Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2022 February Q3
9 marks Standard +0.3
The line \(l_1\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}\). The plane \(\Pi\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} = 4\).
  1. Find the position vector of the point of intersection of \(l_1\) and \(\Pi\). [3]
  2. Find the acute angle between \(l_1\) and \(\Pi\). [3]
\(A\) is the point on \(l_1\) where \(\lambda = 1\). \(l_2\) is the line with the following properties. • \(l_2\) passes through \(A\) • \(l_2\) is perpendicular to \(l_1\) • \(l_2\) is parallel to \(\Pi\)
  1. Find, in vector form, the equation of \(l_2\). [3]
SPS SPS FM Pure 2022 February Q4
9 marks Standard +0.3
  1. \(\mathbf{A}\) is a 2 by 2 matrix and \(\mathbf{B}\) is a 2 by 3 matrix. Giving a reason for your answer, explain whether it is possible to evaluate
    1. \(\mathbf{AB}\)
    2. \(\mathbf{A} + \mathbf{B}\)
    [2]
  2. Given that $$\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix} \begin{pmatrix} 0 & 5 & 0 \\ 2 & 12 & -1 \\ -1 & -11 & 3 \end{pmatrix} = \lambda \mathbf{I}$$ where \(a\), \(b\) and \(\lambda\) are constants,
    1. determine • the value of \(\lambda\) • the value of \(a\) • the value of \(b\)
    2. Hence deduce the inverse of the matrix \(\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}\)
    [3]
  3. Given that $$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leqslant \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf{M}\) is singular. [4]
SPS SPS FM Pure 2022 February Q5
11 marks Standard +0.3
Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).
  1. Find a cartesian equation for \(l\). [3]
\(M\) is the point on \(l\) that is closest to \(C\).
  1. Find the coordinates of \(M\). [4]
  2. Find the exact area of the triangle \(ABC\). [4]
SPS SPS FM Pure 2022 February Q6
13 marks Challenging +1.8
The curve \(C\) has equation $$r = a(p + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\) There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.
  1. Show that the range of possible values for \(p\) is $$2 < p < 4$$ [5]
  2. Sketch the curve with equation $$r = a(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi \quad \text{where } a > 0$$ [1]
John digs a hole in his garden in order to make a pond. The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres. Water flows through a hosepipe into the pond at a rate of 50 litres per minute. Given that the pond is initially empty,
  1. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute. [7]
SPS SPS FM Pure 2022 February Q7
5 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined by \(\mathbf{M} = \begin{pmatrix} 3 & 2 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) $$\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) for all integers \(n \geq 1\) [5 marks]
SPS SPS FM Pure 2022 February Q8
6 marks Challenging +1.8
The complex number \(z\) satisfies the equations $$|z^* - 1 - 2i| = |z - 3|$$ and $$|z - a| = 3$$ where \(a\) is real. Show that \(a\) must lie in the interval \([1 - s\sqrt{t}, 1 + s\sqrt{t}]\), where \(s\) and \(t\) are prime numbers. [6 marks]
SPS SPS FM Pure 2022 February Q9
9 marks Challenging +1.2
The equation \(4x^4 - 4x^3 + px^2 + qx - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha\), \(-\alpha\), \(\beta\) and \(\frac{1}{\beta}\).
  1. Determine the exact roots of the equation. [5]
  2. Determine the values of \(p\) and \(q\). [4]
SPS SPS FM Pure 2022 February Q10
8 marks Standard +0.3
You are given that \(f(x) = 4\sinh x + 3\cosh x\).
  1. Show that the curve \(y = f(x)\) has no turning points. [3]
  2. Determine the exact solution of the equation \(f(x) = 5\). [5]
SPS SPS FM Pure 2022 February Q11
12 marks Challenging +1.2
A particle \(P\) of mass 2 kg can only move along the straight line segment \(OA\), where \(OA\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(OA\) is 0.9 m. When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) ms\(^{-1}\). \(P\) is subject to a force of magnitude \(4e^{-2t}\) N in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2\), \(v = 0.5\) and the resultant force on \(P\) is 0 N.
  1. Show that, according to the model, \(\frac{dv}{dt} + v = 2e^{-2t}\). [3]
  2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\). [5]
  3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\)'s speed must reach a maximum value for some \(t > 0\). [2]
  4. Determine the maximum speed considered in part (c). [2]
SPS SPS FM Pure 2022 February Q12
6 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(e^x > 1 + x\). [2]
  2. Hence, by using a suitable substitution, deduce that \(e^t > et\) for \(t > 1\). [1]
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(e^{\pi}\) or \(\pi^e\). [3]
SPS SPS FM Pure 2023 June Q1
5 marks Easy -1.2
You are given that \(gf(x) = |3x - 1|\) for \(x \in \mathbb{R}\).
  1. Given that \(f(x) = 3x - 1\), express \(g(x)\) in terms of \(x\). [1]
  2. State the range of \(gf(x)\). [1]
  3. Solve the inequality \(|3x - 1| > 1\). [3]
SPS SPS FM Pure 2023 June Q2
6 marks Moderate -0.3
In this question you must show detailed reasoning.
  1. Express \(8\cos x + 5\sin x\) in the form \(R\cos(x - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). [3]
  2. Hence solve the equation \(8\cos x + 5\sin x = 6\) for \(0 \leqslant x < 2\pi\), giving your answers correct to 4 decimal places. [3]
SPS SPS FM Pure 2023 June Q3
6 marks Standard +0.3
You are given that \(f(x) = \ln(2x - 5) + 2x^2 - 30\), for \(x > 2.5\).
  1. Show that \(f(x) = 0\) has a root \(\alpha\) in the interval \([3.5, 4]\). [2]
A student takes 4 as the first approximation to \(\alpha\). Given \(f(4) = 3.099\) and \(f'(4) = 16.67\) to 4 significant figures,
  1. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures. [2]
  2. Show that \(\alpha\) is the only root of \(f(x) = 0\). [2]
SPS SPS FM Pure 2023 June Q4
7 marks Standard +0.3
You are given that \(M = \begin{pmatrix} \frac{1}{\sqrt{3}} & -\sqrt{3} \\ \frac{1}{\sqrt{3}} & 1 \end{pmatrix}\).
  1. Show that \(M\) is non-singular. [2]
The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(M\). Given that the area of hexagon \(R\) is 5 square units,
  1. find the area of hexagon \(S\). [1]
The matrix \(M\) represents an enlargement, with centre \((0,0)\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \((0,0)\).
  1. Find the value of \(k\). [2]
  2. Find the value of \(\theta\). [2]
SPS SPS FM Pure 2023 June Q5
5 marks Standard +0.3
\includegraphics{figure_5} Figure 1 shows a sketch of a triangle \(ABC\). Given \(\overrightarrow{AB} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{BC} = \mathbf{i} - 9\mathbf{j} + 3\mathbf{k}\), show that \(\angle BAC = 105.9°\) to one decimal place. [5]
SPS SPS FM Pure 2023 June Q6
5 marks Standard +0.3
A spherical balloon is inflated so that its volume increases at a rate of \(10\text{ cm}^3\) per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm. [For a sphere of radius \(r\), surface area \(= 4\pi r^2\) and volume \(= \frac{4}{3}\pi r^3\)]. [5]
SPS SPS FM Pure 2023 June Q7
6 marks Challenging +1.2
Prove that for all \(n \in \mathbb{N}\) $$\begin{pmatrix} 3 & 4i \\ i & -1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & 4ni \\ ni & 1-2n \end{pmatrix}$$ [6]
SPS SPS FM Pure 2023 June Q8
7 marks Challenging +1.2
  1. Shade on an Argand diagram the set of points $$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]
The complex number \(w\) satisfies \(|w - 4i| = 3\).
  1. Find the maximum value of \(\arg w\) in the interval \((-\pi, \pi]\). Give your answer in radians correct to 2 decimal places. [2]
SPS SPS FM Pure 2023 June Q9
8 marks Standard +0.3
  1. Use the binomial expansion to show that \((1 - 2x)^{-\frac{1}{4}} \approx 1 + x + \frac{5}{8}x^2\) for sufficiently small values of \(x\). [2]
  2. For what values of \(x\) is the expansion valid? [1]
  3. Find the expansion of \(\sqrt{\frac{1+2x}{1-2x}}\) in ascending powers of \(x\) as far as the term in \(x^2\). [3]
  4. Use \(x = \frac{1}{20}\) in your answer to part (iii) to find an approximate value for \(\sqrt{11}\). [2]
SPS SPS FM Pure 2023 June Q10
6 marks Standard +0.3
The complex number \(z\) is given by \(z = k + 3i\), where \(k\) is a negative real number. Given that \(z + \frac{12}{z}\) is real, find \(k\) and express \(z\) in exact modulus-argument form. [6]
SPS SPS FM Pure 2023 June Q11
7 marks Challenging +1.2
In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$ Show that the gradient of the curve is always negative. [7]
SPS SPS FM Pure 2023 June Q12
7 marks Challenging +1.8
\includegraphics{figure_12} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\). [7]
SPS SPS FM Pure 2023 June Q13
10 marks Challenging +1.2
  1. Solve the differential equation $$\frac{dy}{dx} = y(1 + y)(1 - x),$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f(x)\), where \(f\) is a function to be determined. [7]
  2. By considering the sign of \(\frac{dy}{dx}\) near \((1, 1)\), or otherwise, show that this point is a maximum point on the curve \(y = f(x)\). [3]
SPS SPS FM Pure 2023 June Q14
7 marks Challenging +1.8
A curve \(C\) has equation $$x^3 + y^3 = 3xy + 48$$ Prove that \(C\) has two stationary points and find their coordinates. [7]
SPS SPS FM Pure 2023 June Q15
8 marks Challenging +1.2
In this question you must use detailed reasoning.
  1. Show that \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1+\sin 2x}{-\cos 2x} dx = \ln(\sqrt{a} + b)\), where \(a\) and \(b\) are integers to be determined. [6]
  2. Show that \(\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1+\sin 2x}{-\cos 2x} dx\) is undefined, explaining your reasoning clearly. [2]