Edexcel A Level Maths Paper 3: Statistics and Mechanics October 2020 question paper 9MA0/03

A rough plane is inclined to the horizontal at an angle α, where tanα = A brick P of mass m is placed on the plane. The coefficient of friction between P and the plane is μ Brick P is in equilibrium and on the point of sliding down the plane. Brick P is modelled as a particle. Using the model,

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

For parts (c) and (d), you are not required to do any further calculations. Brick P is now removed from the plane and a much heavier brick Q is placed on the plane. The coefficient of friction between Q and the plane is also

Explain briefly why brick Q will remain at rest on the plane. Brick Q is now projected with speed 0.5 m s⁻¹ down a line of greatest slope of the plane. Brick Q is modelled as a particle. Using the model,

describe the motion of brick Q, giving a reason for your answer.

A particle P moves with acceleration (4i − 5j) m s⁻² At time t = 0, P is moving with velocity (−2i + 2j) m s⁻¹

Find the velocity of P at time t = 2 seconds. At time t = 0, P passes through the origin O. At time t = T seconds, where T 0, the particle P passes through the point A. The position vector of A is (λi − 4.5j)m relative to O, where λ is a constant.

Find the value of T.

Hence find the value of λ

At time t seconds, where t 0, a particle P moves so that its acceleration am s⁻² is given by

At the instant when t = 0, the velocity of P is 36im s⁻¹

Find the velocity of P when t = 4

Find the value of t at the instant when P is moving in a direction perpendicular to i

At time t seconds, where t 0, a particle Q moves so that its position vector r metres, relative to a fixed origin O, is given by

Find the value of t at the instant when the speed of Q is 5 m s⁻¹

A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C. The point C is at a vertical height 4a above the ground. The vertical plane containing AB is perpendicular to the rail. The ladder is inclined to the horizontal at an angle α, where sinα = , as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. The ladder rests in limiting equilibrium. The ladder is modelled as a uniform rod. Using the model,

show that the magnitude of the force exerted on the ladder by the rail at C is

Hence, or otherwise, find the value of μ.

A small ball is projected with speed Um s⁻¹ from a point O at the top of a vertical cliff. The point O is 25 m vertically above the point N which is on horizontal ground. The ball is projected at an angle of 45° above the horizontal. The ball hits the ground at a point A, where AN = 100 m, as shown in Figure 2. The motion of the ball is modelled as that of a particle moving freely under gravity. Using this initial model,

show that U = 28

find the greatest height of the ball above the horizontal ground NA. In a refinement to the model of the motion of the ball from O to A, the effect of air resistance is included. This refined model is used to find a new value of U.

How would this new value of U compare with 28, the value given in part (a)?

State one further refinement to the model that would make the model more realistic.

TOTAL FOR MECHANICS IS 50 MARKS

The Venn diagram shows the probabilities associated with four events, A, B, C and D

Write down any pair of mutually exclusive events from A, B, C and D Given that P(B) = 0.4

find the value of p Given also that A and B are independent

find the value of q Given further that P(Bʹ| C) = 0.64

find

the value of r

the value of s

A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.

Describe the correlation. The variable on the x-axis is Daily Mean Temperature measured in °C.

Using your knowledge of the large data set,

suggest which variable is on the y-axis,

state the units that are used in the large data set for this variable. Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow. He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains r = −0.377

Carry out a suitable test to investigate Stav’s belief at a 5% level of significance. State clearly ● your hypotheses ● your critical value On a random day at Heathrow the Daily Maximum Relative Humidity was 97%

Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.

Each member of a group of 27 people was timed when completing a puzzle. The time taken, x minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot.

Find the range of the times.

Find the interquartile range of the times. For these 27 people Σ x = 607.5 and Σ x² = 17 623.25

calculate the mean time taken to complete the puzzle,

calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.

State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in a minutes and b minutes respectively, where a b. When their times are included with the data of the other 27 people ● the median time increases ● the mean time does not change

Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions.

Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

The discrete random variable D has the following probability distribution

where k is a constant.

Show that the value of k is

The random variables D₁ and D₂ are independent and each have the same distribution as D.

Find P (D₁ + D₂ = 80) Give your answer to 3 significant figures. A single observation of D is made. The value obtained, d, is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral Q

Find the exact probability that the smallest angle of Q is more than 50°

A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.

Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients’ complaint. The health centre also claims that the time a dentist spends with a patient during a routine appointment, T minutes, can be modelled by the normal distribution where T~ N(5, 3.5²)

Using this model,

find the probability that a routine appointment with the dentist takes less than 2 minutes

find P(T < 2 | T 0)

hence explain why this normal distribution may not be a good model for T. The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of T 2

Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.

TOTAL FOR STATISTICS IS 50 MARKS