilal invests £4000 in a savings account for 3 years.
The account pays compound interest at an annual rate of
1.5% for the first year
x % for the second year
x % for the third year
There is a total amount of £4141.61 in the savings account at the end of 3 years.
(a) Work out the rate of interest in the second and third year.
4000 x 1.015 = 4060
(P1)
4060 x x2 = 4141.61
(P1)
√(4141.61/4060) = 1.01...
(P1)
1%
(A1)
1%
(4)
The cost of a jacket decreases by 12.5% in a sale to £140
(b) Work out the cost of the jacket before the sale.
140 ÷ 0.875
(M1)
£160
(A1)
(alt 140 ÷ 87.5 x 100)
£ 160
(2)
(Total 6 marks)
Question 11.
Diagram NOT accurately drawn
B, C and D are points on the circumference of a circle, centre O.
AB and AD are tangents to the circle.
Angle DAB = 50°
Work out the size of angle BCD.
Give a reason for each stage in your working.
ADO and ABO = 90°
(B1)
BOD = 130° (180 - 50)
BCD = 65°
(B1)
Angle between tangent and radius is 90°
Sum of angles in a quadrilateral is 360°
Angle at centre is twice angle at circumference (C2)
65°
(Total 4 marks)
Question 12.
The probability that Rebecca will win any game of snooker is p.
She plays two games of snooker.
(a)
Complete, in terms of p, the probability tree diagram.
(B2)
1-p
p
1-p
1-p
(2)
(b)
Write down an expression, in terms of p, for the probability that Rebecca will win both
games.
p x p or p2
(B1)
(1)
(c)
Write down an expression (and simplify), in terms of p, for the probability that Rebecca will
win exactly one of the games.
p x (1 – p) + p x (1 – p)
(M1)
2p(1-p)
(A1)
(2)
(Total 5 marks)
Question 13.
x is directly proportional to the positive square root of y
!
When x = 2 y = !"
Find the value of x when y = 16
x = k√y
𝟏
2 = k so k = 10
(M1)
x = 10√16
(M1)
x = 40
(A1)
𝟓
x=40
(Total 3 marks)
Question 14.
Prove algebraically that (2n + 3)2 – (2n – 3)2 is a multiple of 8, for all positive integer values of n.
Using difference of two squares
((2n + 3) + (2n – 3))((2n + 3) - (2n – 3)) = 6(4n)
= 24n
= 8(3n)
Therefore is a multiple of 8
(M1) (P1)
(C1)
or multiply out brackets and simplify
(4n2 + 12n + 9) – (4n2 - 12n + 9) = 24n = 8(3n)
(Total 3 marks)
Question 15.
Prove algebraically that the recurring decimal 0.35 has the value
!"
!"
10x = 𝟑. 𝟓
100x = 𝟑𝟓. 𝟓
90x = 32
x=
(M1)
𝟑𝟐
(A1)
𝟗𝟎
(Total 2 marks)
Question 16.
Show that
!
!! ! !!!!!!"
÷
!
simplifies to
!! ! !!"
𝟏
÷
(𝟐𝒙!𝟓)(𝟑𝒙!𝟐)
=
ax + b
where a, b, c and d are integers.
cx + d
𝟏
𝟏
(𝟐𝒙!𝟓)(𝟑𝒙!𝟐)
𝟏
! 𝟐𝒙!𝟓 (𝟑𝒙!𝟐)
𝐱
(M1)
𝟒𝒙𝟐 !𝟐𝟓
÷
𝟏
𝟐𝒙!𝟓 (𝟐𝒙!𝟓)
𝟐𝒙!𝟓 (𝟐𝒙!𝟓)
𝟏
(M1)
(𝟐𝒙!𝟓)
= (𝟑𝒙!𝟐) (A1)
(𝟐𝒙 + 𝟓)
(𝟑𝒙 + 𝟐)
(Total 3 marks)
Question 17.
The diagram shows a sector of a circle of radius 9cm
9
55°
9
Work out the length of AB.
Give your answer to 3 significant figures.
Circumference = π x diameter = 18π
𝟓𝟓
𝟑𝟔𝟎
x 18π =8.64
(M1)
(A1)
8.64cm
(Total 2 marks)
Question 18.
𝒙
g = √𝒚
x = 25.42 correct to 4 significant figures
y = 7.234 correct to 3 decimal places
By considering bounds, work out the value of g to a suitable degree of accuracy
You must show all your working and give a reason for your final answer.
Upper bound
lower bound
𝟐𝟓.𝟒𝟐𝟓
𝟐𝟓.𝟒𝟏𝟓
√𝟕.𝟐𝟑𝟑𝟓
√𝟕.𝟐𝟑𝟒𝟓
= 9.45337....
9.44900.....
9.45 (both rounded to 3sf)
(P1)
(P1) (B1)
(A1)
(C1)
g= . 9.45
(Total 5 marks)
Question 19.
The diagram shows a sketch of the graph of y = cos x°
(a)
Write down the coordinates of the point A.
(90, 0) (B1)
(90 , 0)
(1)
(b) On the same diagram, draw a sketch of the graph of y = 2 cos x°
On graph (B1)
(1)
(Total 2 marks)
Question 20.
Solve algebraically these simultaneous equations.
2x2 + y2 = 51
y=x+6
2x2 + (x + 6)2
(= 51)
2x2 + x2 +12x +36 (= 51)
(M1)
(M1)
3x2 +12x -15 = 0
x2 +4x - 5 = 0
(x – 1 )(x + 5 )
(M1)
x=1, y=7
x=−5, y=1
(A1) (C1)
x=1, y=7
x=−5, y=1
(Total 5 marks)
Question 21.
ABCD is a parallelogram.
9 5
8 cm
12c
mDiagram NOT accurately drawn
AC = 8 cm
DC = 12 cm
Angle DAC = 95°
Calculate the area of the parallelogram.
Give your answer correct to 3 significant figures.
𝐬𝐢𝐧 𝟗𝟓
𝟏𝟐
=
𝐬𝐢𝐧 𝑫
(P1)
𝟖
sin D = 8 x
𝐬𝐢𝐧 𝟗𝟓
𝟏𝟐
ADC = 41.6°
(A1)
ACD = 180 – 95 – 28.2 = 43.4°
(P1)
Area of ACD = ½ x 8 x 12 x sin 43.4° = 32.9
(P1)
Area of parallelogram = 2 x 32.9 = 66.0
(A1)
66.0.cm2
(Total 5 marks)
Question 22.
The diagram shows a container for salt.
The container is a cylinder on top of a cone.
The cylinder has a radius of 3m and a height of hm.
The cone has a base radius of 3m and a vertical height of 4m.
The container is empty.
The container is then filled with salt at a constant rate.
After 5 hours the depth of the salt is 6 metres above the vertex of the cone.
After 9 hours the container is full of salt.
Work out the value of h.
Give your answer as a fraction in its simplest form.
You must show all your working.
𝟏
𝟑
𝐱 𝛑 𝐱 32 x 4 = 12π
(M1)
6 - 4 = 2m
π x 32 x 2 = 18π
(M1)
5 hrs = 30π so 4 hrs = 24π
(M1)
π x 32 x h = 9πh
9πh = 18π + 24π
h=
𝟒𝟐
𝟗
=
𝟏𝟒
𝟑
(M1)
(A1)
h=
𝟏𝟒
𝟑
m
(Total 5 marks)
TOTAL FOR PAPER IS 80 MARKS