Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2023 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}
  1. \(z_1\) is a point on \(L\) such that \(|z_1|\) is a minimum. Find the exact value of \(z_1\) in the form \(a + bi\) [4 marks]
SPS SPS FM Pure 2023 September Q6
8 marks Challenging +1.2
A curve has equation \(y = xe^{\frac{x}{2}}\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]
SPS SPS FM Pure 2023 September Q7
8 marks Standard +0.8
  1. Prove the identity \(\frac{\cos x}{\sec x + 1} + \frac{\cos x}{\sec x - 1} = 2\cot^2 x\) [3 marks]
  2. Hence, solve the equation $$\frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) + 1} = \cot\left(2\theta + \frac{\pi}{3}\right) - \frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) - 1}$$ in the interval \(0 \leq \theta \leq 2\pi\), giving your values of \(\theta\) to three significant figures where appropriate. [5 marks]
SPS SPS FM Pure 2023 September Q8
7 marks Standard +0.8
A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ 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 2023 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_1}
SPS SPS FM Pure 2023 November Q1
4 marks Standard +0.8
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 2023 November Q2
8 marks Standard +0.3
Fig. 5 shows the curve with polar equation \(r = a(3 + 2\cos\theta)\) for \(-\pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \includegraphics{figure_2}
  1. Write down the polar coordinates of the points A and B. [2]
  2. Explain why the curve is symmetrical about the initial line. [2]
  3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve. [4]
SPS SPS FM Pure 2023 November Q3
8 marks Standard +0.8
In this question you must show detailed reasoning. The roots of the equation \(2x^3 - 5x + 7 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\).
  1. Find \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]
  2. Find an equation with integer coefficients whose roots are \(2\alpha - 1\), \(2\beta - 1\) and \(2\gamma - 1\). [4]
SPS SPS FM Pure 2023 November Q4
7 marks Standard +0.8
In this question you must show detailed reasoning.
  1. Given that $$\frac{1}{r(r + 1)(r + 2)} = \frac{A}{r(r + 1)} + \frac{B}{(r + 1)(r + 2)}$$ show that \(A = \frac{1}{2}\) and find the value of \(B\). [3]
  2. Use the method of differences to find $$\sum_{r=10}^{98} \frac{1}{r(r + 1)(r + 2)}$$ giving your answer as a rational number. [4]
SPS SPS FM Pure 2023 November Q5
6 marks Moderate -0.3
  1. Use a Maclaurin series to find a quadratic approximation for \(\ln(1 + 2x)\). [1]
  2. Find the percentage error in using the approximation in part (a) to calculate \(\ln(1.2)\). [3]
  3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid. [2]
SPS SPS FM Pure 2023 November Q6
5 marks Standard +0.8
In this question you must show detailed reasoning. In this question you may assume the results for $$\sum_{r=1}^{n} r^3, \quad \sum_{r=1}^{n} r^2 \quad \text{and} \quad \sum_{r=1}^{n} r.$$ Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n^2(2n^2 - 1).$$ [5]
SPS SPS FM Pure 2023 November Q7
Challenging +1.8
    1. Show on an Argand diagram the locus of points given by the values of \(z\) satisfying $$|z - 4 - 3i| = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that $$\theta \in [\alpha, \alpha + \pi], \text{ where } \alpha = -\arctan\left(\frac{4}{3}\right),$$
    2. show that this locus of points can be represented by the polar curve with equation $$r = 8\cos\theta + 6\sin\theta$$ (6) The set of points \(A\) is defined by $$A = \left\{z : 0 \leqslant \arg z \leqslant \frac{\pi}{3}\right\} \cap \{z : |z - 4 - 3i| \leqslant 5\}$$
    1. Show, by shading on your Argand diagram, the set of points \(A\).
    2. Find the exact area of the region defined by \(A\), giving your answer in simplest form. (7)
SPS SPS FM Pure 2023 November Q8
Challenging +1.8
  1. Use a hyperbolic substitution and calculus to show that $$\int \frac{x^2}{\sqrt{x^2 - 1}} dx = \frac{1}{2}\left[x\sqrt{x^2 - 1} + \arcosh x\right] + k$$ where \(k\) is an arbitrary constant. (6) \includegraphics{figure_8} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac{4}{15}x \arcosh x \quad x \geqslant 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
  2. Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac{1}{15}\left[17\ln\left(3 + 2\sqrt{2}\right) - 6\sqrt{2}\right]$$ (5) This is the last question on the paper.
SPS SPS FM Pure 2024 February Q1
3 marks Moderate -0.5
The plane \(x + 2y + cz = 4\) is perpendicular to the plane \(2x - cy + 6z = 9\), where \(c\) is a constant. Find the value of \(c\). [3]
SPS SPS FM Pure 2024 February Q2
2 marks Easy -1.2
Find the mean value of \(f(x) = x^2 + 6x\) over the interval \([0, 3]\). [2]
SPS SPS FM Pure 2024 February Q3
6 marks Standard +0.3
It is given that \(1 - 3i\) is one root of the quartic equation $$z^4 - 2z^3 + pz^2 + rz + 80 = 0$$ where \(p\) and \(r\) are real numbers.
  1. Express \(z^4 - 2z^3 + pz^2 + rz + 80\) as the product of two quadratic factors with real coefficients. [4 marks]
  2. Find the value of \(p\) and the value of \(r\). [2 marks]
SPS SPS FM Pure 2024 February Q4
6 marks Challenging +1.2
Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series \((1 \times 2 \times 4) + (2 \times 3 \times 5) + (3 \times 4 \times 6) + \ldots\) where \(n\) is a positive integer. Give your answer in fully factorised form. [6]
SPS SPS FM Pure 2024 February Q5
6 marks Standard +0.8
The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 0 \quad u_{n+1} = \frac{5}{6 - u_n}$$ Prove by induction that, for all integers \(n \geq 1\), $$u_n = \frac{5^n - 5}{5^n - 1}$$ [6 marks]
SPS SPS FM Pure 2024 February Q6
7 marks Standard +0.3
  1. Explain why \(\int_1^\infty \frac{1}{x(2x + 5)} dx\) is an improper integral. [1]
  2. Prove that $$\int_1^\infty \frac{1}{x(2x + 5)} dx = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. [6]
SPS SPS FM Pure 2024 February Q7
7 marks Standard +0.8
In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square, \(S\).
  1. Find the area of \(S\). [3]
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]
SPS SPS FM Pure 2024 February Q8
8 marks Challenging +1.2
A linear transformation of the plane is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 1 & -2 \\ \lambda & 3 \end{pmatrix}\), where \(\lambda\) is a constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin. [5]
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines. [3]
SPS SPS FM Pure 2024 February Q9
9 marks Standard +0.8
In this question you must show detailed reasoning. The complex number \(-4 + i\sqrt{48}\) is denoted by \(z\).
  1. Determine the cube roots of \(z\), giving the roots in exponential form. [6]
The points which represent the cube roots of \(z\) are denoted by \(A\), \(B\) and \(C\) and these form a triangle in an Argand diagram.
  1. Write down the angles that any lines of symmetry of triangle \(ABC\) make with the positive real axis, justifying your answer. [3]
SPS SPS FM Pure 2024 February Q10
11 marks Challenging +1.8
The diagram shows the polar curve \(C_1\) with equation \(r = 2\sin\theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_10}
  1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q\sin 4\alpha$$ where \(\alpha = \sin^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]
SPS SPS FM Pure 2024 February Q11
7 marks Challenging +1.8
Three planes have equations \begin{align} (4k + 1)x - 3y + (k - 5)z &= 3
(k - 1)x + (3 - k)y + 2z &= 1
7x - 3y + 4z &= 2 \end{align}
  1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]
SPS SPS FM Pure 2024 February Q12
7 marks Challenging +1.2
Find the general solution of the differential equation $$x\frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]