Edexcel Statistics & Mechanics Past Papers and solutions.

Questions and Worked Solutions for Edexcel Statistics & Mechanics Specimen Paper 3 2018, 9MA0/03.

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Edexcel January 2018 Statistics & Mechanics Specimen Paper 3 (pdf)

- Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90
packets of Kaff coffee from her stock. The results are shown in the table below.

A histogram is drawn and the class 245 ≤ w < 248 is represented by a rectangle of width 1.2 cm and height 10 cm.

(a) Calculate the width and the height of the rectangle representing the class 255 ≤ w < 260

(b) Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.

(c) Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place.

The seller claims that the mean mass of the contents of the packets is more than the stated mass.

Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g,

(d) test, using a 5% level of significance, whether or not the seller’s claim is justified. State your hypotheses clearly.

(You may assume that the mass of the contents of a packet is normally distributed.)

(e) Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed. - A researcher believes that there is a linear relationship between daily
mean temperature and daily total rainfall. The 7 places in the northern
hemisphere from the large data set are used. The mean of the daily mean
temperatures, t °C, and the mean of the daily total rainfall, s mm, for the
month of July in 2015 are shown on the scatter diagram below.

(a) With reference to the scatter diagram, explain why a linear regression model may not be suitable for the relationship between t and s.

The researcher calculated the product moment correlation coefficient for the 7 places and obtained r = 0.658

(b) Stating your hypotheses clearly, test at the 10% level of significance, whether or not the product moment correlation coefficient for the population is greater than zero.

(c) Using your knowledge of the large data set, suggest the names of the 2 places labelled G and H.

(d) Using your knowledge from the large data set, and with reference to the locations of the 2 places labelled G and H, give a reason why these places have the highest temperatures in July.

(e) Suggest how you could make better use of the large data set to investigate the relationship between daily mean temperature and daily total rainfall. - For a particular type of bulb, 36% grow into plants with blue flowers and the remainder
grow into plants with white flowers. Bulbs are sold in mixed bags of 40.

Russell selects a random sample of 5 bags of bulbs.

(a) Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers Maggie takes a random sample of n bulbs.

Using a normal approximation, the probability that more than 244 of these n bulbs will grow into blue flowers is 0.0521 to 4 decimal places.

(b) Find the value of n.

- The Venn diagram shows the probabilities of students’ lunch boxes containing a drink,
sandwiches and a chocolate bar.

D is the event that a lunch box contains a drink,

S is the event that a lunch box contains sandwiches,

C is the event that a lunch box contains a chocolate bar,

u, v and w are probabilities.

(a) Write down P(S ∩ D')

One day, 80 students each bring in a lunch box.

Given that all 80 lunch boxes contain sandwiches and a drink,

(b) estimate how many of these 80 lunch boxes will contain a chocolate bar.

(c) calculate the value of u, the value of v and the value of w. - The lifetimes of batteries sold by company X are normally distributed, with mean 150 hours
and standard deviation 25 hours.

A box contains 12 batteries from company X.

(a) Find the expected number of these batteries that have a lifetime of more than 160 hours.

The equation of the regression line of r on h for these 10 days is r = −12.8 + 0.15h

(d) Give an interpretation of the gradient of this regression line.

The lifetimes of batteries sold by company Y are normally distributed, with mean 160 hours and 80% of these batteries have a lifetime of less than 180 hours.

(b) Find the standard deviation of the lifetimes of batteries from company Y.

Both companies sell their batteries for the same price.

(c) State which company you would recommend. Give reasons for your answer. - A particle, P, moves with constant acceleration (i – 2j) m s
^{-2}.

At time t = 0 seconds, the particle is at the point A with position vector (2i + 5j) m and is moving with velocity u m s^{-1}.

At time t = 3 seconds, P is at the point B with position vector (–2.5i + 8j) m.

Find u. - A particle, P, moves under the action of a single force in such a way that at time t seconds,
where t ≥ 0, its velocity v m s
^{-1}is given by

v = (t^{2}– 3t) i –12t j

The mass of P is 0.5 kg.

Find the time at which the magnitude of the force acting on P is 6.5 N. - A small box of mass 3 kg moves on a rough plane which is inclined at an angle of 20o to
the horizontal.

The box is pulled up a line of greatest slope of the plane using a rope which is attached to the box.

The rope makes an angle of 30o with the plane, as shown in Figure 1.

The rope lies in the vertical plane which contains a line of greatest slope of the plane.

The coefficient of friction between the box and the plane is 0.3.

The tension in the rope is 25 N.

The box is modelled as a particle, the rope is modelled as a light inextensible string and air resistance is ignored.

(a) Using the model, find the acceleration of the box.

(b) Suggest one improvement to the model that would make it more realistic. The rope now breaks and the box slows down and comes to rest.

(c) Show that, after the box comes to rest, it immediately starts to move down the plane. - A beam CD, of mass 20 kg and length 3 m, is smoothly hinged to a vertical wall at one end C.

The beam is held in equilibrium in a horizontal position by a rope of length 1 m.

One end of the rope is fixed to a point F on the wall which is vertically above C.

The other end of the rope is fixed to the point F on the beam so that angle CEF is 30°, as shown in Figure 2. The beam is modelled as a uniform rod and the rope is modelled as a light inextensible string.

Using the model, find - A tennis player serves a ball so as to pass over the net.

The ball is given an initial velocity of 45 m s^{-1}in a direction 10o below the horizontal.

The ball is struck at a point O which is 3.5 m vertically above the point A which is on horizontal ground.

The bottom of the net is the point B which is on the ground and AB = 12 m.

The height of the net is 1 m, as shown in Figure 3.

The ball is modelled as a particle moving freely under gravity.

The ball passes over the net at a point which is vertically above B. Using the model,

(a) find, in centimetres to 2 significant figures, the distance between the ball and the top of the net, as the ball passes over the net,

(b) find, to 2 significant figures, the speed of the ball as it passes over the net.

(c) State two limitations of the model that could affect the reliability of your answers.

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