Edexcel A Level Maths Paper 3: Statistics and Mechanics June 2025 question paper 9MA0/03

A car moves in a straight line along a horizontal road with constant acceleration 2 m s^–2 The car is moving with speed 15 m s^–1 in the direction of the acceleration when it passes a signpost on the road. The car is modelled as a particle.

Use the model to find the speed of the car 4 s after passing the signpost. Figure 1 below shows the horizontal forces acting on the car. Given that

• the car has mass 800 kg

• the driving force of the engine has magnitude D newtons

• the resistance to the motion of the car has magnitude 400 N

• the acceleration of the car is 2 m s^–2 in the direction of the driving force

use the model to find the value of D.

A small box B of mass 2 kg is dragged in a straight line, along a rough horizontal plane, at a constant speed by a force of magnitude 5 N. The line of action of the force makes an angle α with the plane, where sinα = as shown in Figure 2.

Show that the magnitude of the normal reaction of the plane on the box is 16.6 N. At the instant when B is at the point O on the plane, the force of magnitude 5 N is removed.

Describe the motion of the box after the force of magnitude 5 N is removed.

Find the magnitude of the normal reaction of the plane on the box after the force of magnitude 5 N is removed. Given that after the force of magnitude 5 N is removed

• the box is modelled as a particle

• air resistance is modelled as being negligible

• the coefficient of friction between the box and the plane is modelled as 0.2

• the speed of the box as it passes through O is 4 m s^–1

• the box comes to rest at the point X on the plane

use the model to find the length OX.

State one limitation of the model, apart from ignoring air resistance, that could affect your answer to part (d).

[In this question, i and j are horizontal unit vectors due east and due north respectively.] A particle P of mass 0.5 kg moves with constant acceleration (2i – 2.4j) m s^–2 on a smooth horizontal plane under the action of a constant horizontal force F N.

Find F in terms of i and j. At time t = 0, P is moving with velocity (–7i + 7.8j) m s^–1

Find the velocity of P at time t = 2 seconds.

Find the direction of motion of P at time t = 2 seconds, giving your answer as a bearing in degrees. At time t = 0, P passes through the point O. At time t = 5 seconds, P passes through the point A.

Find OA → in terms of i and j.

[In this question, position vectors are given relative to a fixed origin O.] At time t seconds, where t 0, the position vector of a particle P is r metres where

Find the position vector of P at t = 4

Find the exact distance of P from O at t = 4

Find an expression for the velocity of P at time t seconds, where t 0, giving your answer in terms of t, i and j At t = T, the acceleration of P is in a direction that is perpendicular to the line with

Find the value of T.

A small stone is projected with speed 14 m s^–1 from a point O on the top of a cliff. The point O is H metres vertically above the point N. Point N is on horizontal ground. The stone is projected at an angle θ above the horizontal, where tanθ = The stone strikes the horizontal ground at the point A, where NA = 40 m, as shown in Figure 3. The stone is modelled as a particle moving freely under gravity. Using this model, find

the value of H

the maximum height of the stone above the horizontal ground.

A uniform rod AB has mass M and length 2a. A particle of mass 2M is attached to the rod at the point C, where AC = 0.5a The rod rests with end A on rough horizontal ground and end B against a vertical wall. The rod lies in a vertical plane which is perpendicular to the wall. The rod is in equilibrium at an angle α to the wall, as shown in Figure 4. In an initial model

• the vertical wall is modelled as being smooth

• the magnitude of the normal reaction of the ground on the rod at A is R

• the magnitude of the force exerted on the rod by the wall at B is S

Using the model,

find R in terms of M and g

show that S = Mg tanα In a refined model

• the vertical wall is modelled as being rough

• the magnitude of the normal reaction of the ground on the rod at A is R₁

State which is greater, R or R₁, giving a reason for your answer.

A second particle of mass 3M is now attached to the rod at B. The rod again rests with end A on rough horizontal ground and end B against the vertical wall. The rod lies in a vertical plane which is perpendicular to the wall. The rod is now in limiting equilibrium at an angle β to the wall, as shown in Figure 5. The vertical wall is again modelled as being smooth. The coefficient of friction between the rod and the ground is μ

use the model to show that μ =

TOTAL FOR MECHANICS IS 50 MARKS

Bag A contains 5 red, 4 yellow and 3 green beads. Bag B only contains red and yellow beads. A bead is selected at random from bag A and a second bead is selected at random from bag B Given that the probability that both beads selected are yellow is

find the probability of selecting a yellow bead from bag B

Hence complete the tree diagram below by finding the probability on each branch.

Find the exact probability that at least one of the two beads selected is yellow. The event X is that at least one of the beads selected is yellow. The event W is that a green bead is selected.

Find the exact value of P(W|X)

Runners in an athletics club can train with either coach A or coach B for the 400 m race. Coach A trains 120 runners for the 400 m and records the best time, x seconds, for each runner. The results are summarised by the following statistics

Calculate the mean of the best times for the runners trained by coach A

Calculate the standard deviation of the best times for the runners trained by coach A The mean and standard deviation for the best times of the 100 runners trained for the 400 m by coach B are 55.1 seconds and 3.6 seconds respectively. A 400 m race consists of equal numbers of the fastest runners trained by coach A and by coach B

State, giving a reason, which coach is more likely to have trained the winner.

A manufacturer makes components for the car industry. On average 1.5% of the components made are defective.

Using a suitable model, find the probability that in a random sample of 36 components there are

exactly 2 defective components,

more than 3 defective components. Once every six months, the manufacturer tests whether the proportion of defective components has changed using hypotheses H₀:p = 0.015 and H₁:p ≠ 0.015 A random sample of 260 components is taken and 8 are found to be defective.

Using a 5% level of significance, complete the test.

Kay is studying the variables Daily Total Sunshine (x) and Daily Total Rainfall (y) from the large data set for Leeming in 2015 Kay starts with 5th May and then selects every 10th day thereafter.

State the name of the sampling technique Kay uses. Kay wants to find the regression line of y on x for these data.

Using your knowledge of the large data set, explain how Kay might need to clean these data before finding the equation of the regression line. The equation of the regression line Kay finds is y = 0.741 + 0.199x

Using your knowledge of the large data set,

state the units of the gradient of the regression line,

give an interpretation of the y-intercept of the regression line. Kay’s teacher claimed that the greater the amount of sunshine in a day the lower the amount of rain there should be.

State, giving a reason, whether the teacher’s claim is true for Kay’s data. The teacher used all the data for these variables from the large data set for Leeming in 2015, as a sample. The teacher calculated the product moment correlation coefficient for x and y to be – 0.160 In a suitable test to determine whether there is evidence to support the teacher’s claim, the p-value was 0.015

Using a 5% level of significance, state the hypotheses and conclusion for this test.

For the test in part (e) describe

the sample,

a possible population.

The heights of men in a tennis club are normally distributed with a mean of 183 cm and a standard deviation of 4.9 cm. A man from the tennis club is selected at random.

Find the probability that the man’s height is

more than 186 cm,

between 175 cm and 185 cm. The heights of women in the tennis club are normally distributed with a mean of μcm and a standard deviation of σ cm. Given that 40% of the women are shorter than 170 cm and 15% are taller than 175 cm,

find the value of μ and the value of σ Show your working clearly. A man from the tennis club and a woman from the tennis club are selected at random.

Find the probability that both players have heights between 170 cm and 175 cm.

The discrete random variable R takes even integer values from 2 to 2n inclusive. The probability distribution of R is given by

where k is a constant.

Show that k = n(n + 1) When n = 20

find the exact value of P (16 ≤ R < 26) When n = 20, a random value g of R is taken and the quadratic equation in x

is formed.

Find the exact probability that the equation has no real roots.

TOTAL FOR STATISTICS IS 50 MARKS