Questions — SPS (686 questions)

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SPS SPS FM 2025 October Q9
7 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a sketch of a curve C with equation \(y = \text{f}(x)\), where f(x) is a quartic expression in \(x\). The curve • has maximum turning points at \((-1, 0)\) and \((5, 0)\) • crosses the \(y\)-axis at \((0, -75)\) • has a minimum turning point at \(x = 2\)
  1. Find the set of values of \(x\) for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
  2. Find the equation of C. You may leave your answer in factorised form. [3]
The curve \(C_1\) has equation \(y = \text{f}(x) + k\), where \(k\) is a constant. Given that the graph of \(C_1\) intersects the \(x\)-axis at exactly four places,
  1. find the range of possible values for \(k\). [2]
SPS SPS FM 2025 October Q10
4 marks Moderate -0.8
The graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis followed by a translation.
    1. State the scale factor of the stretch. [1]
    2. Give full details of the translation. [2]
Alternatively the graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.
  1. State the scale factor of the stretch parallel to the \(y\)-axis. [1]
SPS SPS FM 2025 October Q11
8 marks Standard +0.3
The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$
  1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
  2. Find the range of the function fg. [2]
  3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]
SPS SPS FM 2025 October Q12
6 marks Standard +0.3
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 2025 October Q13
8 marks Challenging +1.8
In this question you must show detailed reasoning. Solve the following equation for \(x\) in the interval \(0° < x < 180°\) $$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]
SPS SPS SM 2025 October Q1
8 marks Easy -1.8
Express each of the following in the form \(px^q\), where \(p\) and \(q\) are constants.
  1. \(\frac{2}{\sqrt[3]{x}}\) [1]
  2. \((5x\sqrt{x})^3\) [1]
  3. \(\sqrt{2x^3} \times \sqrt{8x^5}\) [1]
  4. \(x^5(27x^6)^{\frac{1}{3}}\) [2]
SPS SPS SM 2025 October Q2
3 marks Moderate -0.3
In this question you must show detailed reasoning. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]
SPS SPS SM 2025 October Q3
5 marks Moderate -0.8
The line \(l\) passes through the points \(A(-3, 0)\) and \(B\left(\frac{5}{3}, 22\right)\)
  1. Find the equation of \(l\) giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are constants. [3]
\includegraphics{figure_2} Figure 2 shows the line \(l\) and the curve \(C\), which intersect at \(A\) and \(B\). Given that
  • \(C\) has equation \(y = 2x^2 + 5x - 3\)
  • the region \(R\), shown shaded in Figure 2, is bounded by \(l\) and \(C\)
  1. use inequalities to define \(R\). [2]
SPS SPS SM 2025 October Q4
6 marks Moderate -0.8
  1. A sequence has terms \(u_1, u_2, u_3, \ldots\) defined by \(u_1 = 3\) and \(u_{n+1} = u_n^2 - 5\) for \(n \geq 1\).
    1. Find the values of \(u_2\), \(u_3\) and \(u_4\). [2]
    2. Describe the behaviour of the sequence. [1]
  2. The second, third and fourth terms of a geometric progression are 12, 8 and \(\frac{16}{3}\). Determine the sum to infinity of this geometric progression. [3]
SPS SPS SM 2025 October Q5
5 marks Standard +0.3
In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the cuboid \(ABCDEFGH\) where \(AD = 3\) cm, \(AF = (2x + 1)\) cm and \(DC = (x - 2)\) cm. The volume of the cuboid is at most 9 cm³. Find the range of possible values of \(x\). Give your answer in interval notation. [5]
SPS SPS SM 2025 October Q6
3 marks Moderate -0.8
Sketch the graph of $$y = (x - k)^2(x + 2k)$$ where \(k\) is a positive constant. Label the coordinates of the points where the graph meets the axes. \includegraphics{figure_6} [3]
SPS SPS SM 2025 October Q7
7 marks Standard +0.3
In this question you must show detailed reasoning. Solve the following equations.
  1. \(y^6 + 7y^3 - 8 = 0\) [3]
  2. \(9^{x+1} + 3^x = 8\) [4]
SPS SPS SM 2025 October Q8
7 marks Moderate -0.3
In this question you must show detailed reasoning. Solutions using calculator technology are not acceptable. Solve the following equations.
  1. \(2\log_3(x + 1) = 1 + \log_3(x + 7)\) [4]
  2. \(\log_y\left(\frac{1}{x}\right) = -\frac{3}{2}\) [3]
SPS SPS SM 2025 October Q9
4 marks Moderate -0.8
  1. Show that the equation \(x^2 + kx - k^2 = 0\) has real roots for all real values of \(k\). [2]
  2. Show that the roots of the equation \(x^2 + kx - k^2 = 0\) are \(\left(\frac{-1 \pm \sqrt{5}}{2}\right)k\). [2]
SPS SPS SM 2025 October Q10
7 marks Moderate -0.3
\(f(x) = x^4 + bx + c\) \((x-2)\) is a factor of \(f(x)\). \(f(-3) = 35\).
  1. Find \(b\) and \(c\). [4]
  2. Hence express \(f(x)\) as the product of linear and cubic factors. [3]
SPS SPS SM 2025 October Q11
9 marks Moderate -0.8
A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M\) kg of salt remaining after \(t\) minutes by \(M = ak^t\) where \(a\) and \(k\) are constants.
  1. Show that the model for \(M\) can be rewritten in the form \(\log_{10} M = t\log_{10} k + \log_{10} a\). [1]
The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.
\(t\) minutes813213550
\(M\) kg0.40.30.20.10.05
The student uses this data and plots \(y = \log_{10} M\) against \(x = t\) using graph drawing software. The software gives \(y = -0.0214x - 0.2403\) for the equation of the line of best fit.
    1. Find the values of \(a\) and \(k\) that follow from the equation of the line. [2]
    2. Interpret the value of \(k\) in context. [1]
  2. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg. Comment on the accuracy when the model is used to estimate the initial mass of the salt. [1]
  3. Use the model to predict the value of \(t\) at which \(M = 0.01\) kg. [2]
  4. Rewrite the model for \(M\) in the form \(M = ae^{-ht}\) where \(h\) is a constant to be determined. [2]
SPS SPS SM 2025 October Q12
5 marks Standard +0.3
An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio \(r\).
    1. Show that \(r = \frac{a + 2d}{a}\). [1]
    2. Find \(d\) in terms of \(a\). [2]
  2. Find the common ratio of the geometric progression. [2]
SPS SPS SM 2025 October Q13
9 marks Standard +0.3
The circle \(C\) has equation $$x^2 + y^2 + 10x - 4y + 1 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the exact radius of \(C\) [2]
    The line with equation \(y = k\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\), giving your answer in set notation. [2]
  3. The line with equation \(y = mx + 4\) is a tangent to \(C\). Find possible exact values of \(m\). [5]
SPS SPS FM 2026 November Q1
6 marks Easy -1.2
  1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
  2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]
SPS SPS FM 2026 November Q2
7 marks Easy -1.2
\(f(x) = x^3 + 4x^2 + x - 6\).
  1. Use the factor theorem to show that \((x + 2)\) is a factor of \(f(x)\). [2]
  2. Factorise \(f(x)\) completely. [4]
  3. Write down all the solutions to the equation $$x^3 + 4x^2 + x - 6 = 0.$$ [1]
SPS SPS FM 2026 November Q3
12 marks Moderate -0.3
The curve \(C\) has equation $$y = \frac{1}{2}x^3 - 9x^2 + \frac{8}{x} + 30, \quad x > 0$$
  1. Find \(\frac{dy}{dx}\). [4]
  2. Show that the point \(P(4, -8)\) lies on \(C\). [2]
  3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]
SPS SPS FM 2026 November Q4
6 marks Standard +0.3
  1. The curves \(e^x - 2e^y = 1\) and \(2e^x + 3e^{2y} = 41\) intersect at the point \(P\). Show that the \(y\)-coordinate of \(P\) satisfies the equation \(3e^{2y} + 4e^y - 39 = 0\). [1]
  2. In this question you must show detailed reasoning. Hence find the exact coordinates of \(P\). [5]
SPS SPS FM 2026 November Q5
8 marks Standard +0.3
  1. Show that the equation $$4\cos\theta - 1 = 2\sin\theta\tan\theta$$ can be written in the form $$6\cos^2\theta - \cos\theta - 2 = 0$$ [4]
  2. Hence solve, for \(0 \leq x < 90°\) $$4\cos 3x - 1 = 2\sin 3x\tan 3x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) [4]
SPS SPS FM 2026 November Q6
5 marks Standard +0.3
Find the values of \(x\) such that $$2\log_3 x - \log_3(x - 2) = 2$$ [5]
SPS SPS FM 2026 November Q7
10 marks Moderate -0.8
\(f(x) = 2x^2 + 4x + 9 \quad x \in \mathbb{R}\)
  1. Write \(f(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are integers to be found. [3]
  2. Sketch the curve with equation \(y = f(x)\) showing any points of intersection with the coordinate axes and the coordinates of any turning point. [3]
    1. Describe fully the transformation that maps the curve with equation \(y = f(x)\) onto the curve with equation \(y = g(x)\) where $$g(x) = 2(x - 2)^2 + 4x - 3 \quad x \in \mathbb{R}$$
    2. Find the range of the function $$h(x) = \frac{21}{2x^2 + 4x + 9} \quad x \in \mathbb{R}$$ [4]