Questions — SPS SPS FM Pure (188 questions)

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SPS SPS FM Pure 2021 June Q1
2 marks Moderate -0.8
A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$ Find the gradient of the curve at the point where \(t = -2\) [2]
SPS SPS FM Pure 2021 June Q2
6 marks Moderate -0.3
The equation \(x^3 - 3x + 1 = 0\) has three real roots.
  1. Show that one of the roots lies between \(-2\) and \(-1\) [2 marks]
  2. Taking \(x_1 = -2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x_2\), the second approximation. [3 marks]
  3. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x_1 = -1\) [1 mark]
SPS SPS FM Pure 2021 June Q3
4 marks Moderate -0.3
Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).
  1. Find the position vector of \(P\). [2]
  2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]
SPS SPS FM Pure 2021 June Q4
8 marks Standard +0.8
Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]
SPS SPS FM Pure 2021 June Q5
5 marks Standard +0.3
\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]
SPS SPS FM Pure 2021 June Q6
5 marks Moderate -0.3
  1. Use the binomial expansion, in ascending powers of \(x\), to show that $$\sqrt{4-x} = 2 - \frac{1}{4}x + kx^2 + ...$$ where \(k\) is a rational constant to be found. [4] A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt{3}\).
  2. State, giving a reason, if the expansion is valid for this value of \(x\). [1]
SPS SPS FM Pure 2021 June Q7
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
SPS SPS FM Pure 2021 June Q8
6 marks Standard +0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\) $$f(n) = 2^{n+2} + 3^{2n+1}$$ is divisible by 7 [6]
SPS SPS FM Pure 2021 June Q9
6 marks Moderate -0.8
\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$
  1. State the range of \(f\) [1]
  2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
  3. state the set of possible values for \(k\). [2]
SPS SPS FM Pure 2021 June Q10
8 marks Challenging +1.8
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_3} Find the maximum vertical height above the platform of the sculpture. [8 marks]
SPS SPS FM Pure 2021 June Q11
7 marks Standard +0.8
  1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
  2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x} dx\) Fully justify your answer. [6 marks]
SPS SPS FM Pure 2021 June Q13
8 marks Standard +0.8
$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ where \(a\) and \(b\) are non-zero constants. Given that the matrix \(\mathbf{A}\) is self-inverse,
  1. determine the value of \(b\) and the possible values for \(a\). [5] The matrix \(\mathbf{A}\) represents a linear transformation \(M\). Using the smaller value of \(a\) from part (a),
  2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line. [3]
SPS SPS FM Pure 2021 June Q14
6 marks Challenging +1.2
\includegraphics{figure_5} Figure 5 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{4\sin 2x}{e^{\sqrt{2}x-1}}, \quad 0 \leq x \leq \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.
  1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2x = \sqrt{2}$$ [4]
  2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2f(x)$$ [2]
SPS SPS FM Pure 2021 June Q15
7 marks Standard +0.3
The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = -\frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
SPS SPS FM Pure 2021 June Q16
7 marks Challenging +1.8
Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]
SPS SPS FM Pure 2021 May Q1
5 marks Moderate -0.3
Points \(A\), \(B\) and \(C\) have coordinates \((0, 1, -4)\), \((1, 1, -2)\) and \((3, 2, 5)\) respectively.
  1. Find the vector product \(\overrightarrow{AB} \times \overrightarrow{AC}\). [3]
  2. Hence find the equation of the plane \(ABC\) in the form \(ax + by + cz = d\). [2]
SPS SPS FM Pure 2021 May Q2
9 marks Standard +0.3
The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_2}
  1. The greatest value of \(r\) on the curve occurs at the point \(P\).
    1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
    2. Find the value of \(r\) at the point \(P\). [1]
    3. Mark the point \(P\) on a copy of the graph. [1]
  2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]
SPS SPS FM Pure 2021 May Q3
5 marks Moderate -0.3
You are given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\).
  1. Find \(\mathbf{A}^4\). [1]
  2. Describe the transformation that \(\mathbf{A}\) represents. [2]
The matrix \(\mathbf{B}\) represents a reflection in the plane \(x = 0\).
  1. Write down the matrix \(\mathbf{B}\). [1]
The point \(P\) has coordinates \((2, 3, 4)\). The point \(P'\) is the image of \(P\) under the transformation represented by \(\mathbf{B}\).
  1. Find the coordinates of \(P'\). [1]
SPS SPS FM Pure 2021 May Q4
3 marks Easy -1.2
Using the formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\), show that \(\sum_{r=1}^{10} r(3r - 2) = 1045\). [3]
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]