Questions — SPS SPS FM (161 questions)

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SPS SPS FM 2020 September Q2
4 marks Challenging +1.2
A sequence of numbers \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 3$$ $$a_{n+1} = \frac{a_n - 3}{a_n - 2}, \quad n \in \mathbb{N}$$
  1. Find \(\sum_{r=1}^{100} a_r\) [3]
  2. Hence find \(\sum_{r=1}^{100} a_r + \sum_{r=1}^{99} a_r\) [1]
SPS SPS FM 2020 September Q3
4 marks Moderate -0.3
Using algebraic integration and making your method clear, find the exact value of $$\int_1^5 \frac{4x + 9}{x + 3} \, dx = a + \ln b$$ where \(a\) and \(b\) are constants to be found [4]
SPS SPS FM 2020 September Q4
5 marks Moderate -0.8
  1. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$$ [3]
Adele uses \(\theta = 5°\) to test the approximation in part (a). Adele's working is shown below.
Using my calculator, \(1 + 4\cos(5°) + 3\cos^2(5°) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5\theta^2\) gives \(8 - 5(5)^2 = -117\)
Therefore, \(1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2\) is not true for \(\theta = 5°\)
    1. Identify the mistake made by Adele in her working.
    2. Show that \(8 - 5\theta^2\) can be used to give a good approximation to \(1 + 4\cos \theta + 3\cos^2 \theta\) for an angle of size \(5°\) [2]
SPS SPS FM 2020 September Q5
7 marks Standard +0.3
\includegraphics{figure_5} Figure 5 shows a sketch of the curve with parametric equations $$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
  1. Show that the area of \(R\) is given $$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
  2. Hence, using algebraic integration, find the exact area of \(R\). [3]
SPS SPS FM 2020 September Q6
5 marks Standard +0.8
A sequence is defined by \(U_n = 2^{n+1} + 9 \times 13^n\) for positive integer values of \(n\). Prove by induction that \(U_n\) is divisible by 11. [5]
SPS SPS FM 2020 September Q7
5 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve with equation $$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]
SPS SPS FM 2020 September Q8
8 marks Challenging +1.2
\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.
  1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
  2. find, showing your working, the integer value of \(k\). [5]
SPS SPS FM 2020 September Q9
7 marks Standard +0.8
Show that $$\int_0^{\pi/2} \frac{\sin 2\theta}{1 + \cos \theta} \, d\theta = 2 - 2\ln 2$$ [7]
SPS SPS FM 2020 September Q10
5 marks Challenging +1.2
A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$ A point \(P\) lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis. Find the exact coordinates of all possible points \(P\), justifying your answer. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]
SPS SPS FM 2020 September Q11
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 2020 September Q12
9 marks Standard +0.3
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$ \includegraphics{figure_9}
  1. Find, in modulus-argument form, the complex number represented by the point P. [2]
  2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    lie within this region. [4]
SPS SPS FM 2022 February Q1
4 marks Easy -1.3
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & 1 \\ 0 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}\).
  1. Find \(\mathbf{A} + 3\mathbf{B}\). [2]
  2. Show that \(\mathbf{A} - \mathbf{B} = k\mathbf{I}\), where \(\mathbf{I}\) is the identity matrix and \(k\) is a constant whose value should be stated. [2]
SPS SPS FM 2022 February Q2
8 marks Moderate -0.8
The complex numbers \(3 - 2i\) and \(2 + i\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + iy\) and showing clearly how you obtain these answers,
  1. \(2z - 3w\), [2]
  2. \((iz)^2\), [3]
  3. \(\frac{z}{w}\). [3]
SPS SPS FM 2022 February Q3
8 marks Moderate -0.3
The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_3}
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
  2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]
SPS SPS FM 2022 February Q4
4 marks Moderate -0.8
The transformation \(S\) is a shear parallel to the \(x\)-axis in which the image of the point \((1, 1)\) is the point \((0, 1)\).
  1. Draw a diagram showing the image of the unit square under \(S\). [2]
  2. Write down the matrix that represents \(S\). [2]
SPS SPS FM 2022 February Q5
11 marks Moderate -0.8
  1. Sketch the curve \(y = \left(\frac{1}{2}\right)^x\), and state the coordinates of any point where the curve crosses an axis. [3]
  2. Use the trapezium rule, with 4 strips of width 0.5, to estimate the area of the region bounded by the curve \(y = \left(\frac{1}{2}\right)^x\), the axes, and the line \(x = 2\). [4]
  3. The point \(P\) on the curve \(y = \left(\frac{1}{2}\right)^x\) has \(y\)-coordinate equal to \(\frac{1}{6}\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac{\log_{10} 3}{\log_{10} 2}.$$ [4]
SPS SPS FM 2022 February Q6
7 marks Moderate -0.8
In an Argand diagram the loci \(C_1\) and \(C_2\) are given by $$|z| = 2 \quad \text{and} \quad \arg z = \frac{1}{4}\pi$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
  2. Hence find, in the form \(x + iy\), the complex number representing the point of intersection of \(C_1\) and \(C_2\). [2]
SPS SPS FM 2022 February Q7
8 marks Moderate -0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\).
  1. Find \(\mathbf{A}^2\) and \(\mathbf{A}^3\). [3]
  2. Hence suggest a suitable form for the matrix \(\mathbf{A}^n\). [1]
  3. Use induction to prove that your answer to part (ii) is correct. [4]
SPS SPS FM 2022 February Q8
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [3]
  2. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 + 2x)^2}{(1 - 3x)^2}\) in ascending powers of \(x\). [4]
SPS SPS FM 2022 February Q9
8 marks Standard +0.3
The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$
  1. Find the angle between \(AB\) and \(AC\). [6]
  2. Find the area of triangle \(ABC\). [2]
SPS SPS FM 2021 November Q1
3 marks Moderate -0.3
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The roots of the equation $$x^3 - 8x^2 + 28x - 32 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the value of $$(\alpha + 2)(\beta + 2)(\gamma + 2).$$ [3 marks]
SPS SPS FM 2021 November Q2
3 marks Standard +0.3
The equation of a curve in polar coordinates is $$r = 11 + 9 \sec \theta.$$ Show that a cartesian equation of the curve is $$(x - 9)\sqrt{x^2 + y^2} = 11x.$$ [3 marks]
SPS SPS FM 2021 November Q3
6 marks Standard +0.3
The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|.$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6 marks]
SPS SPS FM 2021 November Q4
4 marks Moderate -0.3
Prove that $$\sum_{r=1}^{n} 18(r^2 - 4) = n(6n^2 + 9n - 69).$$ [4 marks]
SPS SPS FM 2021 November Q5
4 marks Standard +0.8
Use a trigonometrical substitution to show that $$\int_0^2 \frac{1}{(16 - x^2)^{\frac{3}{2}}} dx = \frac{1}{16\sqrt{3}}$$ [4 marks]