Edexcel A Level Maths Paper 3: Statistics and Mechanics November 2021 question paper 9MA0/03

A particle P moves with constant acceleration (2i − 3j) m s⁻² At time t = 0, P is moving with velocity 4im s⁻¹

Find the velocity of P at time t = 2 seconds. At time t = 0, the position vector of P relative to a fixed origin O is (i + j) m.

Find the position vector of P relative to O at time t = 3 seconds.

A small stone A of mass 3m is attached to one end of a string. A small stone B of mass m is attached to the other end of the string. Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα = The string passes over a pulley P that is fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is Stone A is released from rest and begins to move down the plane. The stones are modelled as particles. The pulley is modelled as being small and smooth. The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer. In reality, the string is not light.

State how this would affect the working in part (b).

A beam AB has mass m and length 2a. The beam rests in equilibrium with A on rough horizontal ground and with B against a smooth vertical wall. The beam is inclined to the horizontal at an angle θ, as shown in Figure 2. The coefficient of friction between the beam and the ground is μ The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall.

Using the model,

A horizontal force of magnitude kmg, where k is a constant, is now applied to the beam at A. This force acts in a direction that is perpendicular to the wall and towards the wall. Given that tanθ = , μ = and the beam is now in limiting equilibrium,

use the model to find the value of k.

A small stone is projected with speed 65 m s⁻¹ from a point O at the top of a vertical cliff. Point O is 70 m vertically above the point N. Point N is on horizontal ground. The stone is projected at an angle α above the horizontal, where tanα = The stone hits the ground at the point A, as shown in Figure 3. The stone is modelled as a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude 10 m s⁻²

Using the model,

find the time taken for the stone to travel from O to A,

find the speed of the stone at the instant just before it hits the ground at A. One limitation of the model is that it ignores air resistance.

State one other limitation of the model that could affect the reliability of your answers.

At time t seconds, a particle P has velocity vm s⁻¹, where

Find the acceleration of P at time t seconds, where t 0

Find the value of t at the instant when P is moving in the direction of i − j At time t seconds, where t 0, the position vector of P, relative to a fixed origin O, is r metres. When t = 1, r = −j

Find an expression for r in terms of t.

Find the exact distance of P from O at the instant when P is moving with

TOTAL FOR MECHANICS IS 50 MARKS

State one disadvantage of using quota sampling compared with simple random sampling. In a university 8% of students are members of the university dance club. A random sample of 36 students is taken from the university. The random variable X represents the number of these students who are members of the dance club.

Using a suitable model for X, find

P(X = 4)

P(X 7) Only 40% of the university dance club members can dance the tango.

Find the probability that a student is a member of the university dance club and can dance the tango. A random sample of 50 students is taken from the university.

Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango.

Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, x, in their last name

• the number of letters, y, in their first name

His results are shown in the scatter diagram on the next page.

Describe the correlation between x and y. Marc suggests that parents with long last names tend to give their children shorter first names.

Using the scatter diagram comment on Marc’s suggestion, giving a reason for your answer. The results from Marc’s random sample of 16 observations are given in the table below.

Use your calculator to find the product moment correlation coefficient between x and y for these data.

Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should

• state your hypotheses clearly

• use a 5% level of significance

Stav is studying the large data set for September 2015 He codes the variable Daily Mean Pressure, x, using the formula y = x − 1010 The data for all 30 days from Hurn are summarised by

State the units of the variable x

Find the mean Daily Mean Pressure for these 30 days.

Find the standard deviation of Daily Mean Pressure for these 30 days. Stav knows that, in the UK, winds circulate

• in a clockwise direction around a region of high pressure

• in an anticlockwise direction around a region of low pressure

The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015

The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order, W NE E You may assume that these 3 locations were under a single region of pressure.

Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table. Give a reason for your answer.

A large college produces three magazines. One magazine is about green issues, one is about equality and one is about sports. A student at the college is selected at random and the events G, E and S are defined as follows G is the event that the student reads the magazine about green issues E is the event that the student reads the magazine about equality S is the event that the student reads the magazine about sports The Venn diagram, where p, q, r and t are probabilities, gives the probability for each

Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and P(G) = 0.25

Find

the value of p

the value of q

find

the value of r

the value of t

Determine whether or not the events (S Eʹ) and G are independent. Show your working clearly.

The heights of females from a country are normally distributed with

• a mean of 166.5 cm

• a standard deviation of 6.1 cm

Given that 1% of females from this country are shorter than kcm,

find the value of k

Find the proportion of females from this country with heights between 150 cm and 175 cm A female, from this country, is chosen at random from those with heights between 150 cm and 175 cm

Find the probability that her height is more than 160 cm The heights of females from a different country are normally distributed with a standard deviation of 7.4 cm Mia believes that the mean height of females from this country is less than 166.5 cm Mia takes a random sample of 50 females from this country and finds the mean of her sample is 164.6 cm

Carry out a suitable test to assess Mia’s belief. You should

• state your hypotheses clearly

• use a 5% level of significance

The discrete random variable X has the following probability distribution

• a, b and c are distinct integers (a < b < c)

• all the probabilities are greater than zero

Find

the value of a

the value of b

the value of c Show your working clearly. The independent random variables X₁ and X₂ each have the same distribution as X

Find P(X₁ = X₂)

TOTAL FOR STATISTICS IS 50 MARKS