Questions — SPS (686 questions)

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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]
SPS SPS FM Mechanics 2022 January Q3
9 marks Standard +0.3
A car of mass 800 kg is driven with its engine generating a power of 15 kW.
  1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds. [2]
  2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac{1}{20}\), find the speed of the car. [3]
  3. The car is now driven at a constant speed of 30 ms\(^{-1}\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
    1. the resistance to motion of the car,
    2. the magnitude of the tension in the towbar
    [4]
SPS SPS FM Mechanics 2022 January Q4
9 marks Challenging +1.3
\includegraphics{figure_4} Two uniform smooth spheres A and B of equal radius are moving on a horizontal surface when they collide. A has mass 0.1 kg and B has mass 0.4 kg. Immediately before the collision A is moving with speed 2.8 ms\(^{-1}\) along the line of centres, and B is moving with speed 1 ms\(^{-1}\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision A is stationary. Find:
  1. the coefficient of restitution between A and B, [5]
  2. the angle turned through by the direction of motion of B as a result of the collision. [4]
SPS SPS FM Mechanics 2022 January Q5
9 marks Challenging +1.2
A right circular cone C of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of C. The other end of the string is attached to a particle P of mass 2.5 kg. P moves in a horizontal circle with constant speed and in contact with the smooth curved surface of C. The extension of the string is 1.5 m.
  1. Find the tension in the string. [2]
  2. Find the speed of P. [7]
SPS SPS FM Mechanics 2022 January Q6
8 marks Challenging +1.8
A uniform rod, PQ, of length \(2a\), rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both P and T is \(\mu\). \includegraphics{figure_6} The rod is on the point of slipping when it is inclined at an angle of 30\(^0\) to the horizontal. Find the value of \(\mu\). [8]
SPS SPS FM Mechanics 2022 January Q7
14 marks Challenging +1.2
The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the length of CD is \((a + 5)\) cm. \includegraphics{figure_7}
  1. Show the distance of the centre of mass of the prism from AD is $$\frac{a^2 + 15a + 75}{3(a + 10)} \text{ cm.}$$ [5]
The prism is placed with the face containing AB in contact with a horizontal surface.
  1. Find the greatest value of \(a\) for which the prism does not topple. [3]
The prism is now placed on an inclined plane which makes an angle \(\theta^o\) with the horizontal. AB lies along a line of greatest slope with B higher than A.
  1. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the great value of \(\theta\) for which the prism does not topple. [6]
SPS SPS FM 2021 November Q1
3 marks Moderate -0.3
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The roots of the equation $$x^3 - 8x^2 + 28x - 32 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the value of $$(\alpha + 2)(\beta + 2)(\gamma + 2).$$ [3 marks]
SPS SPS FM 2021 November Q2
3 marks Standard +0.3
The equation of a curve in polar coordinates is $$r = 11 + 9 \sec \theta.$$ Show that a cartesian equation of the curve is $$(x - 9)\sqrt{x^2 + y^2} = 11x.$$ [3 marks]
SPS SPS FM 2021 November Q3
6 marks Standard +0.3
The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|.$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6 marks]
SPS SPS FM 2021 November Q4
4 marks Moderate -0.3
Prove that $$\sum_{r=1}^{n} 18(r^2 - 4) = n(6n^2 + 9n - 69).$$ [4 marks]
SPS SPS FM 2021 November Q5
4 marks Standard +0.8
Use a trigonometrical substitution to show that $$\int_0^2 \frac{1}{(16 - x^2)^{\frac{3}{2}}} dx = \frac{1}{16\sqrt{3}}$$ [4 marks]
SPS SPS FM 2021 November Q6
7 marks Challenging +1.8
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]
SPS SPS FM 2021 November Q7
7 marks Challenging +1.3
The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]