Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2021 May Q1
7 marks Standard +0.3
In this question you must show detailed reasoning.
  1. By using partial fractions show that \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}\). [5]
  2. Hence determine the value of \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}\). [2]
SPS SPS FM Pure 2021 May Q2
8 marks Standard +0.8
  1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
  2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]
SPS SPS FM Pure 2021 May Q3
5 marks Moderate -0.3
In this question you must show detailed reasoning. Show that $$\int_5^{\infty} (x - 1)^{-\frac{3}{2}} dx = 1$$ [5]
SPS SPS FM Pure 2021 May Q4
6 marks Standard +0.3
You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation \(R\) which is a reflection in 3-D.
  1. State the plane of reflection of \(R\). [1]
  2. Determine the value of \(a\). [3]
  3. With reference to \(R\) explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3 \times 3\) identity matrix. [2]
SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Express \(\frac{5x^2+x+12}{x^3+4x}\) in partial fractions. [5]
SPS SPS FM Pure 2021 May Q6
6 marks Challenging +1.2
A circle \(C\) in the complex plane has equation \(|z - 2 - 5i| = a\). The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(arg(z_1) = \frac{\pi}{4}\). Prove that \(a = \frac{3\sqrt{2}}{2}\). [6]
SPS SPS FM Pure 2021 May Q7
8 marks Challenging +1.2
The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p + x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form
SPS SPS FM Pure 2021 May Q8
8 marks Hard +2.3
Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos(r\theta)\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos(10\theta)\). [8]
SPS SPS FM Pure 2021 May Q9
12 marks Standard +0.8
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x - 0.2y \quad \text{and} \quad \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
  2. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). Comment on this result in the context of the industrial process. [4]
  3. Express \(y\) in terms of \(t\). [5]
SPS SPS FM Pure 2022 June Q1
7 marks Standard +0.3
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int_0^9 \frac{1}{\sqrt{x}} dx\); [3 marks]
    2. \(\int_0^9 \frac{1}{x\sqrt{x}} dx\). [3 marks]
  2. Explain briefly why the integrals in part (a) are improper integrals. [1 mark]
SPS SPS FM Pure 2022 June Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the graph of \(y = f(x)\), \(x \in \mathbb{R}\). The graph consists of two line segments that meet at the point \((1, a)\), \(a < 0\). One line meets the \(x\)-axis at \((3, 0)\). The other line meets the \(x\)-axis at \((-1, 0)\) and the \(y\)-axis at \((0, b)\), \(b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = f(x + 1)\), [2]
  2. \(y = f(|x|)\). [2]
Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(f(x) = |x - 1| - 2\), find
  1. the value of \(a\) and the value of \(b\), [2]
  2. the value of \(x\) for which \(f(x) = 5x\). [3]
SPS SPS FM Pure 2022 June Q3
8 marks Standard +0.3
  1. Show on an Argand diagram the locus of points given by $$|z - 10 - 12i| = 8$$ [2] Set \(A\) is defined by $$A = \left\{z : 0 \leq \arg(z - 10 - 10i) \leq \frac{\pi}{2}\right\} \cap \{z : |z - 10 - 12i| < 8\}$$
  2. Shade the region defined by \(A\) on your Argand diagram. [2]
  3. Determine the area of the region defined by \(A\). [4]
SPS SPS FM Pure 2022 June Q4
8 marks Standard +0.3
The curve with equation \(y = f(x)\) where $$f(x) = x^2 + \ln(2x^2 - 4x + 5)$$ has a single turning point at \(x = \alpha\)
  1. Show that \(\alpha\) is a solution of the equation $$2x^3 - 4x^2 + 7x - 2 = 0$$ [4]
The iterative formula $$x_{n+1} = \frac{1}{7}(2 + 4x_n^2 - 2x_n^3)$$ is used to find an approximate value for \(\alpha\). Starting with \(x_1 = 0.3\)
  1. calculate, giving each answer to 4 decimal places,
    1. the value of \(x_2\)
    2. the value of \(x_4\)
    [2]
Using a suitable interval and a suitable function that should be stated,
  1. show that \(\alpha\) is 0.341 to 3 decimal places. [2]
SPS SPS FM Pure 2022 June Q5
4 marks Standard +0.3
The triangle \(T\) has vertices at the points \((1, k)\), \((3,0)\) and \((11,0)\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T'\) by the matrix $$\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix}$$ Given that the area of triangle \(T'\) is 364 square units, find the value of \(k\). [4]
SPS SPS FM Pure 2022 June Q6
7 marks Moderate -0.3
The complex number \(w\) is given by $$w = 10 - 5i$$
  1. Find \(|w|\). [1]
  2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [1]
The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + i)(z + 3i) = w$$
  1. Use algebra to find \(z\), giving your answer in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
Given that $$\arg(\lambda + 9i + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,
  1. find the value of \(\lambda\). [1]
SPS SPS FM Pure 2022 June Q7
7 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = xe^x\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis. Use integration by parts to find an exact value for the volume of the solid generated. [7]
SPS SPS FM Pure 2022 June Q8
5 marks Standard +0.8
With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$ The point \(A\) lies on \(l\) and has coordinates \((3, -2, 6)\). The point \(P\) has position vector \((-\mathbf{i} + 2\mathbf{k})\) relative to \(O\). Given that vector \(\overrightarrow{PA}\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle BPA = 45°\), find the coordinates of the two possible positions of \(B\). [5]
SPS SPS FM Pure 2022 June Q9
6 marks Standard +0.3
Prove by induction that for \(n \in \mathbb{Z}^+\) $$f(n) = 4^{n+1} + 5^{2n-1}$$ is divisible by 21 [6]
SPS SPS FM Pure 2022 June Q10
8 marks Standard +0.8
The curve defined by the parametric equations $$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$ is shown below. The point \(P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)\) is marked on the curve. \includegraphics{figure_curve}
  1. Show that the equation of the normal to the curve at \(P\) can be written as \(3y - x = \frac{7\sqrt{3}}{2}\) [5]
  2. Show that the Cartesian equation of the curve may be written as \(ay^2 + bx^4 + cx^2 = 0\) where \(a\), \(b\) and \(c\) are integers to be found. [3]
SPS SPS FM Pure 2022 June Q11
8 marks Standard +0.8
Solve the differential equation $$2\cot x \frac{dy}{dx} = (4 - y^2)$$ for which \(y = 0\) at \(x = \frac{\pi}{3}\), giving your answer in the form \(\sec^2 x = g(y)\). [8]
SPS SPS FM Pure 2022 June Q12
8 marks Standard +0.8
A linear transformation T of the \(x\)-\(y\) plane has an associated matrix M, where \(\mathbf{M} = \begin{pmatrix} \lambda & k \\ 1 & \lambda - k \end{pmatrix}\), and \(\lambda\) and \(k\) are real constants.
  1. You are given that \(\det \mathbf{M} > 0\) for all values of \(\lambda\).
    1. Find the range of possible values of \(k\). [3]
    2. What is the significance of the condition \(\det \mathbf{M} > 0\) for the transformation T? [1]
For the remainder of this question, take \(k = -2\).
  1. Determine whether there are any lines through the origin that are invariant lines for the transformation T. [4]
SPS SPS FM Pure 2022 June Q13
8 marks Standard +0.3
  1. Show that \(\sin(2\theta + \frac{1}{2}\pi) = \cos 2\theta\). [2]
  2. Hence solve the equation \(\sin 3\theta = \cos 2\theta\) for \(0 \leq \theta \leq 2\pi\). [6]
SPS SPS FM Pure 2022 June Q14
7 marks Standard +0.8
Using an appropriate substitution, or otherwise, show that $$\int_0^{\frac{\pi}{2}} \frac{\sin 2\theta}{1 + \cos \theta} d\theta = 2 - 2\ln 2$$ [7]
SPS SPS FM Pure 2022 February Q1
7 marks Moderate -0.3
  1. Express \(\frac{1}{(2r-1)(2r+1)}\) in partial fractions. [3]
  2. Hence find \(\sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}\), expressing the result as a single fraction. [4]
SPS SPS FM Pure 2022 February Q2
5 marks Challenging +1.2
\(\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}\) The matrix \(\mathbf{A}\) represents the linear transformation \(M\). Prove that, for the linear transformation \(M\), there are no invariant lines. [5]