Edexcel A Level Maths Paper 3: Statistics and Mechanics June 2024 question paper 9MA0/03

Figure 1 shows a particle P of mass 0.5 kg at rest on a rough horizontal plane.

Find the magnitude of the normal reaction of the plane on P. The coefficient of friction between P and the plane is A horizontal force of magnitude X newtons is applied to P. Given that P is now in limiting equilibrium,

find the value of X.

Figure 2 shows a speed‑time graph for a model of the motion of an athlete running a 200 m race in 24 s. The athlete

• starts from rest at time t = 0 and accelerates at a constant rate, reaching a speed of

• then moves at a constant speed of 10 m s^–1 fromt = 4 tot = 18

• then decelerates at a constant rate from t = 18 to t = 24, crossing the finishing line

Using the model,

find the acceleration of the athlete during the first 4 s of the race, stating the units of your answer,

find the distance covered by the athlete during the first 18 s of the race,

find the value of U.

A particle P of mass m is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

• the plane is inclined to the horizontal at an angle α, where tanα =

• the coefficient of friction between P and the plane is μ, whereμ

The particle P is released from rest and slides down the plane. Air resistance is modelled as being negligible. Using the model,

find, in terms of m and g, the magnitude of the normal reaction of the plane on P,

show that, as P slides down the plane, the acceleration of P down the plane is

State what would happen to P if it is released from rest butμ

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. [ In this question, i is a unit vector due east and j is a unit vector due north. Position vectors are given relative to a fixed origin O. ] At time t seconds, t 1, the position vector of a particle P is r metres, where

and c is a constant. Whent = 4, the bearing of P from O is 135°

Show that c = 3

Find the speed of P whent = 4 Whent = T, P is accelerating in the direction of (–i – 27j).

Find the value of T.

At time t = 0, a small stone is projected with velocity 35 m s^–1 from a point O on horizontal ground. The stone is projected at an angle α to the horizontal, where tanα = In an initial model

• the stone is modelled as a particle P moving freely under gravity

• the stone hits the ground at the point A

Figure 4 shows the path of P from O to A. For the motion of P from O to A

• at time t seconds, the horizontal distance of P from O is x metres

• at time t seconds, the vertical distance of P above the ground is y metres

Using the model, show that

Use the answer to (a), or otherwise, to find the length OA. Using the model, the greatest height of the stone above the ground is found to be H metres.

Use the answer to (a), or otherwise, to find the value of H.

• The model is refined to include air resistance.

Using this new model, the greatest height of the stone above the ground is found to be K metres.

State which is greater, H or K, justifying your answer.

State one limitation of this refined model.

Figure 5 shows a uniform rod AB of mass M and length 2a.

• the rod has its end A on rough horizontal ground

• the rod rests in equilibrium against a small smooth fixed horizontal peg P

• the point C on the rod, where AC = 1.5 a, is the point of contact between the rod and the peg

• the rod is at an angle θ to the ground, where tanθ =

The rod lies in a vertical plane perpendicular to the peg. The magnitude of the normal reaction of the peg on the rod at C is S.

The coefficient of friction between the rod and the ground is μ. Given that the rod is in limiting equilibrium,

find the value of μ.

TOTAL FOR MECHANICS IS 50 MARKS

Xian rolls a fair die 10 times. The random variable X represents the number of times the die lands on a six.

Using a suitable distribution for X, find

P(X = 3)

P(X 3) Xian repeats this experiment each day for 60 days and records the number of days when X = 3

Find the probability that there were at least 12 days when X = 3

Find an estimate for the total number of sixes that Xian will roll during these 60 days.

Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.

Amar is studying the flight of a bird from its nest. He measures the bird’s height above the ground, h metres, at time t seconds for 10 values of t Amar finds the equation of the regression line for the data to be h = 38.6 – 1.28t

Interpret the gradient of this line. The product moment correlation coefficient between h and t is – 0.510

Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight. You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

Jane draws the following scatter diagram for Amar’s data.

With reference to the scatter diagram, state, giving a reason, whether or not the regression line h = 38.6 – 1.28 t is an appropriate model for these data. Jane suggests an improved model using the variable u = t k² where k is a constant. She obtains the equation h = 38.1 – 0.78u

Choose a suitable value for k to write Jane’s improved model for h in terms of t only.

Ming is studying the large data set for Perth in 2015 He intended to use all the data available to find summary statistics for the Daily Mean Air Temperature, x^oC. Unfortunately, Ming selected an incorrect variable on the spreadsheet. This incorrect variable gave a mean of 5.3 and a standard deviation of 12.4

Using your knowledge of the large data set, suggest which variable Ming selected. The correct values for the Daily Mean Air Temperature are summarised as n = 184 x 2801 2. x² 44 695 4.

Calculate the mean and standard deviation for these data. One of the months from the large data set for Perth in 2015 has

for Daily Mean Air Temperature.

Suggest, giving a reason, a month these data may have come from.

The proportion of left-handed adults in a country is 10% Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10% She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.

Find the critical region for a suitable test to assess Freya’s belief. You should

• state your hypotheses clearly

• use a 5% level of significance

• state the probability of rejection in each tail

Write down the actual significance level of your test in part (a) In Freya’s sample 7 adults were left-handed.

With reference to your answer in part (a) comment on Freya’s belief.

The records for a school athletics club show that the height, H metres, achieved by students in the high jump is normally distributed with mean 1.4 metres and standard deviation 0.15 metres.

Find the proportion of these students achieving a height of more than 1.6 metres. The records also show that the time, T seconds, to run 1500 metres is normally distributed with mean 330 seconds and standard deviation 26 seconds. The school’s Head would like to use these distributions to estimate the proportion of students from the school athletics club who can jump higher than 1.6 metres and can run 1500 metres in less than 5 minutes.

State a necessary assumption about H and T for the Head to calculate an estimate of this proportion.

Find the Head’s estimate of this proportion. Students in the school athletics club also throw the discus.

Given that P(D 16.3) = 0.30 and P(D 29.0) = 0.10

calculate the value of µ and the value of σ

The Venn diagram, where p, q and r are probabilities, shows the events A, B, C and D and associated probabilities.

State any pair of mutually exclusive events from A, B, C and D The events B and C are independent.

Find the value of p

Find the greatest possible value of P A | B Given that P B | A 0 5.

find the value of q and the value of r

Find P A B C

Use set notation to write an expression for the event with probability p

TOTAL FOR STATISTICS IS 50 MARKS