Questions — SPS (686 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
SPS SPS FM Pure 2023 September Q5
6 marks Standard +0.3
  1. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg(z + i) = \frac{\pi}{6}$$ [2 marks]
\includegraphics{figure_5}
  1. \(z_1\) is a point on \(L\) such that \(|z_1|\) is a minimum. Find the exact value of \(z_1\) in the form \(a + bi\) [4 marks]
SPS SPS FM Pure 2023 September Q6
8 marks Challenging +1.2
A curve has equation \(y = xe^{\frac{x}{2}}\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]
SPS SPS FM Pure 2023 September Q7
8 marks Standard +0.8
  1. Prove the identity \(\frac{\cos x}{\sec x + 1} + \frac{\cos x}{\sec x - 1} = 2\cot^2 x\) [3 marks]
  2. Hence, solve the equation $$\frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) + 1} = \cot\left(2\theta + \frac{\pi}{3}\right) - \frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) - 1}$$ in the interval \(0 \leq \theta \leq 2\pi\), giving your values of \(\theta\) to three significant figures where appropriate. [5 marks]
SPS SPS FM Pure 2023 September Q8
7 marks Standard +0.8
A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]
SPS SPS FM Pure 2023 September Q9
18 marks Standard +0.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$
    1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
    2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]
  1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
    1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
    2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{-\frac{x}{2}}$$ [2 marks]
    3. Use the recurrence relation $$x_{n+1} = e - e^{-\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
    4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{-\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_1}
SPS SPS FM Pure 2023 November Q1
4 marks Standard +0.8
The complex number \(z\) satisfies the equation \(z^2 - 4iz^* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
SPS SPS FM Pure 2023 November Q2
8 marks Standard +0.3
Fig. 5 shows the curve with polar equation \(r = a(3 + 2\cos\theta)\) for \(-\pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \includegraphics{figure_2}
  1. Write down the polar coordinates of the points A and B. [2]
  2. Explain why the curve is symmetrical about the initial line. [2]
  3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve. [4]
SPS SPS FM Pure 2023 November Q3
8 marks Standard +0.8
In this question you must show detailed reasoning. The roots of the equation \(2x^3 - 5x + 7 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\).
  1. Find \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]
  2. Find an equation with integer coefficients whose roots are \(2\alpha - 1\), \(2\beta - 1\) and \(2\gamma - 1\). [4]
SPS SPS FM Pure 2023 November Q4
7 marks Standard +0.8
In this question you must show detailed reasoning.
  1. Given that $$\frac{1}{r(r + 1)(r + 2)} = \frac{A}{r(r + 1)} + \frac{B}{(r + 1)(r + 2)}$$ show that \(A = \frac{1}{2}\) and find the value of \(B\). [3]
  2. Use the method of differences to find $$\sum_{r=10}^{98} \frac{1}{r(r + 1)(r + 2)}$$ giving your answer as a rational number. [4]
SPS SPS FM Pure 2023 November Q5
6 marks Moderate -0.3
  1. Use a Maclaurin series to find a quadratic approximation for \(\ln(1 + 2x)\). [1]
  2. Find the percentage error in using the approximation in part (a) to calculate \(\ln(1.2)\). [3]
  3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid. [2]
SPS SPS FM Pure 2023 November Q6
5 marks Standard +0.8
In this question you must show detailed reasoning. In this question you may assume the results for $$\sum_{r=1}^{n} r^3, \quad \sum_{r=1}^{n} r^2 \quad \text{and} \quad \sum_{r=1}^{n} r.$$ Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n^2(2n^2 - 1).$$ [5]
SPS SPS FM Pure 2023 November Q7
Challenging +1.8
    1. Show on an Argand diagram the locus of points given by the values of \(z\) satisfying $$|z - 4 - 3i| = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that $$\theta \in [\alpha, \alpha + \pi], \text{ where } \alpha = -\arctan\left(\frac{4}{3}\right),$$
    2. show that this locus of points can be represented by the polar curve with equation $$r = 8\cos\theta + 6\sin\theta$$ (6) The set of points \(A\) is defined by $$A = \left\{z : 0 \leqslant \arg z \leqslant \frac{\pi}{3}\right\} \cap \{z : |z - 4 - 3i| \leqslant 5\}$$
    1. Show, by shading on your Argand diagram, the set of points \(A\).
    2. Find the exact area of the region defined by \(A\), giving your answer in simplest form. (7)
SPS SPS FM Pure 2023 November Q8
Challenging +1.8
  1. Use a hyperbolic substitution and calculus to show that $$\int \frac{x^2}{\sqrt{x^2 - 1}} dx = \frac{1}{2}\left[x\sqrt{x^2 - 1} + \arcosh x\right] + k$$ where \(k\) is an arbitrary constant. (6) \includegraphics{figure_8} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac{4}{15}x \arcosh x \quad x \geqslant 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
  2. Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac{1}{15}\left[17\ln\left(3 + 2\sqrt{2}\right) - 6\sqrt{2}\right]$$ (5) This is the last question on the paper.
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]