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SPS SPS SM 2022 February Q7
10 marks Moderate -0.3
The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}
  1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
  2. Find the area of the shaded segment. [3]
  3. Find the perimeter of the shaded segment. [4]
SPS SPS SM 2022 February Q8
9 marks Moderate -0.8
The diagram shows the graph of \(y = f(x)\), where \(f(x) = 2 - x^2, \quad x \leqslant 0\). \includegraphics{figure_8}
  1. Evaluate \(f(-3)\). [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]
SPS SPS FM Pure 2022 June Q1
7 marks Standard +0.3
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int_0^9 \frac{1}{\sqrt{x}} dx\); [3 marks]
    2. \(\int_0^9 \frac{1}{x\sqrt{x}} dx\). [3 marks]
  2. Explain briefly why the integrals in part (a) are improper integrals. [1 mark]
SPS SPS FM Pure 2022 June Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the graph of \(y = f(x)\), \(x \in \mathbb{R}\). The graph consists of two line segments that meet at the point \((1, a)\), \(a < 0\). One line meets the \(x\)-axis at \((3, 0)\). The other line meets the \(x\)-axis at \((-1, 0)\) and the \(y\)-axis at \((0, b)\), \(b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = f(x + 1)\), [2]
  2. \(y = f(|x|)\). [2]
Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(f(x) = |x - 1| - 2\), find
  1. the value of \(a\) and the value of \(b\), [2]
  2. the value of \(x\) for which \(f(x) = 5x\). [3]
SPS SPS FM Pure 2022 June Q3
8 marks Standard +0.3
  1. Show on an Argand diagram the locus of points given by $$|z - 10 - 12i| = 8$$ [2] Set \(A\) is defined by $$A = \left\{z : 0 \leq \arg(z - 10 - 10i) \leq \frac{\pi}{2}\right\} \cap \{z : |z - 10 - 12i| < 8\}$$
  2. Shade the region defined by \(A\) on your Argand diagram. [2]
  3. Determine the area of the region defined by \(A\). [4]
SPS SPS FM Pure 2022 June Q4
8 marks Standard +0.3
The curve with equation \(y = f(x)\) where $$f(x) = x^2 + \ln(2x^2 - 4x + 5)$$ has a single turning point at \(x = \alpha\)
  1. Show that \(\alpha\) is a solution of the equation $$2x^3 - 4x^2 + 7x - 2 = 0$$ [4]
The iterative formula $$x_{n+1} = \frac{1}{7}(2 + 4x_n^2 - 2x_n^3)$$ is used to find an approximate value for \(\alpha\). Starting with \(x_1 = 0.3\)
  1. calculate, giving each answer to 4 decimal places,
    1. the value of \(x_2\)
    2. the value of \(x_4\)
    [2]
Using a suitable interval and a suitable function that should be stated,
  1. show that \(\alpha\) is 0.341 to 3 decimal places. [2]
SPS SPS FM Pure 2022 June Q5
4 marks Standard +0.3
The triangle \(T\) has vertices at the points \((1, k)\), \((3,0)\) and \((11,0)\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T'\) by the matrix $$\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix}$$ Given that the area of triangle \(T'\) is 364 square units, find the value of \(k\). [4]
SPS SPS FM Pure 2022 June Q6
7 marks Moderate -0.3
The complex number \(w\) is given by $$w = 10 - 5i$$
  1. Find \(|w|\). [1]
  2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [1]
The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + i)(z + 3i) = w$$
  1. Use algebra to find \(z\), giving your answer in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
Given that $$\arg(\lambda + 9i + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,
  1. find the value of \(\lambda\). [1]
SPS SPS FM Pure 2022 June Q7
7 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = xe^x\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis. Use integration by parts to find an exact value for the volume of the solid generated. [7]
SPS SPS FM Pure 2022 June Q8
5 marks Standard +0.8
With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$ The point \(A\) lies on \(l\) and has coordinates \((3, -2, 6)\). The point \(P\) has position vector \((-\mathbf{i} + 2\mathbf{k})\) relative to \(O\). Given that vector \(\overrightarrow{PA}\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle BPA = 45°\), find the coordinates of the two possible positions of \(B\). [5]
SPS SPS FM Pure 2022 June Q9
6 marks Standard +0.3
Prove by induction that for \(n \in \mathbb{Z}^+\) $$f(n) = 4^{n+1} + 5^{2n-1}$$ is divisible by 21 [6]
SPS SPS FM Pure 2022 June Q10
8 marks Standard +0.8
The curve defined by the parametric equations $$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$ is shown below. The point \(P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)\) is marked on the curve. \includegraphics{figure_curve}
  1. Show that the equation of the normal to the curve at \(P\) can be written as \(3y - x = \frac{7\sqrt{3}}{2}\) [5]
  2. Show that the Cartesian equation of the curve may be written as \(ay^2 + bx^4 + cx^2 = 0\) where \(a\), \(b\) and \(c\) are integers to be found. [3]
SPS SPS FM Pure 2022 June Q11
8 marks Standard +0.8
Solve the differential equation $$2\cot x \frac{dy}{dx} = (4 - y^2)$$ for which \(y = 0\) at \(x = \frac{\pi}{3}\), giving your answer in the form \(\sec^2 x = g(y)\). [8]
SPS SPS FM Pure 2022 June Q12
8 marks Standard +0.8
A linear transformation T of the \(x\)-\(y\) plane has an associated matrix M, where \(\mathbf{M} = \begin{pmatrix} \lambda & k \\ 1 & \lambda - k \end{pmatrix}\), and \(\lambda\) and \(k\) are real constants.
  1. You are given that \(\det \mathbf{M} > 0\) for all values of \(\lambda\).
    1. Find the range of possible values of \(k\). [3]
    2. What is the significance of the condition \(\det \mathbf{M} > 0\) for the transformation T? [1]
For the remainder of this question, take \(k = -2\).
  1. Determine whether there are any lines through the origin that are invariant lines for the transformation T. [4]
SPS SPS FM Pure 2022 June Q13
8 marks Standard +0.3
  1. Show that \(\sin(2\theta + \frac{1}{2}\pi) = \cos 2\theta\). [2]
  2. Hence solve the equation \(\sin 3\theta = \cos 2\theta\) for \(0 \leq \theta \leq 2\pi\). [6]
SPS SPS FM Pure 2022 June Q14
7 marks Standard +0.8
Using an appropriate substitution, or otherwise, show that $$\int_0^{\frac{\pi}{2}} \frac{\sin 2\theta}{1 + \cos \theta} d\theta = 2 - 2\ln 2$$ [7]
SPS SPS SM Pure 2022 June Q1
6 marks Moderate -0.8
  1. The expression \((2 + x^2)^3\) can be written in the form $$8 + px^2 + qx^4 + x^6$$ Demonstrate clearly, using the binomial expansion, that \(p = 12\) and find the value of \(q\). [3 marks]
  2. Hence find \(\int \frac{(2 + x^2)^3}{x^4} dx\). [3 marks]
SPS SPS SM Pure 2022 June Q2
5 marks Moderate -0.8
The trapezium \(ABCD\) is shown below. \includegraphics{figure_2} The line \(AB\) has equation \(2x + 3y = 14\) and \(DC\) is parallel to \(AB\). The point D has coordinates \((3, 7)\).
  1. Find an equation of the line DC [2 marks]
  2. The angle BAD is a right angle. Find an equation of the line AD, giving your answer in the form \(mx + ny + p = 0\), where \(m\), \(n\) and \(p\) are integers. [3 marks]
SPS SPS SM Pure 2022 June Q3
10 marks Easy -1.2
A circle has centre \(C(3, -8)\) and radius \(10\).
  1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
  2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
  3. The line with equation \(y = 2x + 1\) intersects the circle.
    1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
    2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt{n}\), where \(m\) and \(n\) are integers. [2 marks]
SPS SPS SM Pure 2022 June Q4
5 marks Moderate -0.3
The function \(f\) is defined by $$f(x) = \frac{5x}{7x - 5}$$
  1. The domain of \(f\) is the set \(\{x \in \mathbb{R} : x \neq a\}\) State the value of \(a\) [1 mark]
  2. Prove that \(f\) is a self-inverse function [3 marks]
  3. Find the range of \(f\) [1 mark]
SPS SPS SM Pure 2022 June Q5
3 marks Moderate -0.8
Relative to a fixed origin \(O\),
  • the point \(A\) has position vector \(-2\mathbf{i} + 3\mathbf{j}\),
  • the point \(B\) has position vector \(3\mathbf{i} + p\mathbf{j}\), where \(p\) is constant,
Given that \(|\overrightarrow{AB}| = 5\sqrt{2}\), find the possible values for \(p\). [3]
SPS SPS SM Pure 2022 June Q6
9 marks Easy -1.2
A small company which makes batteries for electric cars has a 10 year plan for growth. In year 1 the company will make 2600 batteries. In year 10 the company aims to make 12000 batteries. In order to calculate the number of batteries it will need to make each year from year 2 to year 9, the company considers two models. Model A assumes that the number of batteries it will make each year will increase by the same number each year.
  1. According to model A, determine the number of batteries the company will make in year 2. Give your answer to the nearest whole number of batteries. [3]
Model B assumes that the numbers of batteries it will make each year will increase by the same percentage each year.
  1. According to model B, determine the number of batteries the company will make in year 2. Give your answer to the nearest 10 batteries. [3]
Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.
  1. Calculate the difference between the two totals, giving your answer to the nearest 100 batteries. [3]
SPS SPS SM Pure 2022 June Q7
4 marks Moderate -0.5
\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that \(AFE\) is a straight line, \(AF = FE = 10.7\) m, \(BF = FD = 9.2\) m and angle \(BFD = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place. [4]
SPS SPS SM Pure 2022 June Q8
8 marks Moderate -0.3
The function \(f(x)\) is such that \(f(x) = -x^3 + 2x^2 + kx - 10\) The graph of \(y = f(x)\) crosses the \(x\)-axis at the points with coordinates \((a, 0)\), \((2, 0)\) and \((b, 0)\) where \(a < b\)
  1. Show that \(k = 5\) [1 mark]
  2. Find the exact value of \(a\) and the exact value of \(b\) [3 marks]
  3. The functions \(g(x)\) and \(h(x)\) are such that $$g(x) = x^3 + 2x^2 - 5x - 10$$ $$h(x) = -8x^3 + 8x^2 + 10x - 10$$
    1. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = g(x)\) Fully justify your answer. [2 marks]
    2. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = h(x)\) Fully justify your answer. [2 marks]
SPS SPS SM Pure 2022 June Q9
5 marks Standard +0.3
A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u_n\) Find the exact value of \(\sum_{n=5}^{\infty} u_n\) [5 marks]