Questions — SPS (686 questions)

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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 2025 October Q1
3 marks Easy -1.2
Determine the equation of the line that passes through the point \((1, 3)\) and is perpendicular to the line with equation \(3x + 6y - 5 = 0\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]
SPS SPS FM 2025 October Q2
6 marks Moderate -0.8
In a triangle \(ABC\), \(AB = 9\) cm, \(BC = 7\) cm and \(AC = 4\) cm.
  1. Show that \(\cos CAB = \frac{2}{3}\). [2]
  2. Hence find the exact value of \(\sin CAB\). [2]
  3. Find the exact area of triangle \(ABC\). [2]
SPS SPS FM 2025 October Q3
4 marks Moderate -0.5
Given the function \(f(x) = 3x^3 - 7x - 1\), defined for all real values of \(x\), prove from first principles that \(f'(x) = 9x^2 - 7\). [4]
SPS SPS FM 2025 October Q4
8 marks Moderate -0.3
The cubic polynomial \(2x^3 - kx^2 + 4x + k\), where \(k\) is a constant, is denoted by f(x). It is given that f'(2) = 16.
  1. Show that \(k = 3\). [3]
For the remainder of the question, you should use this value of \(k\).
  1. Use the factor theorem to show that \((2x + 1)\) is a factor of f(x). [2]
  2. Hence show that the equation f(x) = 0 has only one real root. [3]
SPS SPS FM 2025 October Q5
4 marks Standard +0.3
In this question you must show detailed reasoning. Consider the expansion of \(\left(\frac{x^2}{2} + \frac{a}{x}\right)^6\). The constant term is 960. Find the possible values of \(a\). [4]
SPS SPS FM 2025 October Q6
6 marks Moderate -0.3
The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$
  1. Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a^b\) where \(a\) and \(b\) are rational numbers. [4]
  2. Find the nature of the stationary point, justifying your answer. [2]
SPS SPS FM 2025 October Q7
7 marks Standard +0.8
The circle \(x^2 + y^2 + 2x - 14y + 25 = 0\) has its centre at the point C. The line \(7y = x + 25\) intersects the circle at points A and B. Prove that triangle ABC is a right-angled triangle. [7]
SPS SPS FM 2025 October Q8
4 marks Standard +0.8
A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]