Questions — SPS (686 questions)

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SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Prove by induction that, for all positive integers \(n\), \(7^n + 3^{n-1}\) is a multiple of \(4\). [5]
SPS SPS FM Pure 2021 May Q6
8 marks Standard +0.3
\(\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}\), where \(k\) is a constant.
  1. Show that the matrix \(\mathbf{A}\) is non-singular for all values of \(k\). [2]
A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\). The point \(P\) has position vector \(\begin{pmatrix} a \\ 2a \end{pmatrix}\) relative to an origin \(O\). The point \(Q\) has position vector \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\) relative to \(O\). Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),
  1. determine the value of \(a\) and the value of \(k\). [3]
Given that, for a different value of \(k\), \(T\) maps the line \(y = 2x\) onto itself,
  1. determine this value of \(k\). [3]
SPS SPS FM Pure 2021 May Q7
9 marks Standard +0.3
Given that \(y = \arcsin x\), \(-1 \leqslant x < 1\),
  1. show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\). [3]
Given that \(f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}\),
  1. show that the mean value of \(f(x)\) over the interval \([0, \sqrt{2}]\), is $$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$ where \(A\) is a constant to be determined. [6]
SPS SPS FM Pure 2021 May Q8
8 marks Challenging +1.3
  1. Using the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
  2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16u^3 + 12u = 3.$$ Give your answer in the form \(\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}\) where \(a\), \(b\) and \(c\) are integers. [5]
SPS SPS FM Pure 2021 May Q9
6 marks Standard +0.8
  1. Using the Maclaurin series for \(\ln(1 + x)\), find the first four terms in the series expansion for \(\ln(1 + 3x^2)\). [2]
  2. Find the range of \(x\) for which the expansion is valid. [1]
  3. Find the exact value of the series $$\frac{3^1}{2 \times 2^2} - \frac{3^2}{3 \times 2^4} + \frac{3^3}{4 \times 2^6} - \frac{3^4}{5 \times 2^8} + \ldots$$ [3]
SPS SPS FM Pure 2021 May Q10
7 marks Standard +0.8
A particular radioactive substance decays over time. A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac{dx}{dt} + \frac{1}{10}x = e^{-0.1t}\cos t.$$
  1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). [3]
Initially there was \(10\) g of the substance.
  1. Find the particular solution of the differential equation. [2]
  2. Find to \(6\) significant figures the amount of substance that would be predicted by the model at
    1. \(6\) hours, [1]
    2. \(6.25\) hours. [1]
SPS SPS FM Pure 2021 May Q1
7 marks Standard +0.3
In this question you must show detailed reasoning.
  1. By using partial fractions show that \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}\). [5]
  2. Hence determine the value of \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}\). [2]
SPS SPS FM Pure 2021 May Q2
8 marks Standard +0.8
  1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
  2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]
SPS SPS FM Pure 2021 May Q3
5 marks Moderate -0.3
In this question you must show detailed reasoning. Show that $$\int_5^{\infty} (x - 1)^{-\frac{3}{2}} dx = 1$$ [5]
SPS SPS FM Pure 2021 May Q4
6 marks Standard +0.3
You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation \(R\) which is a reflection in 3-D.
  1. State the plane of reflection of \(R\). [1]
  2. Determine the value of \(a\). [3]
  3. With reference to \(R\) explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3 \times 3\) identity matrix. [2]
SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Express \(\frac{5x^2+x+12}{x^3+4x}\) in partial fractions. [5]
SPS SPS FM Pure 2021 May Q6
6 marks Challenging +1.2
A circle \(C\) in the complex plane has equation \(|z - 2 - 5i| = a\). The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(arg(z_1) = \frac{\pi}{4}\). Prove that \(a = \frac{3\sqrt{2}}{2}\). [6]
SPS SPS FM Pure 2021 May Q7
8 marks Challenging +1.2
The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p + x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form
SPS SPS FM Pure 2021 May Q8
8 marks Hard +2.3
Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos(r\theta)\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos(10\theta)\). [8]
SPS SPS FM Pure 2021 May Q9
12 marks Standard +0.8
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x - 0.2y \quad \text{and} \quad \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
  2. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). Comment on this result in the context of the industrial process. [4]
  3. Express \(y\) in terms of \(t\). [5]
SPS SPS SM Pure 2021 May Q1
7 marks Moderate -0.8
The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$
  1. Evaluate \(f(169)\). [2]
  2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
  3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [3]
SPS SPS SM Pure 2021 May Q2
4 marks Moderate -0.8
  1. Use the trapezium rule, with four strips each of width \(0.5\), to estimate the value of $$\int_0^2 e^{x^2} dx$$ giving your answer correct to 3 significant figures. [3]
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate. [1]
SPS SPS SM Pure 2021 May Q3
6 marks Moderate -0.8
Vector \(\mathbf{v} = a\mathbf{i} + 0.6\mathbf{j}\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf{v}\) is \(45°\), state the value of \(a\). [1]
  2. Given instead that \(\mathbf{v}\) is parallel to \(8\mathbf{i} + 3\mathbf{j}\), find the value of \(a\). [2]
  3. Given instead that \(\mathbf{v}\) is a unit vector, find the possible values of \(a\). [3]
SPS SPS SM Pure 2021 May Q4
3 marks Standard +0.3
Prove that \(\sqrt{2}\cos(2\theta + 45°) \equiv \cos^2\theta - 2\sin\theta\cos\theta - \sin^2\theta\), where \(\theta\) is measured in degrees. [3]
SPS SPS SM Pure 2021 May Q5
8 marks Standard +0.3
  1. Show that \(\sqrt{\frac{1-x}{1+x}} \approx 1 - x + \frac{1}{2}x^2\), for \(|x| < 1\). [5]
  2. By taking \(x = \frac{2}{7}\), show that \(\sqrt{5} \approx \frac{111}{49}\). [3]
SPS SPS SM Pure 2021 May Q6
4 marks Challenging +1.2
Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]
SPS SPS SM Pure 2021 May Q7
11 marks Standard +0.3
A curve has parametric equations $$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$ for \(-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi\).
  1. Show that \(\frac{dy}{dx} = 1 - 2\sin t\) and hence find the coordinates of the stationary point. [5]
  2. Find the cartesian equation of the curve. [3]
  3. State the set of values that \(x\) can take and hence sketch the curve. [3]
SPS SPS SM Pure 2021 May Q8
12 marks Challenging +1.2
In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]
SPS SPS SM Pure 2021 May Q9
10 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_9} The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]
SPS SPS SM Pure 2021 May Q1
5 marks Standard +0.3
  1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2\cos\theta + (1 - \tan\theta)^2 \approx 3 - 2\theta\). [3]
  2. Hence determine an approximate solution to \(2\cos\theta + (1 - \tan\theta)^2 = 28\sin\theta\). [2]