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SPS SPS SM Statistics 2024 January Q1
4 marks Easy -1.8
At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below. \(n = 205\) \(\sum x = 23042\) \(\sum x^2 = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.
  1. Calculate
    [2]
  2. Use your answers to part (a) to write down a possible Normal model for the distribution of marks. [2]
SPS SPS SM Statistics 2024 January Q2
14 marks Moderate -0.8
The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram. \includegraphics{figure_2} One of the 150 plants is chosen at random, and its height, \(X\) cm, is noted.
  1. Show that P\((20 < X < 30) = 0.147\), correct to 3 significant figures. [2]
Sam suggests that the distribution of \(X\) can be well modelled by the distribution N\((40, 100)\).
    1. Give a brief justification for the use of the normal distribution in this context. [1]
    2. Give a brief justification for the choice of the parameter values 40 and 100. [2]
  2. Use Sam's model to find P\((20 < X < 30)\). [1]
Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N\((m, s^2)\) as her model.
  1. Use Nina's model to find P\((20 < X < 30)\). [4]
    1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model. [2]
    2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution. [2]
SPS SPS SM Statistics 2024 January Q3
12 marks Moderate -0.8
Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.
  1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students. [1]
  2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. [1]
Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.
MeanStandard deviationMedianLower quartileUpper quartile
21.04.2020.518.022.9
  1. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier. [3]
  2. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the 5\% significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test. [7]
SPS SPS SM Statistics 2024 January Q4
6 marks Easy -1.2
The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK. \includegraphics{figure_4} The first two digits of the Geography code give the type of each of the LAs: 06: Unitary authority 07: Non-metropolitan district 08: Metropolitan borough
  1. In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram? [1]
  2. Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. [2]
Now assume the following.
  • The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
  • The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.
  1. Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011. [2]
  2. In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011. [1]
SPS SPS SM Statistics 2024 January Q5
7 marks Standard +0.8
Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.
malefemale
black13
yellow21
chocolate11
Four puppies are chosen at random to train as guide dogs.
  1. Determine the probability that at least 3 black puppies are chosen. [3]
  2. Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen. [3]
  3. Explain whether the 2 events 'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events. [1]
SPS SPS SM Statistics 2024 January Q6
6 marks Standard +0.3
A firm claims that no more than 2\% of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than 2\%. In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the 5\% significance level, using the null hypothesis \(p = 0.02\).
  1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect. [5]
  2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says "18 out of 600 is 3\%, so there is evidence that the actual proportion of underweight bags is greater than 2\%." Criticise this statement. [1]
SPS SPS SM Statistics 2024 January Q7
11 marks Standard +0.8
The probability distribution of a random variable \(X\) is modelled as follows. $$\text{P}(X = x) = \begin{cases} \frac{k}{x} & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{12}{25}\). [2]
  2. Show in a table the values of \(X\) and their probabilities. [1]
  3. The values of three independent observations of \(X\) are denoted by \(X_1\), \(X_2\) and \(X_3\). Find P\((X_1 > X_2 + X_3)\). [3]
In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7.
  1. Determine the probability that a total of exactly 7 is first reached on the 5th observation. [5]
SPS SPS FM Pure 2024 February Q1
3 marks Moderate -0.5
The plane \(x + 2y + cz = 4\) is perpendicular to the plane \(2x - cy + 6z = 9\), where \(c\) is a constant. Find the value of \(c\). [3]
SPS SPS FM Pure 2024 February Q2
2 marks Easy -1.2
Find the mean value of \(f(x) = x^2 + 6x\) over the interval \([0, 3]\). [2]
SPS SPS FM Pure 2024 February Q3
6 marks Standard +0.3
It is given that \(1 - 3i\) is one root of the quartic equation $$z^4 - 2z^3 + pz^2 + rz + 80 = 0$$ where \(p\) and \(r\) are real numbers.
  1. Express \(z^4 - 2z^3 + pz^2 + rz + 80\) as the product of two quadratic factors with real coefficients. [4 marks]
  2. Find the value of \(p\) and the value of \(r\). [2 marks]
SPS SPS FM Pure 2024 February Q4
6 marks Challenging +1.2
Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series \((1 \times 2 \times 4) + (2 \times 3 \times 5) + (3 \times 4 \times 6) + \ldots\) where \(n\) is a positive integer. Give your answer in fully factorised form. [6]
SPS SPS FM Pure 2024 February Q5
6 marks Standard +0.8
The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 0 \quad u_{n+1} = \frac{5}{6 - u_n}$$ Prove by induction that, for all integers \(n \geq 1\), $$u_n = \frac{5^n - 5}{5^n - 1}$$ [6 marks]
SPS SPS FM Pure 2024 February Q6
7 marks Standard +0.3
  1. Explain why \(\int_1^\infty \frac{1}{x(2x + 5)} dx\) is an improper integral. [1]
  2. Prove that $$\int_1^\infty \frac{1}{x(2x + 5)} dx = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. [6]
SPS SPS FM Pure 2024 February Q7
7 marks Standard +0.8
In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square, \(S\).
  1. Find the area of \(S\). [3]
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]
SPS SPS FM Pure 2024 February Q8
8 marks Challenging +1.2
A linear transformation of the plane is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 1 & -2 \\ \lambda & 3 \end{pmatrix}\), where \(\lambda\) is a constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin. [5]
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines. [3]
SPS SPS FM Pure 2024 February Q9
9 marks Standard +0.8
In this question you must show detailed reasoning. The complex number \(-4 + i\sqrt{48}\) is denoted by \(z\).
  1. Determine the cube roots of \(z\), giving the roots in exponential form. [6]
The points which represent the cube roots of \(z\) are denoted by \(A\), \(B\) and \(C\) and these form a triangle in an Argand diagram.
  1. Write down the angles that any lines of symmetry of triangle \(ABC\) make with the positive real axis, justifying your answer. [3]
SPS SPS FM Pure 2024 February Q10
11 marks Challenging +1.8
The diagram shows the polar curve \(C_1\) with equation \(r = 2\sin\theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_10}
  1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q\sin 4\alpha$$ where \(\alpha = \sin^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]
SPS SPS FM Pure 2024 February Q11
7 marks Challenging +1.8
Three planes have equations \begin{align} (4k + 1)x - 3y + (k - 5)z &= 3
(k - 1)x + (3 - k)y + 2z &= 1
7x - 3y + 4z &= 2 \end{align}
  1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]
SPS SPS FM Pure 2024 February Q12
7 marks Challenging +1.2
Find the general solution of the differential equation $$x\frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]
SPS SPS FM Pure 2024 February Q13
7 marks Challenging +1.2
In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh\frac{1}{2}x^2\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360°\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm. [7]
SPS SPS FM Pure 2024 February Q14
6 marks Challenging +1.8
Show that \(\int_0^{\frac{1}{\sqrt{3}}} \frac{4}{1-x^4} dx = \ln(a + \sqrt{b}) + \frac{\pi}{c}\) where \(a\), \(b\) and \(c\) are integers to be determined. [6]
SPS SPS FM Pure 2024 February Q15
8 marks Challenging +1.2
\(y = \cosh^n x\) \quad \(n \geq 5\)
    1. Show that $$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
    2. Determine an expression for \(\frac{d^4y}{dx^4}\) [2]
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer. [2]
SPS SPS FM 2024 October Q1
6 marks Moderate -0.8
Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),
  1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
  2. Find the maximum value of \(f(x)\). [1]
  3. Explain why \(f^{-1}(x)\) does not exist. [1]
SPS SPS FM 2024 October Q2
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS FM 2024 October Q3
6 marks Moderate -0.3
  1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left(2 + \frac{1}{3}kx\right)^6\), where \(k\) is a constant. [3]
  2. In the expansion of \((3 - 4x)\left(2 + \frac{1}{3}kx\right)^6\), the constant term is equal to the coefficient of \(x^2\). Determine the exact value of \(k\), given that \(k\) is positive. [3]