Questions — SPS (686 questions)

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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 FM Mechanics 2021 January Q1
3 marks Moderate -0.5
A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\). A possible model for the kinetic energy \(E\) of the disc is $$E = km^ar^b\omega^c$$ where \(a\), \(b\) and \(c\) are constants and \(k\) is a dimensionless constant. Find the values of \(a\), \(b\) and \(c\). [3 marks]
SPS SPS FM Mechanics 2021 January Q2
11 marks Standard +0.8
The triangular region shown below is rotated through \(360°\) around the \(x\)-axis, to form a solid cone. \includegraphics{figure_1} The coordinates of the vertices of the triangle are \((0, 0)\), \((8, 0)\) and \((0, 4)\). All units are in centimetres.
  1. State an assumption that you should make about the cone in order to find the position of its centre of mass. [1 mark]
  2. Using integration, prove that the centre of mass of the cone is \(2\)cm from its plane face. [5 marks]
  3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding.
    1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. [2 marks]
    2. Find the range of possible values for the coefficient of friction between the cone and the board. [3 marks]
SPS SPS FM Mechanics 2021 January Q3
8 marks Standard +0.8
\includegraphics{figure_2} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W_1\) and \(W_2\) are two fixed parallel vertical walls. The walls are \(3\) metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leq 3\), from \(W_1\). At time \(t = 0\), the particle is projected from \(O\) towards \(W_1\) with speed \(u\text{ms}^{-1}\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac{2}{3}\). The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.
  1. Show that \(T = \frac{45 - 5d}{4u}\). [6]
  2. The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer. [2]
SPS SPS FM Mechanics 2021 January Q4
12 marks Standard +0.3
A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).
  1. Show that \(\lambda = 8\). [4]
  2. Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. Find the acceleration of the car immediately after the towbar breaks. [4]
  3. Use the work-energy principle to find the value of \(d\). [4]
SPS SPS FM Mechanics 2021 January Q5
6 marks Standard +0.3
\includegraphics{figure_3} A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac{1}{4}a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\). [6]
SPS SPS FM Mechanics 2021 January Q6
11 marks Challenging +1.2
Numerical (calculator) integration is not acceptable in this question. \includegraphics{figure_4} The shaded region \(OAB\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac{1}{4}(x-2)^3 + 2\). The point \(A\) has coordinates \((4, 4)\) and the point \(B\) has coordinates \((4, 0)\). A uniform lamina \(L\) has the shape of \(OAB\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \((\bar{x}, \bar{y})\). Given that the area of \(L\) is \(8\)cm²,
  1. show that \(\bar{y} = \frac{8}{7}\). [4]
  2. The lamina is freely suspended from \(A\) and hangs in equilibrium with \(AB\) at an angle \(\theta°\) to the downward vertical. Find the value of \(\theta\). [7]
SPS SPS FM Statistics 2021 January Q1
4 marks Moderate -0.3
Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes.
  1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place. [2 marks]
  2. Alan claims that his mean journey time to work is 30 minutes. State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim. [1 mark]
  3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6. Explain how the working in part (a) would change. [1 mark]
SPS SPS FM Statistics 2021 January Q2
8 marks Standard +0.3
Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.
  1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. [3]
Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
  1. Find the probability that in exactly 3 of these periods there were no calls. [2]
On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
  1. Find the probability that Indre missed exactly 1 call in each of these 2 breaks. [3]
SPS SPS FM Statistics 2021 January Q3
7 marks Moderate -0.3
A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams/m². The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w^2 = 13447 \quad S_{ff} = 42 \quad S_{fw} = 269.5$$
  1. Calculate the product moment correlation coefficient between \(f\) and \(w\) [2]
  2. Interpret the value of your product moment correlation coefficient. [1]
  3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + bf\) [3]
  4. Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams/m² [1]
SPS SPS FM Statistics 2021 January Q4
7 marks Standard +0.3
The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x \leq 0 \\ k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\ 1 & x > 2 \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{2}\) [1]
  2. Showing your working clearly, use calculus to find
    1. E(\(X\))
    2. the mode of \(X\)
    [6]
SPS SPS FM Statistics 2021 January Q5
9 marks Standard +0.3
A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text{and} \quad \sum x^2 = 10198$$ Investigate the shopkeeper's claim using the 5\% level of significance. State any assumptions that you make. [9 marks]
SPS SPS FM Statistics 2021 January Q6
12 marks Standard +0.3
A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac{1}{3}\) that it lands on blue. The spinner is spun repeatedly. The random variable \(B\) represents the number of the spin when the spinner first lands on blue.
  1. Find
    1. P(\(B = 4\))
    2. P(\(B \leq 5\))
    [4]
  2. Find E(\(B^2\)) [3]
Steve invites Tamara to play a game with this spinner. Tamara must choose a colour, either red or blue. Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(e^X\) If Tamara chooses blue, her score is \(X^2\)
  1. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]
SPS SPS FM Statistics 2021 January Q7
7 marks Challenging +1.2
Nine athletes, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\) and \(I\), competed in both the 100m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85 The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes \(B\), \(C\) and \(D\) over their long jump results. The table shows the results that are agreed to be correct.
Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Position in 100m sprint467928315
Position in long jump549312
Given that there were no tied ranks,
  1. find the correct positions of athletes \(B\), \(C\) and \(D\) in the long jump. You must show your working clearly and give reasons for your answers. [5]
  2. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100m sprint and the long jump. [2]
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]