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SPS SPS FM Mechanics 2021 January Q6
11 marks Challenging +1.2
Numerical (calculator) integration is not acceptable in this question. \includegraphics{figure_4} The shaded region \(OAB\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac{1}{4}(x-2)^3 + 2\). The point \(A\) has coordinates \((4, 4)\) and the point \(B\) has coordinates \((4, 0)\). A uniform lamina \(L\) has the shape of \(OAB\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \((\bar{x}, \bar{y})\). Given that the area of \(L\) is \(8\)cm²,
  1. show that \(\bar{y} = \frac{8}{7}\). [4]
  2. The lamina is freely suspended from \(A\) and hangs in equilibrium with \(AB\) at an angle \(\theta°\) to the downward vertical. Find the value of \(\theta\). [7]
SPS SPS FM Statistics 2021 January Q1
4 marks Moderate -0.3
Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes.
  1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place. [2 marks]
  2. Alan claims that his mean journey time to work is 30 minutes. State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim. [1 mark]
  3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6. Explain how the working in part (a) would change. [1 mark]
SPS SPS FM Statistics 2021 January Q2
8 marks Standard +0.3
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. [3]
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.
  1. Find the probability that in exactly 3 of these periods there were no calls. [2]
On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
  1. Find the probability that Indre missed exactly 1 call in each of these 2 breaks. [3]
SPS SPS FM Statistics 2021 January Q3
7 marks Moderate -0.3
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/m². The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w^2 = 13447 \quad S_{ff} = 42 \quad S_{fw} = 269.5$$
  1. Calculate the product moment correlation coefficient between \(f\) and \(w\) [2]
  2. Interpret the value of your product moment correlation coefficient. [1]
  3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + bf\) [3]
  4. Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams/m² [1]
SPS SPS FM Statistics 2021 January Q4
7 marks Standard +0.3
The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x \leq 0 \\ k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\ 1 & x > 2 \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{2}\) [1]
  2. Showing your working clearly, use calculus to find
    1. E(\(X\))
    2. the mode of \(X\)
    [6]
SPS SPS FM Statistics 2021 January Q5
9 marks Standard +0.3
A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text{and} \quad \sum x^2 = 10198$$ Investigate the shopkeeper's claim using the 5\% level of significance. State any assumptions that you make. [9 marks]
SPS SPS FM Statistics 2021 January Q6
12 marks Standard +0.3
A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac{1}{3}\) that it lands on blue. The spinner is spun repeatedly. The random variable \(B\) represents the number of the spin when the spinner first lands on blue.
  1. Find
    1. P(\(B = 4\))
    2. P(\(B \leq 5\))
    [4]
  2. Find E(\(B^2\)) [3]
Steve invites Tamara to play a game with this spinner. Tamara must choose a colour, either red or blue. Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(e^X\) If Tamara chooses blue, her score is \(X^2\)
  1. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]
SPS SPS FM Statistics 2021 January Q7
7 marks Challenging +1.2
Nine athletes, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\) and \(I\), competed in both the 100m 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 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 100m sprint467928315
Position in long jump549312
Given that there were no tied ranks,
  1. 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]
  2. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100m sprint and the long jump. [2]
SPS SPS SM Pure 2020 October Q1
6 marks Easy -1.3
  1. Find $$\int \frac{x}{x^2 + 1} dx$$ [2]
  2. Find. $$\int 2\pi(4x + 3)^{10} dx$$ [2]
  3. Find. $$\int \frac{2}{e^{4x}} dx$$ [2]
SPS SPS SM Pure 2020 October Q2
5 marks Moderate -0.8
  1. Find \(\frac{dy}{dx}\) if \(y = 4\ln(3x)\) [2]
  2. Differentiate \(\frac{2x}{\sqrt{3x+1}}\) giving your answer in the form \(\frac{3x+c}{\sqrt{(3x+1)^p}}\), where \(c, p \in \mathbb{N}\) [3]
SPS SPS SM Pure 2020 October Q3
3 marks Easy -1.8
Expand \((x - 2y)^5\). [3]
SPS SPS SM Pure 2020 October Q4
3 marks Moderate -0.8
What transformations could be used, and in which order, to transform the curve \(y = \sin x\) into the curve \(y = 2 \sin(3x + 30°)\)? [3]
SPS SPS SM Pure 2020 October Q5
5 marks Standard +0.3
Find the equation of the tangent to the curve $$y = 3x^2(x + 2)^6$$ at the point \((-1, 3)\), giving your answer in the form \(y = mx + c\). [5]
SPS SPS SM Pure 2020 October Q6
5 marks Moderate -0.8
  1. Express \(\frac{x}{(x + 1)(x + 2)}\) in partial fractions. [3]
  2. Hence find \(\int \frac{x}{(x + 1)(x + 2)} dx\). [2]
SPS SPS SM Pure 2020 October Q7
7 marks Standard +0.3
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0° < \theta < 360°\). [4]
SPS SPS SM Pure 2020 October Q8
12 marks Challenging +1.3
    1. Sketch the graph of \(y = \cos \sec x\) for \(0 < x < 4\pi\). [3]
    2. It is given that \(\cos \sec \alpha = \cos \sec \beta\), where \(\frac{1}{2}\pi < \alpha < \pi\) and \(2\pi < \beta < \frac{5}{2}\pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\). [2]
    1. Write down the identity giving \(\tan 2\theta\) in terms of \(\tan \theta\). [1]
    2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2\phi \tan 4\phi\), showing all your working. [6]
SPS SPS SM Pure 2020 October Q9
6 marks Standard +0.3
Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = (h^6 + 16)^2 - 4.$$
  1. Find the value of \(\frac{dV}{dh}\) when \(h = 2\). [3]
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures. [3]
SPS SPS SM Pure 2020 October Q10
7 marks Standard +0.3
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{2}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
SPS SPS SM 2021 February Q1
1 marks Easy -2.5
Which of the options below best describes the correlation shown in the diagram below? \includegraphics{figure_1} Tick \((\checkmark)\) one box. [1 mark] moderate positive \(\square\) strong positive \(\square\) moderate negative \(\square\) strong negative \(\square\)
SPS SPS SM 2021 February Q2
1 marks Easy -2.5
Lenny is one of a team of people interviewing shoppers in a town centre. He is asked to survey 50 women between the ages of 18 and 29 Identify the name of this type of sampling. Circle your answer. [1 mark] simple random stratified quota systematic
SPS SPS SM 2021 February Q3
8 marks Standard +0.3
The Venn diagram shows the probabilities associated with four events, \(A\), \(B\), \(C\) and \(D\) \includegraphics{figure_3}
  1. Write down any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) [1]
  2. Given that \(P(B) = 0.4\) find the value of \(p\) [1]
  3. Given also that \(A\) and \(B\) are independent find the value of \(q\) [2]
  4. Given further that \(P(B'|C) = 0.64\) find
    1. the value of \(r\)
    2. the value of \(s\)
    [4]
SPS SPS SM 2021 February Q4
10 marks Easy -1.3
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{figure_4}
  1. Find the range of the times. [1]
  2. Find the interquartile range of the times. [1]
  3. For these 27 people \(\sum x = 607.5\) and \(\sum x^2 = 17623.25\) calculate the mean time taken to complete the puzzle. [1]
  4. calculate the standard deviation of the times taken to complete the puzzle. [2]
  5. 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. [1]
  6. 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
    Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions. [3]
  7. Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d). [1]
SPS SPS SM 2021 February Q5
10 marks Easy -1.3
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 mark]
    2. Find the variance of the number of times he falls off in a day. [1 mark]
    1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
    2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
  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. [2 marks]
    2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]
SPS SPS SM 2021 February Q6
10 marks Standard +0.3
The discrete random variable \(D\) has the following probability distribution
\(d\)1020304050
\(P(D = d)\)\(\frac{k}{10}\)\(\frac{k}{20}\)\(\frac{k}{30}\)\(\frac{k}{40}\)\(\frac{k}{50}\)
where \(k\) is a constant.
  1. Show that the value of \(k\) is \(\frac{600}{137}\) [2]
  2. The random variables \(D_1\) and \(D_2\) are independent and each have the same distribution as \(D\). Find \(P(D_1 + D_2 = 80)\) Give your answer to 3 significant figures. [3]
  3. 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 \(50°\) [5]
SPS SPS SM 2021 February Q7
15 marks Standard +0.3
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.
  1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. [1]
  2. 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. [4]
  3. 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 \(T \sim N(5, 3.5^2)\) Using this model,
    1. find the probability that a routine appointment with the dentist takes less than 2 minutes [1]
    2. find \(P(T < 2 | T > 0)\) [3]
    3. hence explain why this normal distribution may not be a good model for \(T\). [1]
  4. 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. [5]