Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2025 February Q2
4 marks Moderate -0.8
The complex number \(z\) satisfies the equation \(z^2 - 4iz + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
SPS SPS FM Pure 2025 February Q3
6 marks Standard +0.8
Prove by mathematical induction that \(\sum_{r=1}^{n} (r \times r!) = (n+1)! - 1\) for all positive integers \(n\). [6]
SPS SPS FM Pure 2025 February Q4
5 marks Standard +0.3
The cubic equation $$2x^3 + 6x^2 - 3x + 12 = 0$$ has roots \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the cubic equation whose roots are \((\alpha + 3)\), \((\beta + 3)\) and \((\gamma + 3)\), giving your answer in the form \(pw^3 + qw^2 + rw + s = 0\), where \(p\), \(q\), \(r\) and \(s\) are integers to be found. [5]
SPS SPS FM Pure 2025 February Q5
9 marks Standard +0.3
In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).
  1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \quad b \in \mathbb{R}$$ [3]
The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\).
  1. Find the complex numbers \(z_1\) and \(z_2\). [6]
SPS SPS FM Pure 2025 February Q6
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{21}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM Pure 2025 February Q7
10 marks Standard +0.3
Line \(l_1\) has Cartesian equation $$x - 3 = \frac{2y + 2}{3} = 2 - z$$
  1. Write the equation of line \(l_1\) in the form $$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$ where \(\lambda\) is a parameter and \(\mathbf{a}\) and \(\mathbf{b}\) are vectors to be found. [2 marks]
  1. Line \(l_2\) passes through the points \(P(3, 2, 0)\) and \(Q(n, 5, n)\), where \(n\) is a constant.
    1. Show that the lines \(l_1\) and \(l_2\) are not perpendicular. [3 marks]
    2. Explain briefly why lines \(l_1\) and \(l_2\) cannot be parallel. [2 marks]
    3. Given that \(\theta\) is the acute angle between lines \(l_1\) and \(l_2\), show that $$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$ where \(p\) and \(q\) are constants to be found. [3 marks]
SPS SPS FM Pure 2025 February Q8
4 marks Standard +0.3
Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]
SPS SPS FM Pure 2025 February Q9
8 marks Challenging +1.2
$$f(z) = z^3 - 8z^2 + pz - 24$$ where \(p\) is a real constant. Given that the equation \(f(z) = 0\) has distinct roots $$\alpha, \beta \text{ and } \left(\alpha + \frac{12}{\alpha} - \beta\right)$$
  1. solve completely the equation \(f(z) = 0\) [6]
  2. Hence find the value of \(p\). [2]
SPS SPS FM Pure 2025 February Q1
4 marks Standard +0.3
The complex number \(z\) satisfies the equation \(z^2 - 4iz* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
SPS SPS FM Pure 2025 February Q2
6 marks Standard +0.8
Prove by mathematical induction that \(\sum_{r=1}^{n}(r \times r!) = (n + 1)! - 1\) for all positive integers \(n\). [6]
SPS SPS FM Pure 2025 February Q3
7 marks Challenging +1.2
The curve \(C\) has equation $$y = 31\sinh x - 2\sinh 2x \quad x \in \mathbb{R}$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\). [7]
SPS SPS FM Pure 2025 February Q4
9 marks Standard +0.3
The plane \(\Pi_1\) has equation $$\mathbf{r} = 2\mathbf{i} + 4\mathbf{j} - \mathbf{k} + \lambda (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) + \mu(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Find a Cartesian equation for \(\Pi_1\) [4]
The line \(l\) has equation $$\frac{x-1}{5} = \frac{y-3}{-3} = \frac{z+2}{4}$$
  1. Find the coordinates of the point of intersection of \(l\) with \(\Pi_1\) [3]
The plane \(\Pi_2\) has equation $$\mathbf{r}.(2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 5$$
  1. Find, to the nearest degree, the acute angle between \(\Pi_1\) and \(\Pi_2\) [2]
SPS SPS FM Pure 2025 February Q5
9 marks Standard +0.3
In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(-3 + 2i\) and \(5 - 4i\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).
  1. Find the equation of \(C\), giving your answer in the form $$|z - a| = b \quad a \in \mathbb{C}, \, b \in \mathbb{R}$$ [3]
The circle \(D\), with equation \(|z - 2 - 3i| = 2\), intersects \(C\) at the points representing the complex numbers \(z_1\) and \(z_2\)
  1. Find the complex numbers \(z_1\) and \(z_2\) [6]
SPS SPS FM Pure 2025 February Q6
10 marks Standard +0.8
$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)
  1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
  2. find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM Pure 2025 February Q7
8 marks Challenging +1.3
The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta, \quad \text{for } 0 \leq \theta \leq \frac{1}{4}\pi.$$
  1. Sketch the curve. [2]
  2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac{1}{4}\pi\). [6]
SPS SPS FM Pure 2025 February Q8
9 marks Challenging +1.3
  1. Solve the equation \(z^3 = \sqrt{2} - \sqrt{6}i\), giving your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(0 \leq \theta < 2\pi\) [5 marks]
  2. The transformation represented by the matrix \(\mathbf{M} = \begin{bmatrix} 5 & 1 \\ 1 & 3 \end{bmatrix}\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points. [4 marks]
SPS SPS FM Pure 2025 February Q9
5 marks Challenging +1.2
In this question, you must show detailed reasoning. Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
SPS SPS FM Pure 2025 February Q10
8 marks Challenging +1.2
Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.
  1. Find the possible values of \(k\). [3 marks]
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]
SPS SPS FM Pure 2025 February Q11
8 marks Challenging +1.8
The infinite series \(C\) and \(S\) are defined by $$C = \cos \theta + \frac{1}{2}\cos 5\theta + \frac{1}{4}\cos 9\theta + \frac{1}{8}\cos 13\theta + \ldots$$ $$S = \sin \theta + \frac{1}{2}\sin 5\theta + \frac{1}{4}\sin 9\theta + \frac{1}{8}\sin 13\theta + \ldots$$ Given that the series \(C\) and \(S\) are both convergent,
  1. show that $$C + iS = \frac{2e^{i\theta}}{2 - e^{4i\theta}}$$ [4]
  2. Hence show that $$S = \frac{4\sin \theta + 2\sin 3\theta}{5 - 4\cos 4\theta}$$ [4]
SPS SPS FM Pure 2025 February Q12
11 marks Challenging +1.8
The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5. The model uses the differential equation $$\frac{dP}{dt} - \frac{P}{t(1 + t^2)} = \frac{te^{-t}}{\sqrt{1 + t^2}}$$ Find \(P\) as a function of \(t\). [You may assume that \(\lim_{t \to \infty} te^{-t} = 0\)]. [11]
SPS SPS FM Pure 2025 February Q13
6 marks Moderate -0.3
  1. Write down the Maclaurin series of \(e^x\), in ascending power of \(x\), up to and including the term in \(x^3\) [1]
  2. Hence, without differentiating, determine the Maclaurin series of $$e^{(x^3-1)}$$ in ascending powers of \(x\), up to and including the term in \(x^3\), giving each coefficient in simplest form. [5]
SPS SPS FM Pure 2025 September Q1
5 marks Moderate -0.8
\(A = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}\)
  1. Find \(A^{-1}\) [2 marks]
  2. The determinant of \(A^2\) is equal to 4. Find the possible values of \(k\). [3 marks]
SPS SPS FM Pure 2025 September Q2
5 marks Standard +0.3
Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by 3, for all integers \(n \geq 0\). [5]
SPS SPS FM Pure 2025 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 2025 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 \(\frac{dy}{dx}\) 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]