1. PS 04 Mαths Section A: Q1 - 6 90 mins Section B: Q1 - 6 90 mins
Section C: Q mins 04

2. Mαths PS04  Section A Answers
a) By considering the gradient on either side of the stationary point on the curve 𝑦= 𝑥 3 −3 𝑥 2 +3𝑥,
show that this is a point of inflection.
b) Sketch the curve 𝑦= 𝑥 3 −3 𝑥 2 +3𝑥
2. For each of the graphs below, sketch the graph of the corresponding gradient function.
Show the co-ordinates of any points where the curve cuts or meets the x-axis and the give the equations of any asymptotes.
a) b) 
Answers

3. Mαths PS04  Section A Answers
3. The table shows the number of hours of sunshine and the rainfall (in mm) in Heathrow in October 2015
A day is selected at random.
What is the probability that there were less than 6 hours sunshine?
Given that there was some sunshine, what is the probability that there were less than 6 hours of sunshine?
Given that there was less than 0.2 mm of rain, what is the probability that the sun didn’t shine?
Investigate whether there is any correlation between sunshine and rainfall at Heathrow in October 2015. 
Answers
Date
1-Oct
2-Oct
3-Oct
4-Oct
5-Oct
6-Oct
7-Oct
8-Oct
9-Oct
10-Oct
11-Oct
Sunshine
10.6
10.5
2.9
8.1
2.3
0.2 6
6.3
0.1
5.4
Rainfall
0.4 11
5.4 tr
0.2
12-Oct
13-Oct
14-Oct
15-Oct
16-Oct
17-Oct
18-Oct
19-Oct
20-Oct
21-Oct
22-Oct
5.4
3.2
5.3
1.7
0.1
2.5
3.4 tr
0.6
0.4
0.2
3.8
1.6
23-Oct
24-Oct
25-Oct
26-Oct
27-Oct
28-Oct
29-Oct
30-Oct
31-Oct
5.5
2.4
1.7
0.1 5 tr
3.4
4.8
0.2 7
0.4

4. Mαths PS04  Section A Answers
4. Two particles, A and B, have masses m kg and 3 kg respectively, where m > 3.
The particles are connected by a light inextensible string which passes over a smooth fixed pulley.
Initially A is 2.5 m above horizontal ground.
The particles are released from rest with the string taut and the hanging parts of the string vertical,
as shown in the figure.
After A has been descending for 1.25 s, it strikes the ground.
Particle A reaches the ground before B hits the pulley.
Find the acceleration of B as it ascends.
Find the tension in the string as A descends.
Find the value of m.
State how you have used the information that the string is inextensible.
When A strikes the ground it does not rebound and the string becomes slack.
Particle B then moves freely under gravity, without reaching the pulley, until the string becomes taut again.
e) Find the time between the instant when A strikes the ground and the instant when the string becomes taut again. 
Answers

5. Mαths PS04  Section A Answers
5. A firework is launched vertically with a speed of 𝑣 𝑚 𝑠 −1 . When it reaches its maximum height, the firework explodes into two parts, which are projected horizontally in opposite directions, each with speed 2𝑣 𝑚 𝑠 −1 . Show that the two parts of the firework land a distance of 4 𝑣 2 𝑔 𝑚 apart.
6. Find the following integrals
sin 𝑥 cos 𝑥 √(cos 2𝑥+ 3) 𝑑𝑥 b) sin 𝑥 cos 𝑥 cos 2𝑥 𝑑𝑥 
Answers

6. Mαths PS04  Section B Answers 1) 𝑓 𝑥 = 𝑥 2 −5𝑥 −3
Show that 𝑓 𝑥 =0 can be written as
𝑥= 5𝑥+3 ii) 𝑥= 𝑥 2 −3 5
b) Let 𝑥 0 =5. Show that each of the following iterative formulae gives different roots of the equation 𝑓 𝑥 =0
𝑥 𝑛+1 = 5 𝑥 𝑛 ii) 𝑥 𝑛+1 = 𝑥 𝑛 2 −3 5
2. a) Find cos (0.244 rad) correct to 6 decimal places
b) Use the approximation for 𝑐𝑜𝑠𝜃 to find an approximate value for cos(0.244 rad).
c) Calculate the percentage error in your approximation.
d) Calculate the percentage error in the approximation for cos(0.75 rad)
e) Explain the difference between your answers to parts c and d. a 
Answers

7. Mαths PS04  Section B Answers
3. The weights of 31 Jersey cows were recorded to the nearest kilogram. The weights are shown in this table.
Find an estimate for the median weight
Find the lower quartile, 𝑄 1
Find the upper quartile, 𝑄 3
Interpret the meaning of the value you have found in the upper quartile in part c.
4. The diagram shows two intersecting sectors: ABD, with radius 5 cm and angle 1.2 radians, and CBD, with radius 12 cm and angle 0.3 radians.
Find the area of the overlapping section. a 
Weight of cattle
300 – 349
350 – 399
400 – 449
450 – 499
500 – 549
Frequency 3 6 10 7 5
Answers

8. Mαths PS04  Section B Answers
5. A histogram looks like a bar chart but there’s a significant difference.
In a bar chart, the frequency is shown by the height of the column.
In a histogram the frequency 𝛼 the area of the column.
That means that 𝑓=𝑘𝐴, where f = frequency, A = Area and
k is the constant of proportionality.
In this histogram (which shows the masses of a random
sample of young children), the number of children who weighed
between 20 and 24 kg was 80.
How many children weighed between 24 and 26 kg?
How many children weighed between 26 and 34 kg?
How many children were in the sample?
6. Given that 𝜋<𝜃< 3𝜋 2 , find the value of tan 𝜃 2 when 𝜃= 3 4 a 
Answers

9. Mαths PS04  Section C Answers
Ed throws a ball for his dog, Bruce. The vertical height of the ball is modelled by the function
ℎ 𝑡 =40 sin 𝑡 10 −9 cos 𝑡 10 −0.5 𝑡 2 +9, 𝑡≥0
𝑦=ℎ 𝑡 is shown in the diagram.
a) Show that the t - co-ordinate of A is the solution to
𝑡= sin 𝑡 10 −18 cos 𝑡 10
To find an approximation for the t – co-ordinate of A, the iterative formula
𝑡 𝑛+1 = sin 𝑡 𝑛 −18 cos 𝑡 𝑛 is used.
b) Let 𝑡 0 =8. Find the values of 𝑡 1 , 𝑡 2 , 𝑡 3 𝑎𝑛𝑑 𝑡 4 . Give your answers to 3 decimal places.
c) Find ℎ′(𝑡)
d) Taking 8 as a first approximation, apply the Newton-Raphson method once to h(t) to obtain a second approximation for the time when the height of the ball is zero. Give your answer to 3 decimal places.
e) hence suggest an improvement to the range of validity of the model. R 
Answers

10. Mαths PS04  Section C Answers
2. Find the equation of the tangent to the curve with implicit equation 𝑥 2 +3𝑥 𝑦 2 − 𝑦 3 =9 at the point (2,1)
3. For both of the following,
find the binomial expansion up to and including the 𝑥 3 term
state the range of values of x for which the expansion is valid.
a) 1+3𝑥 −3 b) 1−5𝑥 7 3
4. A car is moving along a straight road. When t = 0 s, the car passes a point A with velocity 10 𝑚 𝑠 −1 and this velocity is maintained until t = 30 s. The driver then applies the brakes and the car decelerates uniformly, coming to rest at the point B when t=42 s.
a) Sketch a velocity-time graph to illustrate the motion of the car
b) Find the distance from A to B. 
Answers

11. Mαths PS04  Section C Answers
5. The table shows the masses, in kilograms, of 120 Coulter pine cones.
Draw a cumulative frequency diagram for this data.
Estimate the median mass.
Find i) the interquartile range
ii) the 10th to 90th interpercentile range
d) Draw a box plot to show this data.
6. Given that a = ti + 2j + 3k, and that |a| = 7, find the possible values of t
Mass, m (kg)
Frequency
1.0≤𝑚<1.2 7
1.2≤𝑚<1.4 18
1.4≤𝑚<1.6 34
1.6≤𝑚<1.8 41
1.8≤𝑚<2.0 15
2.0≤𝑚<2.2 5 
Answers

12. Mαths PS04
Answers: Section A
1b) 2)
R1aa2221a11aaaaaaaa 
Section A

13. Mαths PS04
Answers: Section A
3a) b) c) 2 15 3 
Section A

14. Mαths PS04  Answers: Section A Section A
4. a) 3.2 𝑚 𝑠 −2 b) 39 N c) d) The tensions are the same either side of the pulley e) 𝑠 
Section A

15. Mαths PS04  Answers: Section A Section A
6) a) − 1 2 ( cos 2𝑥+3) c b) − 1 4 ln | cos 2𝑥+3|+𝑐 
Section A

16. Mαths PS04  Answers: Section B Section B a) 5.5 b) -0.5
a) b) c) % d) -1.80%
e) The larger the value of 𝜃 the less accurate the approximation is. 1 
Section B

17. Mαths PS02  Answers: Section B Section B
3) a) 432 kg b) 389 kg c) 480 kg
d) Three quarters of the cows weigh 480 kg or less.
4) 4.1 𝑐 𝑚 2 
Section B

18. Mαths PS02
Answers: Section B
5) a) 80 b) 160 c) 420
6) -3 
Section B

19. Mαths PS04  Answers: Section C Section C
b) 𝑡 1 =7.928, 𝑡 2 =7.892, 𝑡 3 =7.882, 𝑡 4 =7.876
c) ℎ ′ 𝑡 =4 cos 𝑡 sin 𝑡 10 −𝑡
d) 7.874
e) Restrict the range of validity to 0≤𝑡≤𝐴 R 
Section C

20. Mαths PS04  Answers: Section C Section C 2) 𝑦=− 7 9 𝑥+ 23 9
3) b) 1− 35𝑥 𝑥 2 9 − 1750 𝑥 , 𝑥 < 1 5
4) a)
b) 360 m R 
Section C

21. Mαths PS04  Answers: Section C Section C
5 a) b) ≈1.6 𝑘𝑔 c) i) ≈0.3 c) ii) ≈0.65
5 d)
6) 6 or -6 R 
Section C