Questions — SPS (1106 questions)

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SPS SPS FM Mechanics 2026 January Q3
3.
\includegraphics[max width=\textwidth, alt={}, center]{b3e54459-0d05-4858-8978-60fe3d4d1719-06_534_533_191_717} A light elastic string has natural length \(8 a\) and modulus of elasticity \(5 m g\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12 a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(A P = B P = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(A B\) it has speed \(\sqrt { 80 a g }\).
  1. Find \(L\) in terms of \(a\).
  2. Find the initial acceleration of \(P\) in terms of \(g\).
    [0pt] [Question 3 Continued] \section*{4.} A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt { \frac { 8 } { 3 } g a }\). Find \(x\) in terms of \(a\).
    [0pt] [Question 4 Continued]
SPS SPS FM Mechanics 2026 January Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{b3e54459-0d05-4858-8978-60fe3d4d1719-10_478_828_178_575}
\(A B\) and \(B C\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(A B C = 60 ^ { \circ }\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(A B\) at an angle \(\theta\) with \(A B\). It then strikes \(B C\) and rebounds at an angle \(\beta\) with \(B C\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is \(40 \%\) of its kinetic energy before the first collision.
  1. Find the value of \(e\).
  2. Find the size of angle \(\beta\).
    [0pt] [Question 5 Continued] \section*{6.}
    \includegraphics[max width=\textwidth, alt={}]{b3e54459-0d05-4858-8978-60fe3d4d1719-12_511_1145_296_452}
    A particle \(P\) of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius 0.8 m with angular speed \(\omega \operatorname { rad~s } ^ { - 1 }\). The particle \(Q\) moves in a horizontal circle of radius 1.4 m also with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centres of the circles are vertically below \(O\), and \(O , P\) and \(Q\) are always in the same vertical plane. The strings \(O P\) and \(P Q\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  3. Find the tension in the string \(O P\).
  4. Find the value of \(\omega\).
  5. Find the value of \(\beta\).
    [0pt] [Question 6 Continued]
SPS SPS FM Mechanics 2026 January Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{b3e54459-0d05-4858-8978-60fe3d4d1719-14_371_880_191_589} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac { 1 } { 2 } m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2 u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 5 } { 8 }\) and \(\alpha + \beta = 90 ^ { \circ }\).
  1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.
  2. Find the value of \(\tan \alpha\).
    [0pt] [Question 7 Continued]
SPS SPS FM Mechanics 2026 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b3e54459-0d05-4858-8978-60fe3d4d1719-16_286_933_201_459} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
  1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
  2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
  3. Find
    1. an expression for \(R\) in terms of \(W\)
    2. the value of \(\theta\)
      (8)
      [0pt] [Question 8 Continued] Spare space for extra working Spare space for extra working Spare space for extra working
SPS SPS SM Statistics 2026 January Q1
1. A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
  1. Explain in detail how the researcher should obtain such a sample.
  2. Give one advantage and one disadvantage of
    1. quota sampling,
    2. systematic sampling.
SPS SPS SM Statistics 2026 January Q2
2. 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]{fdff6575-679e-4d25-ad43-e9d343c1746f-06_346_1383_427_278}
  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 2026 January Q3
3. Researchers investigated the change in the numbers of people in employment using underground, metro, light rail or tram (UMLRT) between 2001 and 2011. The data are combined for those Local Authorities (LAs) with UMLRT stations into five regions: Birmingham, Liverpool, Manchester, Sheffield and Rotherham, and Tyne and Wear. Fig. 1 shows the total numbers of people in employment in those LAs. Fig. 2 shows the total numbers of people in employment who use UMLRT in those LAs. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 1} \includegraphics[alt={},max width=\textwidth]{fdff6575-679e-4d25-ad43-e9d343c1746f-08_834_1694_836_166}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 2} \includegraphics[alt={},max width=\textwidth]{fdff6575-679e-4d25-ad43-e9d343c1746f-08_833_1694_1822_166}
\end{figure}
  1. Use these charts to explain which of Birmingham and Liverpool has the larger proportion of people in employment who used UMLRT in 2011. One of the researchers says, "Between 2001 and 2011, the increase in the number of people in employment who use UMLRT is greatest in Tyne and Wear." Sam says, "But what matters more is which region has the greatest increase in the proportion of people in employment who use UMLRT."
  2. Give a reason why the planners responsible for the building of trains and the maintenance of infrastructure might disagree with Sam.
  3. Explain whether those responsible for encouraging the greater use of public transport would agree with Sam.
  4. The charts are compiled from data in the Large Data Set by using those LAs which contain UMLRT stations in each region. Explain a disadvantage of using these data.
SPS SPS SM Statistics 2026 January Q4
4. Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard.
Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.
    1. Find the mean number of times he falls off in a day.
  1. (ii) Find the variance of the number of times he falls off in a day.
    1. Find the probability that, on a particular day, he falls off exactly 10 times.
  2. (ii) Find the probability that, on a particular day, he falls off 5 or more times.
  3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
    1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.
  4. (ii) Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.
SPS SPS SM Statistics 2026 January Q5
5. The proportion of left-handed adults in a country is 10\%
Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10\%
She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.
  1. Find the critical region for a suitable test to assess Freya's belief. You should
    • state your hypotheses clearly
    • use a \(5 \%\) level of significance
    • state the probability of rejection in each tail
    • Given the null hypothesis is true what is the probability of it being rejected in part (a)?
    In Freya's sample 7 adults were left-handed.
  2. With reference to your answer in part (a) comment on Freya's belief. \section*{6.}
SPS SPS SM Statistics 2026 January Q6
6. Skilled operators make a particular component for an engine. The company believes that the time taken to make this component may be modelled by the normal distribution. They timed one of their operators, Sheila, over a long period. They find that when she makes a component, she takes over 90 minutes to make one \(10 \%\) of the time, and that \(20 \%\) of the time, a component was less than 70 minutes to make. Estimate the mean and standard deviation of the time Sheila takes to make a component.
SPS SPS SM Statistics 2026 January Q7
7. A team game involves solving puzzles to escape from a room.
Using data from the past, the mean time to solve the puzzles and escape from one of these rooms is 65 minutes with a standard deviation of 11.3 minutes. After recent changes to the puzzles in the room, it is claimed that the mean time to solve the puzzles and escape has changed. To test this claim, a random sample of 100 teams is selected.
The total time to solve the puzzles and escape for the 100 teams is 6780 minutes.
Assuming that the times are normally distributed, test at the \(2 \%\) level the claim that the mean time has changed.
SPS SPS SM Statistics 2026 January Q8
8. The discrete random variable \(R\) takes even integer values from 2 to \(2 n\) inclusive.
The probability distribution of \(R\) is given by $$\mathrm { P } ( R = r ) = \frac { r } { k } \quad r = 2,4,6 , \ldots , 2 n$$ where \(k\) is a constant.
  1. Show that \(k = n ( n + 1 )\) When \(n = 20\)
  2. find the exact value of \(\mathrm { P } ( 16 \leqslant R < 26 )\) When \(n = 20\), a random value \(g\) of \(R\) is taken and the quadratic equation in \(x\) $$x ^ { 2 } + g x + 3 g = 5$$ is formed.
  3. Find the exact probability that the equation has no real roots.
SPS SPS SM Statistics 2026 January Q9
9. The Venn diagram, where \(p , q\) and \(r\) are probabilities, shows the events \(A , B , C\) and \(D\) and associated probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{fdff6575-679e-4d25-ad43-e9d343c1746f-22_623_1130_326_438}
  1. State any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) The events \(B\) and \(C\) are independent.
  2. Find the value of \(p\)
  3. Find the greatest possible value of \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\) Given that \(\mathrm { P } \left( B \mid A ^ { \prime } \right) = 0.5\)
  4. find the value of \(q\) and the value of \(r\)
  5. Find \(\mathrm { P } \left( [ A \cup B ] ^ { \prime } \cap C \right)\)
  6. Use set notation to write an expression for the event with probability \(p\)
SPS SPS FM Statistics 2026 January Q1
1. At a wine-tasting competition, two judges give marks out of 100 to 7 wines as follows.
Wine\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Judge I86.387.587.688.889.489.990.5
Judge II85.388.182.787.789.089.491.5
A spectator claims that there is a high level of agreement between the rank orders of the marks given by the two judges. Test the spectator's claim at the \(1 \%\) significance level.
[0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q2
2. At a toy factory, wooden blocks of approximate heights \(20 \mathrm {~mm} , 30 \mathrm {~mm}\) and 50 mm are made in red, yellow and green respectively. The heights of the blocks in mm are modelled by independent random variables which are Normally distributed with means and standard deviations as shown in the table.
ColourMeanStandard deviation
Red200.8
Yellow300.9
Green501.2
In parts (a), (b) and (c), the blocks are selected randomly and independently of one another.
  1. Find the probability that the height of a red block is less than 19 mm .
  2. A tower is made of 15 blocks stacked on top of each other consisting of 5 red blocks, 5 yellow blocks and 5 green blocks. Determine the probability that the tower is at least 495 mm high.
  3. Determine the probability that a tower made of 3 red blocks will be at least 1 mm higher than a tower made of 2 yellow blocks.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q3
4 marks
3. A student is investigating the relationship between different electricity generation methods and cost of electricity in a particular country. The student first checks whether there is any correlation between the cost per unit of electricity, \(x\) euros, and the amount of electricity being generated by wind, \(y \mathrm { GW }\). The data from 30 observations are summarised as follows.
\(n = 30 \quad \sum x = 2.219 \quad \sum y = 357.7 \quad \sum x ^ { 2 } = 0.2368 \quad \sum y ^ { 2 } = 4648 \quad \sum x y = 25.01\)
  1. In this question you must show detailed reasoning. Determine the product moment correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) level to investigate whether there is any correlation between the cost per unit of electricity and the amount of electricity generated by wind.
    [0pt] [4]
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q4
4. The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).
  1. Find
    (a) \(\mathrm { P } ( X + Y ) > 40\),
    (b) \(\operatorname { Var } ( 2 X - Y )\).
  2. State a necessary assumption for your calculations in part (i) to be valid.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q5
5. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 3 ^ { 2 } \right)\). A random sample of 9 observations of \(X\) produced the following values. $$\begin{array} { l l l l l l l l l } 6 & 2 & 3 & 6 & 8 & 11 & 12 & 5 & 10 \end{array}$$
  1. Find a \(90 \%\) confidence interval for \(\mu\).
  2. Explain what is meant by a \(90 \%\) confidence interval in this context.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q6
6. A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(x\) minutes, by an exponential distribution with probability density function (PDF) $$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0
0 & x < 0 \end{cases}$$
  1. In this question you must show detailed reasoning. The mean waiting time is found to be 5.0 minutes. Show that \(\lambda = 0.2\).
    ii) Use the model to calculate the probability that a customer has to wait longer than 20 minutes for a response. In practice it is found that no customer waits for more than 15 minutes for a response. The statistician constructs an improved model to incorporate this fact.
    iii) Sketch the following on the same axis.
    (a) the PDF of the model using the exponential distribution,
    (b) a possible PDF for the improved model.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q7
10 marks
7. A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, \(x\) micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the \(10 \%\) significance level, whether the machine is producing paper of the designed thickness.
[0pt] [10]
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SPS SPS FM Statistics 2026 January Q8
8. Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
  1. How many different arrangements of the letters are possible?
  2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
  3. Find the probability that at least one of these 2 cards has D printed on it.
    [0pt] [BLANK PAGE]
SPS SPS FM Statistics 2026 January Q9
9 A continuous random variable \(X\) has probability density function given by the following function, - where \(a\) is a constant.
\(\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise. } \end{array} \right\}\)
The expected value of \(X\) is 4 .
  1. Show that \(a = 6\). Five independent observations of \(X\) are obtained, and the largest of them is denoted by \(M\).
  2. Find the cumulative distribution function of \(M\).
    [0pt] [BLANK PAGE]
SPS SPS SM Mechanics 2026 January Q1
  1. A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible.
Calculate the speed of projection of the particle and also the maximum height it reaches. \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-05_2688_1886_118_118}
SPS SPS SM Mechanics 2026 January Q2
2. The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-07_2688_1886_118_118}
SPS SPS SM Mechanics 2026 January Q3
3. A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fb36606d-0ee3-4050-af31-1642e5f67a03-08_479_497_296_849} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the upward acceleration.
  2. Find the reaction on the man of the lift floor. \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-09_2688_1886_118_118}