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 2025 February Q7
6 marks Challenging +1.2
Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and \(\text{Re}(z) \geq 9\). [6]
SPS SPS FM 2025 February Q8
4 marks Moderate -0.3
A locus \(C_1\) is defined by \(C_1 = \{z : |z + i| \leq |z - 2i|\}\).
  1. Indicate by shading on the Argand diagram below the region representing \(C_1\). [2] \includegraphics{figure_8}
  2. Find the cartesian equation of the boundary line of the region representing \(C_1\), giving your answer in the form \(ax + by + c = 0\). [2]
SPS SPS FM 2025 February Q9
9 marks Standard +0.8
\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
  1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
  2. Use the model to find the volume of water that the bottle can contain. [7]
SPS SPS SM 2025 February Q1
2 marks Easy -1.2
Given that \((x - 2)\) is a factor of \(2x^3 + kx - 4\), find the value of the constant \(k\). [2]
SPS SPS SM 2025 February Q2
13 marks Standard +0.3
  1. \includegraphics{figure_2} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(ABCDEF\). The base \(ACFD\) of the roof is a horizontal rectangle, and the cross-section \(ABC\) of the roof is an isosceles triangle with \(AB = BC\). The lengths of \(AC\) and \(CF\) are \(2x\) cm and \(y\) cm respectively, and the height of \(BE\) above the base of the roof is \(x\) cm. The total surface area of the five faces of the roof is \(600\) cm\(^2\) and the volume of the roof is \(V\) cm\(^3\). Show that \(V = kx (300 - x^2)\), where \(k = \sqrt{a + b}\) and \(a\) and \(b\) are integers to be determined. [6]
  2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum. [4]
  3. Find the maximum volume of the roof. Give your answer in cm\(^3\), correct to the nearest integer. [1]
  4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state. [2]
SPS SPS SM 2025 February Q3
6 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a sector \(AOB\) of a circle with centre \(O\). The length of the arc \(AB\) is \(6\) cm and the area of the sector \(AOB\) is \(24\) cm\(^2\). Find the area of the shaded segment enclosed by the arc \(AB\) and the chord \(AB\), giving your answer correct to \(3\) significant figures. [6]
SPS SPS SM 2025 February Q4
6 marks Moderate -0.3
  1. The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than \(2\). Given that \(n^3 + 1 = (n + 1) (n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
  2. Prove that \(K\) has at least two distinct factors other than \(1\) and \(K\). [5]
SPS SPS SM 2025 February Q5
7 marks Standard +0.3
A curve has the following properties: • The gradient of the curve is given by \(\frac{dy}{dx} = -2x\). • The curve passes through the point \((4, -13)\). Determine the coordinates of the points where the curve meets the line \(y = 2x\). [7]
SPS SPS SM 2025 February Q6
9 marks Standard +0.3
For all real values of \(x\), the functions \(f\) and \(g\) are defined by \(f (x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.
  1. Find \(fg(x)\). Determine the range of \(fg(x)\) in terms of \(a\). [4]
  2. If \(fg(2) = 144\), find the value of \(a\). [3]
  3. Determine whether the function \(fg\) has an inverse. [2]
SPS SPS SM 2025 February Q7
8 marks Standard +0.3
  1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ [3]
  2. Hence solve the equation $$2 \sin 2\theta \tan 2\theta = \cos 2\theta + 5,$$ giving all values of \(\theta\) between \(0°\) and \(180°\), correct to \(1\) decimal place. [5]
SPS SPS SM 2025 February Q8
7 marks Challenging +1.2
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
SPS SPS FM Statistics 2025 April Q1
8 marks Moderate -0.3
It is known that, under standard conditions, 12% of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.
  1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample. [1]
  2. Test, at the 5% significance level, whether the proportion of defective light bulbs has decreased under the new process. [7]
SPS SPS FM Statistics 2025 April Q2
13 marks Moderate -0.8
In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: 20 | 3 5 7 9 21 | 0 2 5 6 8 22 | 1 3 4 5 7 9 23 | 0 2 5 8 24 | 1 4 6 7 25 | 2 5 Key: 21 | 0 represents a reaction time of 210 milliseconds
  1. State the median reaction time. [1]
  2. Calculate the interquartile range of these reaction times. [2]
  3. Find the mean and standard deviation of these reaction times. [3]
  4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table. [1]
  5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
    1. the median
    2. the mean
    3. the standard deviation
    [4]
  6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context [2]
SPS SPS FM Statistics 2025 April Q3
9 marks Standard +0.8
Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.
  1. Show that \(P(M = 6) = \frac{1}{3}\) [2]
The table shows the probability distribution of \(M\)
\(m\)45678
\(P(M = m)\)\(\frac{1}{15}\)\(\frac{4}{15}\)\(\frac{1}{3}\)\(\frac{4}{15}\)\(\frac{1}{15}\)
Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.
  1. Find \(P(M = S)\) [3]
  2. Find \(P(S = 7 | M = S)\) [4]
SPS SPS FM Statistics 2025 April Q4
6 marks Standard +0.8
The discrete random variable \(X\) has a geometric distribution. It is given that \(\text{Var}(X) = 20\). Determine \(P(X \geq 7)\). [6]
SPS SPS FM Statistics 2025 April Q5
7 marks Standard +0.3
An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.
  1. If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions. [2]
Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7-mark questions are the first four questions on the paper, but are arranged in random order. The 9-mark questions are the last four questions, but are arranged in random order. Find the probability that
  1. the questions on geometric distributions and on binomial distributions are next to one another, [2]
  2. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions. [3]
SPS SPS FM Statistics 2025 April Q6
11 marks Standard +0.3
The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.
  1. Find the 90th percentile for the weights of these dogs. [2]
  2. Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg. [3]
The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg.
  1. Given that 5% of female dogs of this breed weigh more than 30 kg, find the standard deviation of their weights. [3]
  2. Sketch the distributions of the weights of male and female dogs of this breed on a single diagram. [3]
SPS SPS FM Statistics 2025 April Q7
9 marks Standard +0.3
The random variable \(y\) has probability density function f(y) given by $$f(y) = \begin{cases} ky(a - y) & 0 \leq y \leq 3 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(a\) are positive constants.
    1. Explain why \(a \geq 3\) [1]
    2. Show that \(k = \frac{2}{9(a - 2)}\) [3]
Given that \(E(Y) = 1.75\)
  1. Find the values of a and k. [4]
  2. Write down the mode of Y [1]
SPS SPS SM Statistics 2025 April Q2
5 marks Easy -1.2
The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species. \includegraphics{figure_2} The number of worms in the sample with lengths in the class \(3 \leq l < 4\) is 30.
  1. Find the number of worms in the sample with lengths in the class \(0 \leq l < 2\). [2]
  2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leq l < 5.5\). [3]
SPS SPS SM Statistics 2025 April Q3
5 marks Moderate -0.8
A researcher has collected data on the heights of a sample of adults but has encoded the actual values using a linear transformation of the form \(aX + b\), where \(X\) represents the original height in centimetres. Given the following information about the encoded data: The mean of the encoded heights is 5.4 cm The standard deviation of the encoded heights is 2.0 cm The researcher knows that the transformation used was \(0.2X - 30\)
  1. Find the mean of the original heights in the sample. [2]
  2. Find the standard deviation of the original heights in the sample. [2]
  3. If an encoded height value is 6.8, what was the original height in centimetres? [1]
SPS SPS SM Statistics 2025 April Q4
8 marks Moderate -0.3
A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that 15% of boards fail the mechanical inspection. Of those that pass the mechanical inspection, 8% fail the electrical test. Of those that fail the mechanical inspection, 60% fail the electrical test.
  1. If a board is randomly selected from production, what is the probability that it passes both inspections? [2]
  2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection? [3]
  3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections? [3]
SPS SPS SM Statistics 2025 April Q5
13 marks Easy -1.3
In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: 20 | 3 5 7 9 21 | 0 2 5 6 8 22 | 1 3 4 5 7 9 23 | 0 2 5 8 24 | 1 4 6 7 25 | 2 5 Key: 21 | 0 represents a reaction time of 210 milliseconds
  1. State the median reaction time. [1]
  2. Calculate the interquartile range of these reaction times. [2]
  3. Find the mean and standard deviation of these reaction times. [3]
  4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table. [1]
  5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on: a. the median b. the mean c. the standard deviation [4]
  6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context [2]
SPS SPS SM Statistics 2025 April Q6
11 marks Moderate -0.8
A retail bakery makes cherry muffins where, due to the production process, 15% of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.
  1. State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid. [4]
  2. Using your model from part (a), find the probability that a randomly selected box contains:
    1. exactly 3 muffins with a lower than expected quantity of cherries, [2]
    2. at least 5 muffins with a lower than expected quantity of cherries. [2]
  3. The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries. [3]
SPS SPS SM Statistics 2025 April Q7
9 marks Standard +0.3
Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.
  1. Show that \(P(M = 6) = \frac{1}{3}\) [2]
The table shows the probability distribution of \(M\)
\(m\)45678
\(P(M = m)\)\(\frac{1}{15}\)\(\frac{4}{15}\)\(\frac{1}{3}\)\(\frac{4}{15}\)\(\frac{1}{15}\)
Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.
  1. Find \(P(M = S)\) [3]
  2. Find \(P(S = 7 | M = S)\) [4]
SPS SPS FM Pure 2025 June Q1
5 marks Moderate -0.3
The complex number \(z\) satisfies the equation \(z + 2iz^* = 12 + 9i\). Find \(z\), giving your answer in the form \(x + iy\). [5]