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SPS SPS SM 2020 October Q5
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z^2}{x})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS SM 2020 October Q6
5 marks Moderate -0.8
  1. A student was asked to solve the equation \(2^{2x+4} - 9(2^x) = 0\). The student's attempt is written out below. $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ $$\text{Let } y = 2^x$$ $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$ Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2^{2x+4} - 9(2^x) = 0\), giving your answer in exact form. [3]
SPS SPS SM 2020 October Q7
11 marks Moderate -0.3
  1. Sketch the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\) on the axes provided below. \includegraphics{figure_1} [3]
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\). [6]
  3. Hence, solve the inequality \(\frac{3}{x^2} \leq x^2 - 2\), giving your answer in interval notation. [2]
SPS SPS SM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM 2020 October Q9
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]
SPS SPS SM 2020 October Q10
8 marks Standard +0.8
In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]
SPS SPS FM 2021 March Q1
10 marks Moderate -0.8
Differentiate the following with respect to \(x\), simplifying your answers fully
  1. \(y = e^{3x} + \ln 2x\) [1]
  2. \(y = (5 + x^2)^{\frac{3}{2}}\) [2]
  3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [4]
  4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [3]
SPS SPS FM 2021 March Q2
5 marks Standard +0.3
  1. Express \(2 \tan^2 \theta - \frac{1}{\cos \theta}\) in terms of \(\sec \theta\). [1]
  2. Hence solve, for \(0° < \theta < 360°\), the equation $$2 \tan^2 \theta - \frac{1}{\cos \theta} = 4.$$ [4]
SPS SPS FM 2021 March 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 March Q4
7 marks Standard +0.8
\includegraphics{figure_4} The diagram shows the curves \(y = e^{3x}\) and \(y = (2x - 1)^4\). The shaded region is bounded by the two curves and the line \(x = \frac{1}{2}\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [7]
SPS SPS FM 2021 March 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 March Q6
8 marks Standard +0.3
$$\text{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}$$ [2] Prove by induction that \(\text{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]
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]