Edexcel A Level Maths Paper 3: Statistics and Mechanics June 2023 question paper 9MA0/03

A car is initially at rest on a straight horizontal road. The car then accelerates along the road with a constant acceleration of 3.2 m s^–2 Find

the speed of the car after 5 s,

the distance travelled by the car in the first 5 s.

A particle P has mass 5 kg. The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N. The only resistance to motion is a frictional force of magnitude F newtons, as shown in Figure 1.

Find the magnitude of the normal reaction of the plane on P The particle is accelerating along the plane at 1.4 m s^–2

Find the value of F The coefficient of friction between P and the plane is μ

Find the value of μ, giving your answer to 2 significant figures.

At time t seconds, where t 0, a particle P has velocity vm s^–1 where

Find

the speed of P at time t = 0

the value of t when P is moving parallel to (i + j)

the acceleration of P at time t seconds

the value of t when the direction of the acceleration of P is perpendicular to i

[In this question, i and j are horizontal unit vectors and position vectors are given relative to a fixed origin O ] A particle P is moving on a smooth horizontal plane. The particle has constant acceleration (2.4i + j) m s^–2 At time t = 0, P passes through the point A. At time t = 5 s, P passes through the point B. The velocity of P as it passes through A is (–16i – 3j) m s^–1

Find the speed of P as it passes through B. The position vector of A is (44i – 10j) m. At time t = T seconds, where T 5, P passes through the point C. The position vector of C is (4i + cj) m.

Find the value of T.

Find the value of c.

A small ball is projected with speed 28 m s^–1 from a point O on horizontal ground. After moving for T seconds, the ball passes through the point A. The point A is 40 m horizontally and 20 m vertically from the point O, as shown in Figure 2. The motion of the ball from O to A is modelled as that of a particle moving freely under gravity. Given that the ball is projected at an angle α to the ground, use the model to

show that tan²α – 4 tanα + 3 = 0

find the greatest possible height, in metres, of the ball above the ground as the ball moves from O to A. The model does not include air resistance.

State one other limitation of the model. _____________________________________________________________________________________

A rod AB has mass M and length 2a. The rod has its end A on rough horizontal ground and its end B against a smooth vertical wall. The rod makes an angle θ with the ground, as shown in Figure 3. The rod is at rest in limiting equilibrium.

State the direction (left or right on Figure 3 above) of the frictional force acting on the rod at A. Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at B is S. In an initial model, the rod is modelled as being uniform. Use this initial model to answer parts (b), (c) and (d).

By taking moments about A, show that

The coefficient of friction between the rod and the ground is μ

find the value of μ

find, in terms of M and g, the magnitude of the resultant force acting on the rod at A. In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to B than it is to A. A new value for S is calculated using this new model, with tanθ =

State whether this new value for S is larger, smaller or equal to the value that S would take using the initial model. Give a reason for your answer.

TOTAL FOR MECHANICS IS 50 MARKS

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

Find P(A) The events B and C are independent.

Find the value of p and the value of q

Find P(AB′)

A machine fills packets with sweets and of the packets also contain a prize. The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable T represents the number of packets of sweets that contain a prize in each box.

State a condition needed for T to be modelled by B(40, A box is selected at random.

the probability that the box has exactly 6 packets containing a prize,

the probability that the box has fewer than 3 packets containing a prize. Kamil’s sweet shop buys 5 boxes of these sweets.

Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than A random sample of 110 packets is taken and 9 packets contain a prize.

Use a suitable test to assess Kamil’s claim. You should

• state your hypotheses clearly

• use a 5% level of significance

Ben is studying the Daily Total Rainfall, xmm, in Leeming for 1987 He used all the data from the large data set and summarised the information in the following table.

Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation.

Calculate estimates for

the mean Daily Total Rainfall,

the standard deviation of the Daily Total Rainfall. Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987

Using your knowledge of the large data set,

give a reason why these data would not be suitable,

state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987

A study was made of adult men from region A of a country. It was found that their heights were normally distributed with a mean of 175.4 cm and standard deviation 6.8 cm.

Find the proportion of these men that are taller than 180 cm. A student claimed that the mean height of adult men from region B of this country was different from the mean height of adult men from region A. A random sample of 52 adult men from region B had a mean height of 177.2 cm The student assumed that the standard deviation of heights of adult men was 6.8 cm both for region A and region B.

Use a suitable test to assess the student’s claim. You should

• state your hypotheses clearly

• use a 5% level of significance

Find the p-value for the test in part (b)

Tisam is playing a game. She uses a ball, a cup and a spinner. The random variable X represents the number the spinner lands on when it is spun. The probability distribution of X is given in the following table

where a, b, c and d are probabilities. To play the game

• the spinner is spun to obtain a value of x

• Tisam then stands x cm from the cup and tries to throw the ball into the cup

The event S represents the event that Tisam successfully throws the ball into the cup. To model this game Tisam assumes that

• P(S {X = x}) should be the same whatever value of x is obtained from the spinner

Using Tisam’s model,

find the probability distribution of X Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm. He successfully gets the ball in the cup 30% of the time.

State, giving a reason, why Tisam’s model of this game is not suitable to describe Nav playing the game for all values of X

A medical researcher is studying the number of hours, T, a patient stays in hospital following a particular operation. The histogram on the page opposite summarises the results for a random sample of 90 patients.

Use the histogram to estimate P(10 < T < 30) For these 90 patients the time spent in hospital following the operation had

• a mean of 14.9 hours

• a standard deviation of 9.3 hours

Tomas suggests that T can be modelled by N(14.9, 9.3²)

With reference to the histogram, state, giving a reason, whether or not Tomas’ model could be suitable. Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation

where

• x is measured in tens of hours

• k is a constant

Use algebraic integration to show that

Show that, for Xiang’s model, k = 99 to the nearest integer.

Estimate P(10 < T < 30) using

Tomas’ model of T ~ N(14.9, 9.3²)

Xiang’s curve with equation y = 99xe^–x and the answer to part (c) The researcher decides to use Xiang’s curve to model P(a T b)

State one limitation of Xiang’s model.

TOTAL FOR STATISTICS IS 50 MARKS