Questions — SPS (686 questions)

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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]
SPS SPS SM Pure 2022 June Q10
6 marks Moderate -0.8
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leq x \leq 2\pi\). State the values of the intercepts with the coordinate axes. [2 marks]
    1. Given that $$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$ prove that \(\cos \theta = \frac{1}{2}\). [2 marks]
    2. Hence solve the equation $$\sin^2 2x = \cos 2x(2 - \cos 2x)$$ in the interval \(0 \leq x \leq \pi\) [2 marks]
SPS SPS SM Pure 2022 June Q11
7 marks Standard +0.3
A sequence is defined by $$u_1 = 600$$ $$u_{n+1} = pu_n + q$$ where \(p\) and \(q\) are constants. It is given that \(u_2 = 500\) and \(u_4 = 356\)
  1. Find the two possible values of \(u_3\) [5 marks]
  2. When \(u_n\) is a decreasing sequence, the limit of \(u_n\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\). [2 marks]
SPS SPS SM Pure 2022 June Q12
5 marks Moderate -0.8
A curve is defined for \(x \geq 0\) by the equation $$y = 6x - 2x^{\frac{1}{2}}$$
  1. Find \(\frac{dy}{dx}\). [2 marks]
  2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer. [3 marks]
SPS SPS SM Pure 2022 June Q13
4 marks Moderate -0.3
$$\frac{1 + 11x - 6x^2}{(x - 3)(1 - 2x)} \equiv A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}$$ Find the values of the constants \(A\), \(B\) and \(C\). [4]
SPS SPS SM Pure 2022 June Q14
6 marks Moderate -0.3
A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)
  1. Sketch a graph to show the region R. Shade the region R.
  2. Find the area of R [6 marks]
SPS SPS SM Pure 2022 June Q15
6 marks Standard +0.8
  1. Prove that $$n - 1 \text{ is divisible by } 3 \Rightarrow n^3 - 1 \text{ is divisible by } 9$$ [3 marks]
  2. Show that if \(\log_2 3 = \frac{p}{q}\), then $$2^p = 3^q.$$ Use proof by contradiction to prove that \(\log_2 3\) is irrational. [3 marks]
SPS SPS SM Pure 2022 June Q16
7 marks Standard +0.8
\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.
  1. Find the exact value of \(a\). [3]
The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).
  1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]
SPS SPS SM Pure 2022 June Q17
4 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = 2x^2 - x\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = -0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).
  1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). [1]
The estimate for R is found to be 2.58. Using this value, and showing your working,
  1. estimate the value of \(\int_{-0.5}^{1.5} (2x^2 + 1 + 2x) \, dx\). [3]
SPS SPS FM Mechanics 2021 September Q1
7 marks Moderate -0.8
A car is initially travelling with a constant velocity of \(15 \text{ m s}^{-1}\) for \(T\) s. It then decelerates at a constant rate for \(\frac{T}{2}\) s, reaching a velocity of \(10 \text{ m s}^{-1}\). It then immediately accelerates at a constant rate for \(\frac{3T}{2}\) s reaching a velocity of \(20 \text{ m s}^{-1}\).
  1. Sketch a velocity–time graph to illustrate the motion. [3]
  2. Given that the car travels a total distance of 1312.5 m over the journey described, find the value of \(T\). [4]
SPS SPS FM Mechanics 2021 September Q2
7 marks Standard +0.3
A particle \(P\) moves in a straight line. At time \(t\) s the displacement \(s\) cm from a fixed point \(O\) is given by: $$s = \frac{1}{6}\left(8t^3 - 105t^2 + 144t + 540\right).$$ Find the distance between the points at which the particle is instantaneously at rest. [7]
SPS SPS FM Mechanics 2021 September Q3
9 marks Standard +0.3
A cylindrical object with mass 8 kg rests on two cylindrical bars of equal radius. The lines connecting the centre of each of the bars to the centre of the object make an angle of \(40°\) to the vertical. \includegraphics{figure_2}
  1. Draw a diagram showing all the forces acting on the object. Describe each of the forces using words. [2]
  2. Calculate the magnitude of the force on each of the bars due to the cylindrical object. [7]
SPS SPS FM Mechanics 2021 September Q4
8 marks Standard +0.3
A box \(A\) of mass 0.8 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a sphere \(B\) of mass 1.2 kg, which hangs freely below the pulley. The magnitude of the frictional force between \(A\) and the table is \(F\) N. The system is released from rest when the string is taut. After release, \(B\) descends a distance of 0.9 m in 0.8 s. Modelling \(A\) and \(B\) as particles, calculate
  1. the acceleration of \(B\), [2]
  2. the tension in the string, [3]
  3. the value of \(F\). [3]