Questions — SPS SPS SM Pure (97 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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SPS SPS SM Pure 2021 May Q2
3 marks Easy -1.2
Solve the equation \(|2x - 1| = |x + 3|\). [3]
SPS SPS SM Pure 2021 May Q3
6 marks Standard +0.3
Solve the equation \(2^{4x-1} = 3^{5-2x}\), giving your answer in the form \(x = \frac{\log_{10} a}{\log_{10} b}\). [6]
SPS SPS SM Pure 2021 May Q4
3 marks Moderate -0.8
A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]
SPS SPS SM Pure 2021 May Q5
8 marks Standard +0.3
A curve has equation \(x^3 - 3x^2y + y^2 + 1 = 0\).
  1. Show that \(\frac{dy}{dx} = \frac{6xy - 3x^2}{2y - 3x^2}\). [4]
  2. Find the equation of the normal to the curve at the point \((1, 2)\). [4]
SPS SPS SM Pure 2021 May Q6
7 marks Challenging +1.2
In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]
SPS SPS SM Pure 2021 May Q7
13 marks Challenging +1.2
It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation $$3\tan 2x - 8\tan x = 4.$$
  1. Show that \(t = \sqrt[3]{\frac{1}{2} + \frac{1}{3}t - \frac{1}{3}t^2}\), where \(t = \tan x\). [3]
  2. Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8. [2]
  3. Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration. [3]
  4. Solve the equation \(3\tan 4y - 8\tan 2y = 4\) for \(0 < y < \frac{1}{4}\pi\). [2]
SPS SPS SM Pure 2021 May Q8
9 marks Challenging +1.2
Find the general solution of the differential equation $$(2x^3 - 3x^2 - 11x + 6)\frac{dy}{dx} = y(20x - 35).$$ Give your answer in the form \(y = f(x)\). [9]
SPS SPS SM Pure 2021 May Q9
14 marks Challenging +1.3
  1. Show that the two non-stationary points of inflection on the curve \(y = \ln(1 + 4x^2)\) are at \(x = \pm\frac{1}{2}\). [6]
\includegraphics{figure_9} The diagram shows the curve \(y = \ln(1 + 4x^2)\). The shaded region is bounded by the curve and a line parallel to the \(x\)-axis which meets the curve where \(x = \frac{1}{2}\) and \(x = -\frac{1}{2}\).
  1. Show that the area of the shaded region is given by $$\int_0^{\ln 2} \sqrt{e^y - 1} \, dy.$$ [3]
  2. Show that the substitution \(e^y = \sec^2\theta\) transforms the integral in part (ii) to \(\int_0^{\frac{\pi}{4}} 2\tan^2\theta \, d\theta\). [2]
  3. Hence find the exact area of the shaded region. [3]
SPS SPS SM Pure 2020 October Q1
6 marks Easy -1.3
  1. Find $$\int \frac{x}{x^2 + 1} dx$$ [2]
  2. Find. $$\int 2\pi(4x + 3)^{10} dx$$ [2]
  3. Find. $$\int \frac{2}{e^{4x}} dx$$ [2]
SPS SPS SM Pure 2020 October Q2
5 marks Moderate -0.8
  1. Find \(\frac{dy}{dx}\) if \(y = 4\ln(3x)\) [2]
  2. Differentiate \(\frac{2x}{\sqrt{3x+1}}\) giving your answer in the form \(\frac{3x+c}{\sqrt{(3x+1)^p}}\), where \(c, p \in \mathbb{N}\) [3]
SPS SPS SM Pure 2020 October Q3
3 marks Easy -1.8
Expand \((x - 2y)^5\). [3]
SPS SPS SM Pure 2020 October Q4
3 marks Moderate -0.8
What transformations could be used, and in which order, to transform the curve \(y = \sin x\) into the curve \(y = 2 \sin(3x + 30°)\)? [3]
SPS SPS SM Pure 2020 October Q5
5 marks Standard +0.3
Find the equation of the tangent to the curve $$y = 3x^2(x + 2)^6$$ at the point \((-1, 3)\), giving your answer in the form \(y = mx + c\). [5]
SPS SPS SM Pure 2020 October Q6
5 marks Moderate -0.8
  1. Express \(\frac{x}{(x + 1)(x + 2)}\) in partial fractions. [3]
  2. Hence find \(\int \frac{x}{(x + 1)(x + 2)} dx\). [2]
SPS SPS SM Pure 2020 October Q7
7 marks Standard +0.3
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0° < \theta < 360°\). [4]
SPS SPS SM Pure 2020 October Q8
12 marks Challenging +1.3
    1. Sketch the graph of \(y = \cos \sec x\) for \(0 < x < 4\pi\). [3]
    2. It is given that \(\cos \sec \alpha = \cos \sec \beta\), where \(\frac{1}{2}\pi < \alpha < \pi\) and \(2\pi < \beta < \frac{5}{2}\pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\). [2]
    1. Write down the identity giving \(\tan 2\theta\) in terms of \(\tan \theta\). [1]
    2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2\phi \tan 4\phi\), showing all your working. [6]
SPS SPS SM Pure 2020 October Q9
6 marks Standard +0.3
Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = (h^6 + 16)^2 - 4.$$
  1. Find the value of \(\frac{dV}{dh}\) when \(h = 2\). [3]
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures. [3]
SPS SPS SM Pure 2020 October Q10
7 marks Standard +0.3
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{2}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
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]