Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2024 February Q13
7 marks Challenging +1.2
In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh\frac{1}{2}x^2\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360°\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm. [7]
SPS SPS FM Pure 2024 February Q14
6 marks Challenging +1.8
Show that \(\int_0^{\frac{1}{\sqrt{3}}} \frac{4}{1-x^4} dx = \ln(a + \sqrt{b}) + \frac{\pi}{c}\) where \(a\), \(b\) and \(c\) are integers to be determined. [6]
SPS SPS FM Pure 2024 February Q15
8 marks Challenging +1.2
\(y = \cosh^n x\) \quad \(n \geq 5\)
    1. Show that $$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
    2. Determine an expression for \(\frac{d^4y}{dx^4}\) [2]
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer. [2]
SPS SPS FM Pure 2025 January Q1
4 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the curve \(y = 6x - x^2\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning. [4]
SPS SPS FM Pure 2025 January Q2
8 marks Standard +0.3
  1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]
SPS SPS FM Pure 2025 January Q3
8 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) is constant. A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).
  1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
  2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
  3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [2]
SPS SPS FM Pure 2025 January Q4
12 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$
  1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]
It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).
  1. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]
The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).
  1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
  2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]
SPS SPS FM Pure 2025 January Q5
11 marks Standard +0.3
With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
  2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]
The point \(A\), with position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\), lies on \(l_1\) The point \(B\) is the image of \(A\) after reflection in the line \(l_2\)
  1. Find the position vector of \(B\). [3]
SPS SPS FM Pure 2025 January Q6
12 marks Standard +0.3
You are given the complex number \(w = 2 + 2\sqrt{3}i\).
  1. Express \(w\) in modulus-argument form. [3]
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]
SPS SPS FM Pure 2025 January Q7
8 marks Standard +0.8
A candlestick has base diameter \(8\) cm and height \(28\) cm, as shown in Figure \(9\). A model of the candlestick is shown in Figure \(10\), together with the equations that were used to create the model. \includegraphics{figure_7}
  1. Show that the volume generated by rotating the shaded region (in Figure \(10\)) \(2\pi\) radians about the \(y\)-axis is \(\frac{112}{15}\pi\). [4]
  2. Hence find the volume of metal needed to create the candlestick. [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]
SPS SPS FM Pure 2025 June Q2
10 marks Standard +0.3
  1. Use binomial expansions to show that \(\sqrt{\frac{1 + 4x}{1 - x}} \approx 1 + \frac{5}{2}x - \frac{5}{8}x^2\) [6]
A student substitutes \(x = \frac{1}{2}\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt{6}\)
  1. Give a reason why the student should not use \(x = \frac{1}{2}\) [1]
  2. Substitute \(x = \frac{1}{11}\) into $$\sqrt{\frac{1 + 4x}{1 - x}} = 1 + \frac{5}{2}x - \frac{5}{8}x^2$$ to obtain an approximation to \(\sqrt{6}\). Give your answer as a fraction in its simplest form. [3]
SPS SPS FM Pure 2025 June Q3
3 marks Moderate -0.8
Describe a sequence of transformations which maps the graph of $$y = |2x - 5|$$ onto the graph of $$y = |x|$$ [3 marks]
SPS SPS FM Pure 2025 June Q4
5 marks Standard +0.8
Given that $$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ show that $$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ where \(A\) is a rational constant to be found. [5]
SPS SPS FM Pure 2025 June Q5
3 marks Standard +0.3
Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = I + 2\mathbf{A}$$ where \(I\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]
SPS SPS FM Pure 2025 June Q6
9 marks Standard +0.3
  1. Prove that $$1 - \cos 2\theta = \tan \theta \sin 2\theta, \quad \theta \neq \frac{(2n + 1)\pi}{2}, \quad n \in \mathbb{Z}$$ [3]
  2. Hence solve, for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), the equation $$(\sec^2 x - 5)(1 - \cos 2x) = 3\tan^2 x \sin 2x$$ Give any non-exact answer to 3 decimal places where appropriate. [6]
SPS SPS FM Pure 2025 June Q7
6 marks Standard +0.8
Fig. 10 shows the graph of \(x^3 + y^3 = xy\). \includegraphics{figure_10}
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
  2. P is the maximum point on the curve. The parabola \(y = kx^2\) intersects the curve at P. Find the value of the constant \(k\). [2]
SPS SPS FM Pure 2025 June Q8
7 marks Standard +0.8
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_8}
  1. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
    1. [(b) (i)] Find the value of \(\alpha\). [2 marks]
    2. [(b) (ii)] Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). [4 marks]
SPS SPS FM Pure 2025 June Q9
9 marks Challenging +1.2
\includegraphics{figure_9} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x, \quad x > 0\) The line \(l\) is the normal to \(C\) at the point \(P(e, e)\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis. Show that the exact area of \(R\) is \(Ae^2 + B\) where \(A\) and \(B\) are rational numbers to be found. [9]
SPS SPS FM Pure 2025 June Q10
5 marks Standard +0.3
Prove by induction that \(f(n) = 2^{4n} + 5^{2n} + 7^n\) is divisible by 3 for all positive integers \(n\). [5]
SPS SPS FM Pure 2025 June Q11
11 marks Challenging +1.2
Fig. 15 shows the graph of \(f(x) = 2x + \frac{1}{x} + \ln x - 4\). \includegraphics{figure_11}
  1. Show that the equation $$2x + \frac{1}{x} + \ln x - 4 = 0$$ has a root, \(\alpha\), such that \(0.1 < \alpha < 0.9\). [2]
  2. Obtain the following Newton-Raphson iteration for the equation in part (i). $$x_{r+1} = x_r - \frac{2x_r^3 + x_r + x_r^2(\ln x_r - 4)}{2x_r^2 - 1 + x_r}$$ [3]
  3. Explain why this iteration fails to find \(\alpha\) using each of the following starting values.
    1. \(x_0 = 0.4\) [2]
    2. \(x_0 = 0.5\) [2]
    3. \(x_0 = 0.6\) [2]
SPS SPS FM Pure 2025 June Q12
13 marks Challenging +1.2
In this question you must show detailed reasoning. \includegraphics{figure_12} The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{2 + t}}, \quad y = \ln(1 + t), \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac{1}{2}\).
  1. Find the exact \(y\)-coordinate of \(P\). [1]
The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).
  1. Determine the exact coordinates of \(Y\). [4]
The curve \(C\) and the line segment \(PY\) are rotated \(2\pi\) radians about the \(y\)-axis.
  1. Determine the exact volume of the solid generated. Give your answer in the form \(\pi(\ln p + q)\), where \(p\) and \(q\) are rational numbers. [8]
[You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\)]
SPS SPS FM Pure 2025 June Q13
9 marks Challenging +1.8
  1. Using a suitable substitution, find $$\int \sqrt{1 - x^2} \, dx.$$ [4]
  2. Show that the differential equation $$\frac{dy}{dx} = 2\sqrt{1 - x^2 - y^2 + x^2y^2},$$ given that \(y = 0\) when \(x = 0\), \(|x| < 1\) and \(|y| < 1\), has the solution $$y = x \cos\left(x\sqrt{1 - x^2}\right) + \sqrt{1 - x^2} \sin\left(x\sqrt{1 - x^2}\right).$$ [5]
SPS SPS FM Pure 2025 June Q14
5 marks Challenging +1.8
The three dimensional non-zero vector \(\mathbf{u}\) has the following properties:
  • The angle \(\theta\) between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 1 \\ 5 \\ 9 \end{pmatrix}\) is acute.
  • The (non-reflex) angle between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 9 \\ 5 \\ 1 \end{pmatrix}\) is \(2\theta\).
  • \(\mathbf{u}\) is perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\).
Find the angle \(\theta\). [5]
SPS SPS FM Pure 2025 February Q1
4 marks Moderate -0.5
The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = (1 \quad 3).$$ Calculate all possible products formed from two of these three matrices. [4]