Questions — SPS (1106 questions)

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SPS SPS FM Mechanics 2023 January Q4
4. A uniform ladder \(A B\) of length 5 m and mass 8 kg is placed at an angle \(\theta\) to the horizontal, with \(A\) on rough horizontal ground and \(B\) against a smooth vertical wall. The coefficient of friction between the ladder and the ground is 0.4 .
  1. By taking moments, find the smallest value of \(\theta\) for which the ladder is in equilibrium.
  2. A man of mass 75 kg stands on the ladder when \(\theta = 60 ^ { \circ }\). Find the greatest distance from \(A\) that he can stand without the ladder slipping.
    [0pt] [Question 4 Continued]
SPS SPS FM Mechanics 2023 January Q5
5. Two smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide
  • the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
  • the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 6).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15f4500a-8eb8-4b5f-896c-de730272a35b-12_451_961_406_255} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision find the coefficient of restitution between \(A\) and \(B\).
[0pt] [Question 5 Continued] \section*{6.}
SPS SPS FM Mechanics 2023 January Q6
6. A light elastic string of natural length \(a\) has modulus of elasticity \(k m g\), where \(k\) is a constant. One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle of mass \(m\). The particle moves, with the string stretched, in a horizontal circle with constant angular speed \(\omega\), with the centre of the circle vertically below \(O\).
  1. Show that, if the string makes a constant angle \(\theta\) with the vertical, $$\cos \theta = \frac { k g - a \omega ^ { 2 } } { k a \omega ^ { 2 } }$$
  2. Show that \(\omega < \sqrt { \frac { k g } { a } }\)
    [0pt] [Question 6 Continued] Spare space for extra working Spare space for extra working Spare space for extra working Spare space for extra working
    [0pt] [End of Examination]
SPS SPS FM Statistics 2023 January Q1
1. Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
  1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. Indre is allowed 20 minutes of break time during each 4 -hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
  2. Find the probability that in exactly 3 of these periods there were no calls.
SPS SPS FM Statistics 2023 January Q2
2. A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams. To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows $$\begin{array} { l l l l l l l l } 508 & 510 & 500 & 500 & 498 & 503 & 508 & 505 \end{array}$$ Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,
  1. find a \(95 \%\) confidence interval for the mean weight, \(\mu\) grams, of yoghurt in a pot. Give your answers to 2 decimal places.
SPS SPS FM Statistics 2023 January Q3
3. A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams \(/ \mathrm { m } ^ { 2 }\). The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w ^ { 2 } = 13447 \quad \mathrm {~S} _ { f f } = 42 \quad \mathrm {~S} _ { f w } = 269.5$$
  1. Calculate the product moment correlation coefficient between \(f\) and \(w\)
  2. Interpret the value of your product moment correlation coefficient.
  3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + b f\)
SPS SPS FM Statistics 2023 January Q4
4. Sweet pea plants grown using a standard plant food have a mean height of 1.6 m . A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$\begin{aligned} n & = 49
\Sigma x & = 74.48
\Sigma x ^ { 2 } & = 120.8896 \end{aligned}$$
  1. Test, at the \(5 \%\) significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m .
  2. State with a reason whether you needed to use the Central Limit Theorem to carry out the test in part (a).
SPS SPS FM Statistics 2023 January Q5
5. Nine athletes, \(A , B , C , D , E , F , G , H\) and \(I\), competed in both the 100 m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85
  1. Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher an athlete's position is in the 100 m sprint, the higher their position is in the long jump. Use a \(5 \%\) level of significance. The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes \(B , C\) and \(D\) over their long jump results. The table shows the results that are agreed to be correct.
    Athlete\(A\)B\(C\)D\(E\)\(F\)G\(H\)I
    Position in \(\mathbf { 1 0 0 ~ m }\) sprint467928315
    Position in long jump549312
    Given that there were no tied ranks,
  2. find the correct positions of athletes \(B , C\) and \(D\) in the long jump. You must show your working clearly and give reasons for your answers.
    (5)
  3. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100 m sprint and the long jump.
    (2)
SPS SPS FM Statistics 2023 January Q6
6. A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, \(W \mathrm {~kg}\), of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, \(X \mathrm {~kg}\), of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables \(W\) and \(X\) are independent.
  1. Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy.
    (6) The manufacturer packs \(n\) of these wooden toys and \(2 n\) of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
  2. Calculate the value of \(n\). END OF TEST
SPS SPS SM Mechanics 2023 January Q1
  1. 10 seconds after passing a warning signal, a train is travelling at \(18 m s ^ { - 1 }\) and has gone 215 m beyond the signal. Find the acceleration (assumed to be constant) of the train during the 10 seconds and its velocity as it passed the signal.
\section*{BLANK PAGE FOR WORKING}
SPS SPS SM Mechanics 2023 January Q2
2. A particle of mass \(m\) is placed on a rough inclined plane.
The plane makes an angle \(\theta\) with the horizontal.
The coefficient of friction between the particle and the plane is \(\mu\) where \(\mu < \tan \theta\). The particle is released from rest and accelerates down the plane.
  1. Draw a fully labelled diagram to show the forces acting on the particle.
  2. Find an expression in terms of \(g , \theta\) and \(\mu\) for the acceleration of the particle.
  3. Explain what would happen to the particle if \(\mu > \tan \theta\). \section*{BLANK PAGE FOR WORKING}
SPS SPS SM Mechanics 2023 January Q3
  1. A parcel \(P\) of weight 50 N is being held in equilibrium by two light, inextensible strings \(A P\) and \(B P\). The string \(A P\) is attached to a wall at \(A\), and string \(B P\) passes over a smooth pulley which is at the same height as \(A\), as shown in the diagram.
When the tension in \(B P\) is 40 N , the strings are at right angles to each other.
\includegraphics[max width=\textwidth, alt={}, center]{4109fba0-077e-472b-b37f-7ac2e45aacc7-08_531_768_479_699}
  1. Find the tension in string \(A P\).
  2. Explain why the parcel can never be in equilibrium, with both strings horizontal. \section*{BLANK PAGE FOR WORKING}
SPS SPS SM Mechanics 2023 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4109fba0-077e-472b-b37f-7ac2e45aacc7-10_680_1218_141_466} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,
  1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~ms} ^ { - 2 }\)
  2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
  3. State one limitation of the model that will affect the accuracy of your answer to part (a). \section*{BLANK PAGE FOR WORKING}
SPS SPS SM Mechanics 2023 January Q5
  1. At time \(t\) seconds, where \(0 \leq t \leq T\), a particle, \(P\), moves so that its velocity \(v m s ^ { - 1 }\) is given by
$$v = 7.2 t - 0.45 t ^ { 2 }$$ When \(t = 0\) the particle is sitting stationary at a displacement of \(\mathrm { d } m\) from a point O .
The particle's acceleration is zero when \(t = T\).
  1. Find the value of \(T\). For \(t \geq T\), the particle moves with a velocity \(v = 48 - 2.4 t m s ^ { - 1 }\).
  2. Find the time when \(P\) is at its maximum displacement from O . The particle passes through the point O when \(t = 38\).
  3. Find \(d\). \section*{BLANK PAGE FOR WORKING}
SPS SPS SM Mechanics 2023 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4109fba0-077e-472b-b37f-7ac2e45aacc7-14_334_787_212_680} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\)
The string passes over a small smooth pulley, \(P\), 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.
Block \(B\) hangs freely below \(P\), as shown in Figure 1.
The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\)
The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.
  1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
  2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
  3. Suggest two refinements to the model that would make it more realistic. \section*{BLANK PAGE FOR WORKING}
SPS SPS SM Statistics 2023 January Q1
1.
  1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.
    Age group0 to 4
    Frequency2143
    Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
  2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.
SPS SPS SM Statistics 2023 January Q2
2. Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town \(B\).
  1. In the Printed Answer Booklet, complete the two-way frequency table.
    \cline { 2 - 4 } \multicolumn{1}{c|}{}Town
    \cline { 2 - 4 } \multicolumn{1}{c|}{}ABCTotal
    adult
    child
    Total
  2. One of the people is chosen at random.
    1. Find the probability that this person is an adult from town A .
    2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 .
      Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756
      Jane used the same figures to write down the following sample of five student numbers.
      \(114,287,562,81\) and 817
    1. State, with a reason, which one of these samples is not random.
    2. Explain why Jane omitted the number 859 from her sample.
SPS SPS SM Statistics 2023 January Q3
3. Pierre is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.
SPS SPS SM Statistics 2023 January Q4
4.
  1. The masses, in grams, of plums of a certain kind have the distribution \(\mathrm { N } ( 55,18 )\). The heaviest \(5 \%\) of plums are classified as extra large. Find the minimum mass of extra large plums.
  2. The masses, in grams, of apples of a certain kind have the distribution \(\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)\). It is given that half of the apples have masses between 62 g and 72 g . Determine \(\sigma\).
SPS SPS SM Statistics 2023 January Q5
5. 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.
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-08_381_1557_504_264}
  1. Find the range of the times.
  2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
  3. calculate the mean time taken to complete the puzzle,
  4. 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.
  5. 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).
SPS SPS SM Statistics 2023 January Q6
6. The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 .
This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the \(5 \%\) significance level, whether the mean level of pollutant has changed.
SPS SPS SM Statistics 2023 January Q7
7. 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 subset.
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-12_533_903_790_548}
  1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
  2. Find
    1. the value of \(p\)
    2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
  3. find
    1. the value of \(r\)
    2. the value of \(t\)
  4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. END OF TEST
SPS SPS FM 2022 November Q1
1.
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-05_1031_938_262_529} The Argand diagram above shows a half-line \(l\) and a circle \(C\). The circle has centre 3 i and passes through the origin.
  1. Write down, in complex number form, the equations of \(l\) and \(C\).
    [0pt]
  2. Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]
SPS SPS FM 2022 November Q2
2.
  1. Show that \(\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }\).
  2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
  3. State, giving a brief reason, whether the series \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }\) converges.
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 November Q3
3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of \(a\).
[0pt] [BLANK PAGE]