Questions — SPS SPS FM (161 questions)

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SPS SPS FM 2020 October Q1
7 marks Moderate -0.8
  1. Find the binomial expansion of \((2 + x)^5\), simplifying the terms. [4]
  2. Hence find the coefficient of \(y^3\) in the expansion of \((2 + 3y + y^2)^5\). [3]
SPS SPS FM 2020 October Q2
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z}{y})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS FM 2020 October Q3
8 marks Standard +0.3
  1. Give full details of a sequence of two transformations needed to transform the graph \(y = |x|\) to the graph of \(y = |2(x + 3)|\). [3]
  2. Solve \(|x| > |2(x + 3)|\), giving your answer in set notation. [5]
SPS SPS FM 2020 October Q4
5 marks Standard +0.3
Prove by induction that, for \(n \geq 1\), \(\sum_{r=1}^n r(3r + 1) = n(n + 1)^2\). [5]
SPS SPS FM 2020 October Q5
6 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows triangle \(ABC\), with \(AB = x\) cm, \(AC = (x + 2)\) cm, \(BC = 2\sqrt{7}\) cm and angle \(CAB = 60°\).
  1. Find the value of \(x\). [4]
  2. Find the area of triangle \(ABC\), giving your answer in an exact form as simply as possible. [2]
SPS SPS FM 2020 October Q6
5 marks Moderate -0.3
Prove by contradiction that \(\sqrt{7}\) is irrational. [5]
SPS SPS FM 2020 October Q7
7 marks Moderate -0.3
A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).
  1. Find \(\frac{dy}{dx}\). [1]
  2. Hence sketch the gradient function for the curve. [4]
  3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]
SPS SPS FM 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 FM 2020 October Q9
8 marks Challenging +1.8
In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 5 - 2n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [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 2020 September Q1
3 marks Moderate -0.3
Vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are given by $$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$ where \(p\) and \(q\) are constants. Given that \(\overrightarrow{AC}\) is parallel to \(\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}\), find the value of \(p\) and the value of \(q\). [3]