Questions — SPS (686 questions)

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SPS SPS FM 2026 November Q8
5 marks Moderate -0.3
Prove by induction that \(7 \times 9^n - 15\) is divisible by \(4\), for all integers \(n \geq 0\). [5]
SPS SPS FM 2026 November Q9
8 marks Moderate -0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio \(r\) and first term \(a\). Given \(r \neq 1\) and \(a \neq 0\)
  1. prove that $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]
Given also that \(S_{10}\) is four times \(S_5\)
  1. find the exact value of \(r\). [4]
SPS SPS FM 2026 November Q10
6 marks Moderate -0.3
  1. Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$(1 + kx)^{10}$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. [3] Given that in the expansion of \((1 + kx)^{10}\) the coefficient \(x^3\) is 3 times the coefficient of \(x\),
  2. find the possible values of \(k\). [3]
SPS SPS SM 2025 November Q1
7 marks Easy -1.3
Do not use a calculator for this question
  1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
    1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
    2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]
SPS SPS SM 2025 November Q2
12 marks Easy -1.3
  1. Write \(4x^2 - 24x + 27\) in the form \(a(x - b)^2 + c\). [4]
  2. State the coordinates of the minimum point on the curve \(y = 4x^2 - 24x + 27\). [2]
  3. Solve the equation \(4x^2 - 24x + 27 = 0\). [3]
  4. Sketch the graph of the curve \(y = 4x^2 - 24x + 27\). [3]
SPS SPS SM 2025 November Q3
5 marks Standard +0.3
The equation \(kx^2 + 4x + (5 - k) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\). Find the set of possible values of \(k\). Write your answer using set notation. [5]
SPS SPS SM 2025 November Q4
9 marks Moderate -0.8
Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of
  1. \(\log_2 (16x)\), [2]
  2. \(\log_2 \left(\frac{x^4}{2}\right)\) [3]
  3. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]
(Total 9 marks)
SPS SPS SM 2025 November Q5
Moderate -0.8
An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 29 terms is 1102.
  1. Show that \(a + 14d = 38\). (3 marks)
  2. The sum of the second term and the seventh term is 13. Find the value of \(a\) and the value of \(d\). (4 marks)
SPS SPS SM 2025 November Q6
8 marks Standard +0.8
A sequence \(t_1, t_2, t_3, t_4, t_5, \ldots\) is given by $$t_{n+1} = at_n + 3n + 2, \quad t \in \mathbb{N}, \quad t_1 = -2,$$ where \(a\) is a non zero constant.
  1. Given that \(\sum_{r=1}^{3} (r^3 + t_r) = 12\), determine the possible values of \(a\). [4]
  2. Evaluate \(\sum_{r=8}^{31} (t_{r+1} - at_r)\). [4]
SPS SPS SM 2025 November Q7
11 marks Moderate -0.8
There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
  2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]
\(t\)12345678910
\(\log_{10} y\)1.511.581.982.68
SPS SPS SM 2025 November Q8
11 marks Standard +0.3
The circles \(C_1\) and \(C_2\) have respective equations $$x^2 + y^2 - 6x - 2y = 15$$ $$x^2 + y^2 - 18x + 14y = 95.$$
  1. By considering the coordinates of the centres and the lengths of the radii of \(C_1\) and \(C_2\), show that \(C_1\) and \(C_2\) touch internally at some point \(P\). [4]
  2. Determine the coordinates of \(P\). [3]
  3. Find the equation of the common tangent to the circles at P. [4]
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]
SPS SPS FM Pure 2025 September Q5
6 marks Standard +0.3
  1. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg(z + i) = \frac{\pi}{6}$$ [2 marks] \includegraphics{figure_5}
  2. \(z_1\) is a point on \(L\) such that \(|z|\) is a minimum. Find the exact value of \(z_1\) in the form \(a + bi\) [4 marks]
SPS SPS FM Pure 2025 September Q8
7 marks Standard +0.8
A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]
SPS SPS FM Pure 2025 September Q9
18 marks Standard +0.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$
    1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
    2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]
  1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
    1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
    2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{\frac{x}{2}}$$ [2 marks]
    3. Use the recurrence relation $$x_{n+1} = e - e^{\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
    4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_9}
SPS SPS FM Pure 2026 November Q1
4 marks Moderate -0.3
The complex number \(z\) satisfies the equation \(z + 2iz^* + 1 - 4i = 0\). You are given that \(z = x + iy\), where \(x\) and \(y\) are real numbers. Determine the values of \(x\) and \(y\). [4]
SPS SPS FM Pure 2026 November Q2
6 marks Moderate -0.8
Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]
SPS SPS FM Pure 2026 November Q3
9 marks Challenging +1.2
The figure below shows the curve with cartesian equation \((x^2 + y^2)^2 = xy\). \includegraphics{figure_3}
  1. Show that the polar equation of the curve is \(r^2 = a \sin b\theta\), where \(a\) and \(b\) are positive constants to be determined. [3]
  2. Determine the exact maximum value of \(r\). [2]
  3. Determine the area enclosed by one of the loops. [4]
SPS SPS FM Pure 2026 November Q4
9 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. The curves with equations $$y = \frac{3}{4}\sinh x \text{ and } y = \tanh x + \frac{1}{5}$$ intersect at just one point \(P\)
    1. Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation $$15e^{4x} - 48e^{3x} + 32e^x - 15 = 0$$ [3]
    2. Show that \(e^x = 3\) is a solution of this equation. [1]
    3. Hence state the exact coordinates of \(P\). [1]
  2. Show that $$\int_{-4}^{0} \frac{e^x}{x^2} dx = e^{-\frac{1}{4}}$$ [4]
SPS SPS FM Pure 2026 November Q5
5 marks Standard +0.8
Use the method of differences to prove that for \(n > 2\) $$\sum_{r=2}^{n} \frac{4}{r^2-1} = \frac{(pn+q)(n-1)}{n(n+1)}$$ where \(p\) and \(q\) are constants to be determined. [5]
SPS SPS FM Pure 2026 November Q6
10 marks Standard +0.3
  1. \(z_1 = a + bi\) and \(z_2 = c + di\) where \(a\), \(b\), \(c\) and \(d\) are real constants. Given that
    • \(b > d\)
    • \(z_1 + z_2\) is real
    • \(|z_1| = \sqrt{13}\)
    • \(|z_2| = 5\)
    • \(\text{Re}(z_2 - z_1) = 2\)
    show that \(a = 2\) and determine the value of each of \(b\), \(c\) and \(d\) [5]
    1. On the same Argand diagram
      showing the coordinates of any points of intersection with the axes. [2]
    2. Determine the range of possible values of \(|z - w|\) [3]
SPS SPS FM Pure 2026 November Q7
4 marks Standard +0.8
In this question you must show detailed reasoning. Evaluate \(\int_0^{\frac{1}{2}} \frac{2}{x^2 - x + 1} dx\). Give your answer in exact form. [4]