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 2021 March Q7
6 marks Challenging +1.2
This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_7} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a, b, c \in Q\) [6]
SPS SPS FM 2021 March Q8
4 marks Standard +0.3
The function f is defined, for any complex number \(z\), by $$\text{f}(z) = \frac{iz - 1}{iz + 1}.$$ Suppose throughout that \(x\) is a real number.
  1. Show that $$\text{Re f}(x) = \frac{x^2 - 1}{x^2 + 1} \quad \text{and} \quad \text{Im f}(x) = \frac{2x}{x^2 + 1}.$$ [2]
  2. Show that \(\text{f}(x)\text{f}(x)^* = 1\), where \(\text{f}(x)^*\) is the complex conjugate of \(\text{f}(x)\). [2]
SPS SPS FM 2021 April Q1
11 marks Moderate -0.3
  1. Differentiate the following with respect to \(x\), simplifying your answers fully
    1. \(y = e^{3x} + \ln 2x\) [1]
    2. \(y = (5 + x^2)^{\frac{3}{2}}\) [1]
    3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [2]
    4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [2]
  2. Integrate with respect to \(x\)
    1. \(\frac{7}{(2x-5)^8} - \frac{3}{2x-5}\) [2]
    2. \(\frac{4x^2+5x-3}{2x-5}\) [3]
SPS SPS FM 2021 April Q2
4 marks Challenging +1.2
solve, for \(0° < \theta < 360°\), the equation $$2 \tan^2 \theta - \frac{1}{\cos \theta} = 4.$$ [4]
SPS SPS FM 2021 April Q3
9 marks Moderate -0.3
$$\text{f}(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$
  1. Write down the domain and range of \(\text{f}^{-1}\) [2]
  2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
  3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = -\frac{5}{3}\) [3]
SPS SPS FM 2021 April Q5
13 marks Standard +0.3
  1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5\sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]
SPS SPS FM 2021 April Q6
6 marks Challenging +1.2
This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_6} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a,b,c \in Q\) [6]
SPS SPS FM 2021 April Q7
6 marks Standard +0.3
$$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]
SPS SPS FM 2021 April Q8
4 marks Standard +0.3
The function f is defined, for any complex number \(z\), by $$\text{f}(z) = \frac{iz - 1}{iz + 1}.$$ Suppose throughout that \(x\) is a real number.
  1. Show that $$\text{Re f}(x) = \frac{x^2 - 1}{x^2 + 1} \quad \text{and} \quad \text{Im f}(x) = \frac{2x}{x^2 + 1}.$$ [2]
  2. Show that \(\text{f}(x)\text{f}(x)^* = 1\), where \(\text{f}(x)^*\) is the complex conjugate of \(\text{f}(x)\). [2]
SPS SPS FM Statistics 2021 June Q1
4 marks Moderate -0.8
Employees at a company were asked how long their average commute to work was. The table below gives information about their answers.
Time taken (\(t\) minutes)Number of people
\(0 < t \leq 10\)\(x\)
\(10 < t \leq 20\)30
\(20 < t \leq 30\)35
\(30 < t \leq 50\)28
\(50 < t \leq 90\)12
The company estimates that the mean time for employees commuting to work is 28 minutes. Work out the value of \(x\), showing your working clearly. [4]
SPS SPS FM Statistics 2021 June Q2
8 marks Moderate -0.3
Events \(A\) and \(B\) are such that \(P(A \cup B) = 0.95\), \(P(A \cap B) = 0.6\) and \(P(A|B) = 0.75\).
  1. Find \(P(B)\). [3]
  2. Find \(P(A)\). [3]
  3. Show that the events \(A'\) and \(B\) are independent. [2]
SPS SPS FM Statistics 2021 June Q3
4 marks Standard +0.3
The letters of the word CHAFFINCH are written on cards.
  1. In how many ways can the letters be rearranged with no restrictions. [1]
  2. In how many difference ways can the letters be rearranged if the vowels are to have at least one consonant between them. [3]
SPS SPS FM Statistics 2021 June Q4
9 marks Standard +0.3
The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6kg and three in five sacks weigh more than 5.5kg.
  1. Find the mean and standard deviation of the weights of potato sacks. [5]
  2. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6kg? You must explain your reasoning clearly in this question. [4]
SPS SPS FM Statistics 2021 June Q5
7 marks Moderate -0.3
Eleven students in a class sit a Mathematics exam and their average score is 67% with a standard deviation of 12%. One student from the class is absent and sits the paper later, achieving a score of 85%.
  1. Find the mean score for the whole class and the standard deviation for the whole class. [5]
  2. Comment, with justification, on whether the score for the paper sat later should be considered as an outlier. [2]
SPS SPS FM Statistics 2021 June Q6
6 marks Standard +0.3
Only two airlines fly daily into an airport. AMP Air has 70 flights per day and Volt Air has 65 flights per day. Passengers flying with AMP Air have an 18% probability of losing their luggage and passengers flying with Volt Air have a 23% probability of losing their luggage. You overhear a passenger in the airport complaining about her luggage being lost. Find the exact probability that she travelled with Volt Air, giving your answer as a rational number. [6]
SPS SPS FM Statistics 2021 June Q7
12 marks Standard +0.8
A continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{x^2}{a} + b, & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are positive constants. It is given that \(P(X \geq 2) = 0.75\).
  1. Show that \(a = 32\) and \(b = \frac{1}{12}\). [5]
  2. Find \(E(X)\). [3]
  3. Find \(P(X > E(X)|X > 2)\) [4]
SPS SPS FM Pure 2021 June Q1
2 marks Moderate -0.8
A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$ Find the gradient of the curve at the point where \(t = -2\) [2]
SPS SPS FM Pure 2021 June Q2
6 marks Moderate -0.3
The equation \(x^3 - 3x + 1 = 0\) has three real roots.
  1. Show that one of the roots lies between \(-2\) and \(-1\) [2 marks]
  2. Taking \(x_1 = -2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x_2\), the second approximation. [3 marks]
  3. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x_1 = -1\) [1 mark]
SPS SPS FM Pure 2021 June Q3
4 marks Moderate -0.3
Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).
  1. Find the position vector of \(P\). [2]
  2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]
SPS SPS FM Pure 2021 June Q4
8 marks Standard +0.8
Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]
SPS SPS FM Pure 2021 June Q5
5 marks Standard +0.3
\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]
SPS SPS FM Pure 2021 June Q6
5 marks Moderate -0.3
  1. Use the binomial expansion, in ascending powers of \(x\), to show that $$\sqrt{4-x} = 2 - \frac{1}{4}x + kx^2 + ...$$ where \(k\) is a rational constant to be found. [4] A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt{3}\).
  2. State, giving a reason, if the expansion is valid for this value of \(x\). [1]
SPS SPS FM Pure 2021 June Q7
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
SPS SPS FM Pure 2021 June Q8
6 marks Standard +0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\) $$f(n) = 2^{n+2} + 3^{2n+1}$$ is divisible by 7 [6]
SPS SPS FM Pure 2021 June Q9
6 marks Moderate -0.8
\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$
  1. State the range of \(f\) [1]
  2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
  3. state the set of possible values for \(k\). [2]