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SPS SPS FM Pure 2022 February Q4
9 marks Standard +0.3
  1. \(\mathbf{A}\) is a 2 by 2 matrix and \(\mathbf{B}\) is a 2 by 3 matrix. Giving a reason for your answer, explain whether it is possible to evaluate
    1. \(\mathbf{AB}\)
    2. \(\mathbf{A} + \mathbf{B}\)
    [2]
  2. Given that $$\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix} \begin{pmatrix} 0 & 5 & 0 \\ 2 & 12 & -1 \\ -1 & -11 & 3 \end{pmatrix} = \lambda \mathbf{I}$$ where \(a\), \(b\) and \(\lambda\) are constants,
    1. determine • the value of \(\lambda\) • the value of \(a\) • the value of \(b\)
    2. Hence deduce the inverse of the matrix \(\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}\)
    [3]
  3. Given that $$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leqslant \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf{M}\) is singular. [4]
SPS SPS FM Pure 2022 February Q5
11 marks Standard +0.3
Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).
  1. Find a cartesian equation for \(l\). [3]
\(M\) is the point on \(l\) that is closest to \(C\).
  1. Find the coordinates of \(M\). [4]
  2. Find the exact area of the triangle \(ABC\). [4]
SPS SPS FM Pure 2022 February Q6
13 marks Challenging +1.8
The curve \(C\) has equation $$r = a(p + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\) There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.
  1. Show that the range of possible values for \(p\) is $$2 < p < 4$$ [5]
  2. Sketch the curve with equation $$r = a(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi \quad \text{where } a > 0$$ [1]
John digs a hole in his garden in order to make a pond. The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres. Water flows through a hosepipe into the pond at a rate of 50 litres per minute. Given that the pond is initially empty,
  1. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute. [7]
SPS SPS FM Pure 2022 February Q7
5 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined by \(\mathbf{M} = \begin{pmatrix} 3 & 2 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) $$\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) for all integers \(n \geq 1\) [5 marks]
SPS SPS FM Pure 2022 February Q8
6 marks Challenging +1.8
The complex number \(z\) satisfies the equations $$|z^* - 1 - 2i| = |z - 3|$$ and $$|z - a| = 3$$ where \(a\) is real. Show that \(a\) must lie in the interval \([1 - s\sqrt{t}, 1 + s\sqrt{t}]\), where \(s\) and \(t\) are prime numbers. [6 marks]
SPS SPS FM Pure 2022 February Q9
9 marks Challenging +1.2
The equation \(4x^4 - 4x^3 + px^2 + qx - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha\), \(-\alpha\), \(\beta\) and \(\frac{1}{\beta}\).
  1. Determine the exact roots of the equation. [5]
  2. Determine the values of \(p\) and \(q\). [4]
SPS SPS FM Pure 2022 February Q10
8 marks Standard +0.3
You are given that \(f(x) = 4\sinh x + 3\cosh x\).
  1. Show that the curve \(y = f(x)\) has no turning points. [3]
  2. Determine the exact solution of the equation \(f(x) = 5\). [5]
SPS SPS FM Pure 2022 February Q11
12 marks Challenging +1.2
A particle \(P\) of mass 2 kg can only move along the straight line segment \(OA\), where \(OA\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(OA\) is 0.9 m. When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) ms\(^{-1}\). \(P\) is subject to a force of magnitude \(4e^{-2t}\) N in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2\), \(v = 0.5\) and the resultant force on \(P\) is 0 N.
  1. Show that, according to the model, \(\frac{dv}{dt} + v = 2e^{-2t}\). [3]
  2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\). [5]
  3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\)'s speed must reach a maximum value for some \(t > 0\). [2]
  4. Determine the maximum speed considered in part (c). [2]
SPS SPS FM Pure 2022 February Q12
6 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(e^x > 1 + x\). [2]
  2. Hence, by using a suitable substitution, deduce that \(e^t > et\) for \(t > 1\). [1]
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(e^{\pi}\) or \(\pi^e\). [3]
SPS SPS SM Mechanics 2022 February Q1
3 marks Easy -1.8
Find $$\int (x^4 - 6x^2 + 7) dx$$ giving your answer in simplest form. [3]
SPS SPS SM Mechanics 2022 February Q2
4 marks Easy -1.8
Given that $$f(x) = x^2 - 4x + 5 \quad x \in \mathbb{R}$$
  1. express \(f(x)\) in the form \((x + a)^2 + b\) where \(a\) and \(b\) are integers to be found. [2]
The curve with equation \(y = f(x)\) • meets the \(y\)-axis at the point \(P\) • has a minimum turning point at the point \(Q\)
  1. Write down
    1. the coordinates of \(P\)
    2. the coordinates of \(Q\)
    [2]
SPS SPS SM Mechanics 2022 February Q3
6 marks Standard +0.3
The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_{n+1} = k - \frac{24}{u_n} \quad u_1 = 2$$ where \(k\) is an integer. Given that \(u_1 + 2u_2 + u_3 = 0\)
  1. show that $$3k^2 - 58k + 240 = 0$$ [3]
  2. Find the value of \(k\), giving a reason for your answer. [2]
  3. Find the value of \(u_3\) [1]
SPS SPS SM Mechanics 2022 February Q4
6 marks Moderate -0.8
Relative to a fixed origin \(O\), • the point \(A\) has position vector \(\mathbf{5i + 3j - 2k}\) • the point \(B\) has position vector \(\mathbf{7i + j + 2k}\) • the point \(C\) has position vector \(\mathbf{4i + 8j - 3k}\)
  1. Find \(|\overrightarrow{AB}|\) giving your answer as a simplified surd. [2]
Given that \(ABCD\) is a parallelogram,
  1. find the position vector of the point \(D\). [2]
The point \(E\) is positioned such that • \(ACE\) is a straight line • \(AC:CE = 2:1\)
  1. Find the coordinates of the point \(E\). [2]
SPS SPS SM Mechanics 2022 February Q5
9 marks Standard +0.3
In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x^3 - 10x^2 + 27x - 23$$ The point \(P(5, -13)\) lies on \(C\) The line \(l\) is the tangent to \(C\) at \(P\)
  1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [4]
  2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. [1]
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
  1. Use algebraic integration to find the exact area of \(R\). [4]
SPS SPS SM Mechanics 2022 February Q6
9 marks Moderate -0.3
A scientist is studying the growth of two different populations of bacteria. The number of bacteria, \(N\), in the first population is modelled by the equation $$N = Ae^{kt} \quad t \geq 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that • there were 1000 bacteria in this population at the start of the study • it took exactly 5 hours from the start of the study for this population to double
  1. find a complete equation for the model. [4]
  2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. [2]
The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500e^{1.4t} \quad t \geq 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study. Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,
  1. find the value of \(T\). [3]
SPS SPS SM Mechanics 2022 February Q7
7 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\frac{1 - \cos 2\theta}{\sin^2 2\theta} = k \sec^2 \theta \quad \theta = \frac{n\pi}{2} \quad n \in \mathbb{Z}$$ where \(k\) is a constant to be found. [3]
  2. Hence solve, for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\) $$\frac{1 - \cos 2x}{\sin^2 2x} = (1 + 2\tan x)^2$$ Give your answers to 3 significant figures where appropriate. [4]
SPS SPS SM Mechanics 2022 February Q8
3 marks Challenging +1.2
Show that $$\sum_{n=2}^{\infty} \left(\frac{1}{4}\right)^n \cos(180n)^{\circ} = \frac{9}{28}$$ [3]
SPS SPS SM Mechanics 2022 February Q9
7 marks Standard +0.3
The function \(f\) is defined by $$f(x) = \frac{(x + 5)(x + 1)}{(x + 4)} - \ln(x + 4) \quad x \in \mathbb{R} \quad x > k$$
  1. State the smallest possible value of \(k\). [1]
  2. Show that $$f'(x) = \frac{ax^2 + bx + c}{(x + 4)^2}$$ where \(a\), \(b\) and \(c\) are integers to be found. [4]
  3. Hence show that \(f\) is an increasing function. [2]
SPS SPS SM Mechanics 2022 February Q10
10 marks Standard +0.3
\includegraphics{figure_4} Figure 4 Figure 4 shows a sketch of the graph with equation $$y = |2x - 3k|$$ where \(k\) is a positive constant.
  1. Sketch the graph with equation \(y = f(x)\) where $$f(x) = k - |2x - 3k|$$ stating • the coordinates of the maximum point • the coordinates of any points where the graph cuts the coordinate axes [4]
  2. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
  3. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ [2]
SPS SPS SM Mechanics 2022 February Q11
6 marks Challenging +1.2
The curve \(C\) has parametric equations $$x = \sin 2\theta \quad y = \cos\text{ec}^3 \theta \quad 0 < \theta < \frac{\pi}{2}$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(\theta\) [3]
  2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\) [3]
SPS SPS SM Mechanics 2022 February Q12
10 marks Standard +0.3
Answer all the questions. Two cyclists, \(A\) and \(B\), are cycling along the same straight horizontal track. The cyclists are modelled as particles and the motion of the cyclists is modelled as follows: • At time \(t = 0\), cyclist \(A\) passes through the point \(O\) with speed \(2\text{ms}^{-1}\) • Cyclist \(A\) is moving in a straight line with constant acceleration \(2\text{ms}^{-2}\) • At time \(t = 2\) seconds, cyclist \(B\) starts from rest at \(O\) • Cyclist \(B\) moves with constant acceleration \(6\text{ms}^{-2}\) along the same straight line and in the same direction as cyclist \(A\) • At time \(t = T\) seconds, \(B\) overtakes \(A\) at the point \(X\) Using the model,
  1. sketch, on the same axes, for the interval from \(t = 0\) to \(t = T\) seconds, • a velocity-time graph for the motion of \(A\) • a velocity-time graph for the motion of \(B\) [2]
  2. explain why the two graphs must cross before time \(t = T\) seconds, [1]
  3. find the time when \(A\) and \(B\) are moving at the same speed, [2]
  4. find the distance \(OX\) [5]
SPS SPS SM Mechanics 2022 February Q13
9 marks Standard +0.3
\includegraphics{figure_13} A golfer hits a ball from a point \(A\) with a speed of \(25\text{ms}^{-1}\) at an angle of \(15°\) above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be \(10\text{ms}^{-2}\). The ball first lands at a point \(B\) which is \(4\text{m}\) below the level of \(A\) (see diagram).
  1. Determine the time taken for the ball to travel from \(A\) to \(B\). [3]
  2. Determine the horizontal distance of \(B\) from \(A\). [2]
  3. Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball. [4]
SPS SPS SM Mechanics 2022 February Q14
11 marks Challenging +1.2
\includegraphics{figure_14} One end of a light inextensible string is attached to a particle \(A\) of mass \(2\text{kg}\). The other end of the string is attached to a second particle \(B\) of mass \(3\text{kg}\). Particle \(A\) is in contact with a smooth plane inclined at \(30°\) to the horizontal and particle \(B\) is in contact with a rough horizontal plane. A second light inextensible string is attached to \(B\). The other end of this second string is attached to a third particle \(C\) of mass \(4\text{kg}\). Particle \(C\) is in contact with a smooth plane \(\Pi\) inclined at an angle of \(60°\) to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from \(A\) to the pulley, and from \(C\) to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between \(B\) and the horizontal plane is \(\mu\). The system is released from rest and in the subsequent motion \(C\) moves down \(\Pi\) with acceleration \(a\text{ms}^{-2}\).
  1. By considering an equation involving \(\mu\), \(a\) and \(g\) show that \(a < \frac{5}{9}g(2\sqrt{3} - 1)\). [7]
  2. Given that \(a = \frac{1}{5}g\), determine the magnitude of the contact force between \(B\) and the horizontal plane. Give your answer correct to 3 significant figures. [4]
SPS SPS FM Mechanics 2022 January Q1
6 marks Standard +0.8
A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32N. The rope has natural length 4 m and modulus of elasticity of 470 N. By considering energy, determine the total distance she falls before first coming to instantaneous rest. [6]
SPS SPS FM Mechanics 2022 January Q2
5 marks Challenging +1.3
\includegraphics{figure_2} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac{1}{3}\). The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta^0\) with the upward vertical, as shown in the diagram above. Find, to one decimal place, the value of \(\theta\). [5]