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SPS SPS FM Pure 2025 January Q2
8 marks Standard +0.3
  1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]
SPS SPS FM Pure 2025 January Q3
8 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) is constant. A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).
  1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
  2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
  3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [2]
SPS SPS FM Pure 2025 January Q4
12 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$
  1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]
It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).
  1. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]
The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).
  1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
  2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]
SPS SPS FM Pure 2025 January Q5
11 marks Standard +0.3
With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
  2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]
The point \(A\), with position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\), lies on \(l_1\) The point \(B\) is the image of \(A\) after reflection in the line \(l_2\)
  1. Find the position vector of \(B\). [3]
SPS SPS FM Pure 2025 January Q6
12 marks Standard +0.3
You are given the complex number \(w = 2 + 2\sqrt{3}i\).
  1. Express \(w\) in modulus-argument form. [3]
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]
SPS SPS FM Pure 2025 January Q7
8 marks Standard +0.8
A candlestick has base diameter \(8\) cm and height \(28\) cm, as shown in Figure \(9\). A model of the candlestick is shown in Figure \(10\), together with the equations that were used to create the model. \includegraphics{figure_7}
  1. Show that the volume generated by rotating the shaded region (in Figure \(10\)) \(2\pi\) radians about the \(y\)-axis is \(\frac{112}{15}\pi\). [4]
  2. Hence find the volume of metal needed to create the candlestick. [4]
SPS SPS FM 2025 February Q1
7 marks Challenging +1.2
The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). \includegraphics{figure_1} The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is 48 units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
SPS SPS FM 2025 February Q2
5 marks Moderate -0.3
  1. Find the first three terms in the expansion of \((1-2x)^{-1}\) in ascending powers of \(x\), where \(|x| < \frac{1}{2}\). [3]
  2. Hence find the coefficient of \(x^2\) in the expansion of \(\frac{x+3}{\sqrt{1-2x}}\). [2]
SPS SPS FM 2025 February Q3
4 marks Moderate -0.3
Express \(\frac{9x^2+43x+8}{(3+x)(1-x)(2x+1)}\) in partial fractions. [4]
SPS SPS FM 2025 February Q4
5 marks Moderate -0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a constant.
  1. Find \(\mathbf{A}^{-1}\). [2]
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\).
  1. Given that \(\mathbf{PA} = \mathbf{B}\), find the matrix \(\mathbf{P}\). [3]
SPS SPS FM 2025 February Q5
10 marks Moderate -0.8
  1. \(P\), \(Q\) and \(T\) are three transformations in 2-D. \(P\) is a reflection in the \(x\)-axis. \(\mathbf{A}\) is the matrix that represents \(P\). Write down the matrix \(\mathbf{A}\). [1]
  2. \(Q\) is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). \(\mathbf{B}\) is the matrix that represents \(Q\). Find the matrix \(\mathbf{B}\). [2]
  3. \(T\) is \(P\) followed by \(Q\). \(\mathbf{C}\) is the matrix that represents \(T\). Determine the matrix \(\mathbf{C}\). [2]
  4. \(L\) is the line whose equation is \(y = x\). Explain whether or not \(L\) is a line of invariant points under \(T\). [2]
  5. An object parallelogram, \(M\), is transformed under \(T\) to an image parallelogram, \(N\). Explain what the value of the determinant of \(\mathbf{C}\) means about • the area of \(N\) compared to the area of \(M\). • the orientation of \(N\) compared to the orientation of \(M\). [3]
SPS SPS FM 2025 February Q6
8 marks Standard +0.3
The equations of two lines are \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})\).
  1. Show that these lines meet, and find the coordinates of the point of intersection. [5]
  2. Find the acute angle between these lines. [3]
SPS SPS FM 2025 February Q7
6 marks Challenging +1.2
Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and \(\text{Re}(z) \geq 9\). [6]
SPS SPS FM 2025 February Q8
4 marks Moderate -0.3
A locus \(C_1\) is defined by \(C_1 = \{z : |z + i| \leq |z - 2i|\}\).
  1. Indicate by shading on the Argand diagram below the region representing \(C_1\). [2] \includegraphics{figure_8}
  2. Find the cartesian equation of the boundary line of the region representing \(C_1\), giving your answer in the form \(ax + by + c = 0\). [2]
SPS SPS FM 2025 February Q9
9 marks Standard +0.8
\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
  1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
  2. Use the model to find the volume of water that the bottle can contain. [7]
SPS SPS SM 2025 February Q1
2 marks Easy -1.2
Given that \((x - 2)\) is a factor of \(2x^3 + kx - 4\), find the value of the constant \(k\). [2]
SPS SPS SM 2025 February Q2
13 marks Standard +0.3
  1. \includegraphics{figure_2} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(ABCDEF\). The base \(ACFD\) of the roof is a horizontal rectangle, and the cross-section \(ABC\) of the roof is an isosceles triangle with \(AB = BC\). The lengths of \(AC\) and \(CF\) are \(2x\) cm and \(y\) cm respectively, and the height of \(BE\) above the base of the roof is \(x\) cm. The total surface area of the five faces of the roof is \(600\) cm\(^2\) and the volume of the roof is \(V\) cm\(^3\). Show that \(V = kx (300 - x^2)\), where \(k = \sqrt{a + b}\) and \(a\) and \(b\) are integers to be determined. [6]
  2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum. [4]
  3. Find the maximum volume of the roof. Give your answer in cm\(^3\), correct to the nearest integer. [1]
  4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state. [2]
SPS SPS SM 2025 February Q3
6 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a sector \(AOB\) of a circle with centre \(O\). The length of the arc \(AB\) is \(6\) cm and the area of the sector \(AOB\) is \(24\) cm\(^2\). Find the area of the shaded segment enclosed by the arc \(AB\) and the chord \(AB\), giving your answer correct to \(3\) significant figures. [6]
SPS SPS SM 2025 February Q4
6 marks Moderate -0.3
  1. The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than \(2\). Given that \(n^3 + 1 = (n + 1) (n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
  2. Prove that \(K\) has at least two distinct factors other than \(1\) and \(K\). [5]
SPS SPS SM 2025 February Q5
7 marks Standard +0.3
A curve has the following properties: • The gradient of the curve is given by \(\frac{dy}{dx} = -2x\). • The curve passes through the point \((4, -13)\). Determine the coordinates of the points where the curve meets the line \(y = 2x\). [7]
SPS SPS SM 2025 February Q6
9 marks Standard +0.3
For all real values of \(x\), the functions \(f\) and \(g\) are defined by \(f (x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.
  1. Find \(fg(x)\). Determine the range of \(fg(x)\) in terms of \(a\). [4]
  2. If \(fg(2) = 144\), find the value of \(a\). [3]
  3. Determine whether the function \(fg\) has an inverse. [2]
SPS SPS SM 2025 February Q7
8 marks Standard +0.3
  1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ [3]
  2. Hence solve the equation $$2 \sin 2\theta \tan 2\theta = \cos 2\theta + 5,$$ giving all values of \(\theta\) between \(0°\) and \(180°\), correct to \(1\) decimal place. [5]
SPS SPS SM 2025 February Q8
7 marks Challenging +1.2
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
SPS SPS FM Statistics 2025 April Q1
8 marks Moderate -0.3
It is known that, under standard conditions, 12% of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.
  1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample. [1]
  2. Test, at the 5% significance level, whether the proportion of defective light bulbs has decreased under the new process. [7]
SPS SPS FM Statistics 2025 April Q2
13 marks Moderate -0.8
In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: 20 | 3 5 7 9 21 | 0 2 5 6 8 22 | 1 3 4 5 7 9 23 | 0 2 5 8 24 | 1 4 6 7 25 | 2 5 Key: 21 | 0 represents a reaction time of 210 milliseconds
  1. State the median reaction time. [1]
  2. Calculate the interquartile range of these reaction times. [2]
  3. Find the mean and standard deviation of these reaction times. [3]
  4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table. [1]
  5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
    1. the median
    2. the mean
    3. the standard deviation
    [4]
  6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context [2]