Questions — SPS SPS SM Mechanics (18 questions)

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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 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]