The continuous random variable \(X\) is uniformly distributed over the interval \([ 1 , d ]\).
a) The 90 th percentile of \(X\) is 19 . Find the value of \(d\).
b) Calculate the mean and standard deviation of \(X\).
| \(\mathbf { 0 }\) | \(\mathbf { 4 } \quad\) A bakery produces large loaves with masses, in grams, that are normally distributed |
with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
It is found that \(11 \%\) of the large loaves weigh more than 805 g and that \(20 \%\) of the large loaves weigh less than 795 g .
a) Find the values of \(\mu\) and \(\sigma\).
The bakery also produces small loaves with masses, in grams, that are normally distributed with mean 400 and standard deviation 9 .
Following a change of management at the bakery, a customer suspects that the mean mass of the small loaves has decreased. The customer weighs the next 15 small loaves that he purchases and calculates their mean mass to be 397 g .
b) Perform a hypothesis test at the \(5 \%\) significance level to investigate the customer's suspicion, assuming the standard deviation, in grams, is still 9.
c) State another assumption you have made in part (b).
5 A medical researcher is investigating possible links between diet and a particular disease. She selects a random sample of 22 countries and records the average daily calorie intake per capita from sugar and the percentage of the population who suffer from this disease.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Sugar consumption and rate of disease}
\includegraphics[alt={},max width=\textwidth]{9c111615-42d5-4804-8eb3-c20fe8d9faee-05_654_1264_591_461}
\end{figure}
There are 22 data points and the product moment correlation coefficient is \(0 \cdot 893\).
a) Stating your hypotheses clearly, show that these data could be used to suggest that there is a link between the disease and sugar consumption.
The medical researcher realises that her data is from the year 2000. She repeats her investigation with a random sample of 13 countries using new data from the year 2020. She produces the following graph.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Sugar consumption and rate of disease}
\includegraphics[alt={},max width=\textwidth]{9c111615-42d5-4804-8eb3-c20fe8d9faee-05_700_1273_1763_461}
\end{figure}
b) How should the researcher interpret the new data in the light of the data from 2000?
\section*{Section B: Differential Equations and Mechanics}
A particle \(P\) moves on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in directions east and north respectively. At time \(t\) seconds, the position vector of \(P\) is given by \(\mathbf { r }\) metres, where
$$\mathbf { r } = \left( t ^ { 3 } - 7 t ^ { 2 } \right) \mathbf { i } + \left( 2 t ^ { 2 } - 15 t + 11 \right) \mathbf { j }$$
a) i) Find an expression for the velocity vector of \(P\) at time \(t \mathrm {~s}\).
ii) Determine the value of \(t\) when \(P\) is moving north-east and hence write down the velocity of \(P\) at this value of \(t\).
b) Find the acceleration vector of \(P\) when \(t = 7\).
| \(\mathbf { 0 }\) | \(\mathbf { 7 } \quad\) A rod \(A B\), of mass 20 kg and length 3.2 m , is resting horizontally in equilibrium on two |
smooth supports at points \(X\) and \(Y\), where \(A X = 0.4 \mathrm {~m}\) and \(A Y = 2.4 \mathrm {~m}\). A particle of mass 8 kg is attached to the rod at a point \(C\), where \(B C = 0.2 \mathrm {~m}\). The reaction of the support at \(Y\) is four times the reaction of the support at \(X\).
You may not assume that the rod \(A B\) is uniform.
a) i) Find the magnitude of each of the reaction forces exerted on the rod at \(X\) and \(Y\).
ii) Show that the weight of the rod acts at the midpoint of \(A B\).
b) Is it now possible to determine whether the rod is uniform or non-uniform? Give a reason for your answer.
A boy kicks a ball from a point \(O\) on horizontal ground towards a vertical wall \(A B\). The initial speed of the ball is \(23 \mathrm {~ms} ^ { - 1 }\) in a direction that is \(18 ^ { \circ }\) above the horizontal. The diagram below shows a window \(C D\) in the wall \(A B\), such that \(B D = 1.1 \mathrm {~m}\) and \(B C = 2 \cdot 2 \mathrm {~m}\). The horizontal distance from \(O\) to \(B\) is 8 m .
\includegraphics[max width=\textwidth, alt={}, center]{9c111615-42d5-4804-8eb3-c20fe8d9faee-07_567_1540_605_274}
You may assume that the window will break if the ball strikes it with a speed of at least \(21 \mathrm {~ms} ^ { - 1 }\).
a) Show that the ball strikes the window and determine whether or not the window breaks.
b) Give one reason why your answer to part (a) may be unreliable.
The diagram below shows a wooden crate of mass 35 kg being pushed on a rough horizontal floor, by a force of magnitude 380 N inclined at an angle of \(30 ^ { \circ }\) below the horizontal. The crate, which may be modelled as a particle, is moving at a constant speed.
\includegraphics[max width=\textwidth, alt={}, center]{9c111615-42d5-4804-8eb3-c20fe8d9faee-08_394_665_573_701}
a) The coefficient of friction between the crate and the floor is \(\mu\). Show that
$$\mu = \frac { 190 \sqrt { 3 } } { 533 } .$$
Suppose instead that the crate is pulled with the same force of 380 N inclined at an angle of \(30 ^ { \circ }\) above the horizontal, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{9c111615-42d5-4804-8eb3-c20fe8d9faee-08_392_663_1425_701}
b) Without carrying out any further calculations, explain why the crate will no longer move at a constant speed.