Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2023 February Q1
4 marks Moderate -0.8
Find \(\sum_{r=1}^{n}(2r^2 - 1)\), expressing your answer in fully factorised form. [4]
SPS SPS FM Pure 2023 February Q2
4 marks Moderate -0.8
Solve the equation \(2z - 5iz^* = 12\). [4]
SPS SPS FM Pure 2023 February Q3
3 marks Standard +0.8
In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac{1}{2}x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{2}\pi\). \includegraphics{figure_4} This region is rotated through \(2\pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated. [3]
SPS SPS FM Pure 2023 February Q4
8 marks Standard +0.3
The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) is perpendicular to \(\Pi\). [2]
  2. Hence find a Cartesian equation of \(\Pi\). [2]
The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)
  1. determine the possible coordinates of \(A\). [4]
SPS SPS FM Pure 2023 February Q5
6 marks Standard +0.3
Prove by induction that for all positive integers \(n\) $$f(n) = 3^{2n+4} - 2^{2n}$$ is divisible by 5 [6]
SPS SPS FM Pure 2023 February Q6
4 marks Standard +0.8
In this question you must show detailed reasoning. Find \(\int_{2}^{\infty} \frac{1}{4+x^2} \, dx\). [4]
SPS SPS FM Pure 2023 February Q7
10 marks Challenging +1.3
  1. Prove that $$\tanh^{-1}(x) = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad -k < x < k$$ stating the value of the constant \(k\). [5]
  2. Hence, or otherwise, solve the equation $$2x = \tanh\left(\ln \sqrt{2-3x}\right)$$ [5]
SPS SPS FM Pure 2023 February Q8
6 marks Challenging +1.8
The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]
SPS SPS FM Pure 2023 February Q9
7 marks Standard +0.8
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation \(|z - 3| = 2\) [1]
\includegraphics{figure_9}
  1. There is a unique complex number \(w\) that satisfies both \(|w - 3| = 2\) and \(\arg(w + 1) = \alpha\) where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).
    1. Find the value of \(\alpha\). [2]
    2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4]
SPS SPS FM Pure 2023 February Q10
10 marks Challenging +1.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + \frac{2y}{x} = \frac{x+3}{x(x-1)(x^2+3)} \quad (x > 1)$$ [8]
  2. Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f(x)\). [2]
SPS SPS FM Pure 2023 February Q11
9 marks Challenging +1.2
In an Argand diagram, the points \(A\), \(B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2i\).
  1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [6]
The points \(D\), \(E\) and \(F\) are the midpoints of the sides of triangle \(ABC\).
  1. Find the exact area of triangle \(DEF\). [3]
SPS SPS FM Pure 2023 February Q12
11 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$
  1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
  2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
    3x - 6y + 4z &= 1
    3x + 2y - z &= 0 \end{align} [5]
    1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
    2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]
SPS SPS FM Pure 2023 February Q13
11 marks Challenging +1.8
In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin \theta} e^{\frac{1}{2}\cos \theta}\) for \(0 \leqslant \theta \leqslant \pi\). \includegraphics{figure_13}
  1. Find the exact area enclosed by the curve. [4]
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}} e^{\frac{1}{6}}\). [7]
SPS SPS FM Pure 2023 February Q14
7 marks Challenging +1.3
  1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln\left(\frac{1}{2} + \cos x\right)\). [4]
  2. By considering the root of the equation \(\ln\left(\frac{1}{2} + \cos x\right) = 0\) deduce that \(\pi \approx 3\sqrt{3 \ln\left(\frac{3}{2}\right)}\). [3]
SPS SPS FM Pure 2024 January Q1
7 marks Standard +0.8
Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at \(P(1, 2)\). \includegraphics{figure_1} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7]
SPS SPS FM Pure 2024 January Q2
6 marks Standard +0.3
  1. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
  2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ where $$f(x) = k - |2x - 3k|$$ [2]
SPS SPS FM Pure 2024 January Q3
4 marks Moderate -0.5
Find the value of the integral: $$\int_0^1 \frac{x^{\frac{1}{2}} + x^{-\frac{1}{3}}}{x} \, dx$$ [4]
SPS SPS FM Pure 2024 January Q4
13 marks Standard +0.3
The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.
  1. Find the value of \(c\) and the value of \(d\). [3]
The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.
  1. Find the position vector of \(P\). [6]
  2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]
SPS SPS FM Pure 2024 January Q5
13 marks Standard +0.8
Let $$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$
  1. Express \(f(x)\) in terms of partial fractions [5]
  2. Hence, or otherwise, find the series expansion of \(f(x)\), in ascending powers of \(x\), up to and including the term in \(x^2\). Simplify each term. [6]
  3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac{1}{2}\). [2]
SPS SPS FM Pure 2024 January Q6
10 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} -2 & 5 \\ 6 & k \end{pmatrix}$$ where \(k\) is a constant. Given that $$\mathbf{M}^2 + 11\mathbf{M} = a\mathbf{I}$$ where \(a\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix,
    1. determine the value of \(a\)
    2. show that \(k = -9\) [3]
  2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf{M}\). [6]
  3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer. [1]
SPS SPS FM Pure 2024 January Q7
14 marks Standard +0.3
The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]
The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$\arg(z - 6) = -\frac{3\pi}{4}$$
  1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies. [4]
  2. Find the complex number for which both \(|z - 6i| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]
SPS SPS FM Pure 2023 September Q1
5 marks Moderate -0.8
$$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}$$
  1. Find \(\mathbf{A}^{-1}\) [2 marks]
  2. The determinant of \(\mathbf{A}^2\) is equal to 4. Find the possible values of \(k\). [3 marks]
SPS SPS FM Pure 2023 September Q2
5 marks Moderate -0.3
A sequence \(u_n\) is defined by \(u_{n+1} = 2u_n + 3\) and \(u_1 = 1\). Prove by induction that \(u_n = 4 \times 2^{n-1} - 3\) for all positive integers \(n\). [5]
SPS SPS FM Pure 2023 September Q3
5 marks Standard +0.3
A finite region is bounded by the curve with equation \(y = x + x^{-\frac{3}{2}}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi\left(a\sqrt{2} + b\right)\), where \(a\) and \(b\) are rational numbers to be determined. [5 marks]
SPS SPS FM Pure 2023 September Q4
13 marks Standard +0.8
The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$
  1. Find an expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). [2]
The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).
  1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]
The line \(l\) intersects the curve \(C\) again at the point \(Q\).
  1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]