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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]
SPS SPS FM Mechanics 2021 September Q5
8 marks Standard +0.3
In this question use \(g = 10 \text{ m s}^{-2}\). A particle of mass 3 kg rests in limiting equilibrium on a rough plane inclined at \(30°\) to the horizontal.
  1. Find the exact value of the coefficient of friction between the particle and the plane. [2]
A horizontal force of 36 N is now applied to the particle.
  1. Find how far down the plane the particle travels after the force has been applied for 4 s. [6]
SPS SPS FM Statistics 2021 September Q1
6 marks Moderate -0.8
  1. 5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy. [3 marks]
  2. A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers. How many different teams can be selected? [3 marks]
SPS SPS FM Statistics 2021 September Q2
9 marks Moderate -0.3
\(P(E) = 0.25\), \(P(F) = 0.4\) and \(P(E \cap F) = 0.12\)
  1. Find \(P(E'|F')\) [2 marks]
  2. Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer. [2 marks]
The event \(G\) has \(P(G) = 0.15\) The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
  1. Draw a Venn diagram to illustrate the events \(E\), \(F\) and \(G\), giving the probabilities for each region. [3 marks]
  2. Find \(P([F \cup G]')\) [2 marks]
SPS SPS FM Statistics 2021 September Q3
11 marks Moderate -0.8
A group of students were surveyed by a principal and \(\frac{2}{3}\) were found to always hand in assignments on time. When questioned about their assignments \(\frac{3}{5}\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac{11}{20}\) hand them in on time.
  1. Draw a tree diagram to represent this information. [3 marks]
  2. Find the probability that a randomly selected student:
    1. always start their assignments on the day they are issued and hand them in on time. [2 marks]
    2. does not always hand in assignments on time and does not start their assignments on the day they are issued. [4 marks]
  3. Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer. [2 marks]
SPS SPS FM Statistics 2021 September Q4
4 marks Moderate -0.8
In a town, 54% of the residents are female and 46% are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female. [4 marks]
SPS SPS FM Statistics 2021 September Q5
9 marks Standard +0.3
The heights of a population of men are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 20% of the men are taller than 180 cm and 5% are shorter than 170 cm.
  1. Sketch a diagram to show the distribution of heights represented by this information. [2 marks]
  2. Find the value of \(\mu\) and \(\sigma\). [5 marks]
  3. Three men are selected at random, find the probability that they are all taller than 175 cm. [2 marks]
SPS SPS SM Mechanics 2021 September Q1
8 marks Easy -1.3
A racing car starts from rest at the point \(A\) and moves with constant acceleration of \(11 \text{ m s}^{-2}\) for \(8 \text{ s}\). The velocity it has reached after \(8 \text{ s}\) is then maintained for \(7 \text{ s}\). The racing car then decelerates from this velocity to \(40 \text{ m s}^{-1}\) in a further \(2 \text{ s}\), reaching point \(B\).
  1. Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch. [5]
  2. Given that the distance between \(A\) and \(B\) is \(1404 \text{ m}\), find the value of \(T\). [3]
SPS SPS SM Mechanics 2021 September Q2
7 marks Easy -1.3
A particle \(P\) is acted upon by three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) given by \(\mathbf{F}_1 = (6\mathbf{i} - 4\mathbf{j}) \text{ N}\), \(\mathbf{F}_2 = (-3\mathbf{i} + 9\mathbf{j}) \text{ N}\) and \(\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j}) \text{ N}\), where \(a\) and \(b\) are constants. Given that \(P\) is in equilibrium,
  1. find the value of \(a\) and the value of \(b\). [2]
The force \(\mathbf{F}_3\) is now removed. The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\).
  1. Find the magnitude of \(\mathbf{R}\). [3]
  2. Find the angle, to \(0.1°\), that \(\mathbf{R}\) makes with \(\mathbf{i}\). [2]
SPS SPS SM Mechanics 2021 September Q3
12 marks Moderate -0.3
A car of mass \(1200 \text{ kg}\) pulls a trailer of mass \(400 \text{ kg}\) along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of \(3200 \text{ N}\). The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is \(0.4 \text{ m s}^{-2}\),
  1. find the resistance to motion on the trailer, [4]
  2. find the tension in the tow-rope. [3]
When the car and trailer are travelling at \(25 \text{ m s}^{-1}\) the tow-rope breaks. Assuming that the resistances to motion remain unchanged,
  1. find the distance the trailer travels before coming to a stop, [4]
  2. state how you have used the modelling assumption that the tow-rope is inextensible. [1]
SPS SPS SM Mechanics 2021 September Q4
13 marks Moderate -0.3
A car starts from the point \(A\). At time \(t\) s after leaving \(A\), the distance of the car from \(A\) is \(s\) m, where \(s = 30t - 0.4t^2\), \(0 \leq t \leq 25\). The car reaches the point \(B\) when \(t = 25\).
  1. Find the distance \(AB\). [2]
  2. Show that the car travels with a constant acceleration and state the value of this acceleration. [3]
A runner passes through \(B\) when \(t = 0\) with an initial velocity of \(2 \text{ m s}^{-1}\) running directly towards \(A\). The runner has a constant acceleration of \(0.1 \text{ m s}^{-2}\).
  1. Find the distance from \(A\) at which the runner and the car pass one another. [8]
SPS SPS FM Pure 2022 February Q1
7 marks Moderate -0.3
  1. Express \(\frac{1}{(2r-1)(2r+1)}\) in partial fractions. [3]
  2. Hence find \(\sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}\), expressing the result as a single fraction. [4]
SPS SPS FM Pure 2022 February Q2
5 marks Challenging +1.2
\(\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}\) The matrix \(\mathbf{A}\) represents the linear transformation \(M\). Prove that, for the linear transformation \(M\), there are no invariant lines. [5]
SPS SPS FM Pure 2022 February Q3
9 marks Standard +0.3
The line \(l_1\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}\). The plane \(\Pi\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} = 4\).
  1. Find the position vector of the point of intersection of \(l_1\) and \(\Pi\). [3]
  2. Find the acute angle between \(l_1\) and \(\Pi\). [3]
\(A\) is the point on \(l_1\) where \(\lambda = 1\). \(l_2\) is the line with the following properties. • \(l_2\) passes through \(A\) • \(l_2\) is perpendicular to \(l_1\) • \(l_2\) is parallel to \(\Pi\)
  1. Find, in vector form, the equation of \(l_2\). [3]