Questions — Edexcel M2 (623 questions)

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Edexcel M2 2010 January Q7
11 marks Standard +0.8
[The centre of mass of a semi-circular lamina of radius \(r\) is \(\frac{4r}{3\pi}\) from the centre] \includegraphics{figure_3} A template \(T\) consists of a uniform plane lamina \(PQRQS\), as shown in Figure 3. The lamina is bounded by two semicircles, with diameters \(SQ\) and \(QR\), and by the sides \(SP\), \(PQ\) and \(QR\) of the rectangle \(PQRS\). The point \(O\) is the mid-point of \(SR\), \(PQ = 12\) cm and \(QR = 2\) cm.
  1. Show that the centre of mass of \(T\) is a distance \(\frac{4|2x^2 - 3|}{8x + 3\pi}\) cm from \(SR\). [7]
The template \(T\) is freely suspended from the point \(P\) and hangs in equilibrium. Given that \(x = 2\) and that \(\theta\) is the angle that \(PQ\) makes with the horizontal,
  1. show that \(\tan \theta = \frac{48 + 9\pi}{22 + 6\pi}\). [4]
Edexcel M2 2010 January Q8
17 marks Standard +0.3
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in a horizontal and upward vertical direction respectively] A particle \(P\) is projected from a fixed point \(O\) on horizontal ground with velocity \(u(\mathbf{i} + c\mathbf{j}) \text{ ms}^{-1}\), where \(c\) and \(u\) are positive constants. The particle moves freely under gravity until it strikes the ground at \(A\), where it immediately comes to rest. Relative to \(O\), the position vector of a point on the path of \(P\) is \((x\mathbf{i} + y\mathbf{j})\) m.
  1. Show that $$y = cx - \frac{4.9x^2}{u^2}.$$ [5]
Given that \(u = 7\), \(OA = R\) m and the maximum vertical height of \(P\) above the ground is \(H\) m,
  1. using the result in part (a), or otherwise, find, in terms of \(c\),
    1. \(R\)
    2. \(H\).
    [6]
Given also that when \(P\) is at the point \(Q\), the velocity of \(P\) is at right angles to its initial velocity,
  1. find, in terms of \(c\), the value of \(x\) at \(Q\). [6]
Edexcel M2 2012 January Q1
4 marks Easy -1.2
A tennis ball of mass \(0.1\) kg is hit by a racquet. Immediately before being hit, the ball has velocity \(30\mathbf{i}\) m s\(^{-1}\). The racquet exerts an impulse of \((-2\mathbf{i} - 4\mathbf{j})\) N s on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit. [4]
Edexcel M2 2012 January Q2
10 marks Moderate -0.3
A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf{v}\) m s\(^{-1}\), where \(\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}\). Find
  1. the speed of \(P\) when \(t = 4\) [2]
  2. the acceleration of \(P\) when \(t = 4\) [3]
Given that \(P\) is at the point with position vector \((-4\mathbf{i} + \mathbf{j})\) m when \(t = 1\),
  1. find the position vector of \(P\) when \(t = 4\) [5]
Edexcel M2 2012 January Q3
10 marks Standard +0.3
A cyclist and her cycle have a combined mass of \(75\) kg. The cyclist is cycling up a straight road inclined at \(5°\) to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude \(20\) N. At the instant when the cyclist has a speed of \(12\) m s\(^{-1}\), she is decelerating at \(0.2\) m s\(^{-2}\).
  1. Find the rate at which the cyclist is working at this instant. [5]
When the cyclist passes the point \(A\) her speed is \(8\) m s\(^{-1}\). At \(A\) she stops working but does not apply the brakes. She comes to rest at the point \(B\). The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude \(20\) N.
  1. Use the work-energy principle to find the distance \(AB\). [5]
Edexcel M2 2012 January Q4
10 marks Standard +0.3
\includegraphics{figure_1} The trapezium \(ABCD\) is a uniform lamina with \(AB = 4\) m and \(BC = CD = DA = 2\) m, as shown in Figure 1.
  1. Show that the centre of mass of the lamina is \(\frac{4\sqrt{3}}{9}\) m from \(AB\). [5]
The lamina is freely suspended from \(D\) and hangs in equilibrium.
  1. Find the angle between \(DC\) and the vertical through \(D\). [5]
Edexcel M2 2012 January Q5
11 marks Standard +0.3
\includegraphics{figure_2} A uniform rod \(AB\) has mass \(4\) kg and length \(1.4\) m. The end \(A\) is resting on rough horizontal ground. A light string \(BC\) has one end attached to \(B\) and the other end attached to a fixed point \(C\). The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at \(20°\) to the ground, as shown in Figure 2.
  1. Find the tension in the string. [4]
Given that the rod is about to slip,
  1. find the coefficient of friction between the rod and the ground. [7]
Edexcel M2 2012 January Q6
15 marks Standard +0.3
Three identical particles, \(A\), \(B\) and \(C\), lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The mass of each particle is \(m\). Particle \(A\) is projected towards \(B\) with speed \(u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac{2}{3}\).
  1. Find, in terms of \(u\),
    1. the speed of \(A\) after this collision,
    2. the speed of \(B\) after this collision.
    [7]
  2. Show that the kinetic energy lost in this collision is \(\frac{5}{36}mu^2\) [4]
After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\).
  1. Find, in terms of \(u\), the speed of \(C\) immediately after this collision between \(B\) and \(C\). [4]
Edexcel M2 2012 January Q7
15 marks Standard +0.3
[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((6\mathbf{i} + 12\mathbf{j})\) m s\(^{-1}\), and passes through the point \(A\) at time \(t\) seconds after projection. The point \(B\) is on the horizontal plane vertically below \(A\), as shown in Figure 3. It is given that \(OB = 2AB\). Find
  1. the value of \(t\), [7]
  2. the speed, \(V\) m s\(^{-1}\), of the ball at the instant when it passes through \(A\). [5]
At another point \(C\) on the path the speed of the ball is also \(V\) m s\(^{-1}\).
  1. Find the time taken for the ball to travel from \(O\) to \(C\). [3]
Edexcel M2 2014 January Q1
4 marks Easy -1.2
Simplify fully
  1. \((2\sqrt{x})^2\) [1]
  2. \(\frac{5 + \sqrt{7}}{2 + \sqrt{7}}\) [3]
Edexcel M2 2014 January Q2
5 marks Easy -1.2
\(y = 2x^2 - \frac{4}{\sqrt{x}} + 1\), \(x > 0\)
  1. Find \(\frac{dy}{dx}\), giving each term in its simplest form. [3]
  2. Find \(\frac{d^2y}{dx^2}\), giving each term in its simplest form. [2]
Edexcel M2 2014 January Q3
7 marks Moderate -0.5
Solve the simultaneous equations $$x - 2y - 1 = 0$$ $$x^2 + 4y^2 - 10x + 9 = 0$$ [7]
Edexcel M2 2014 January Q4
4 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of a curve with equation \(y = f(x)\). The curve crosses the \(y\)-axis at \((0, 3)\) and has a minimum at \(P(4, 2)\). On separate diagrams, sketch the curve with equation
  1. \(y = f(x + 4)\), [2]
  2. \(y = 2f(x)\). [2]
On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.
Edexcel M2 2014 January Q5
5 marks Moderate -0.8
Given that for all positive integers \(n\), $$\sum_{r=1}^{n} a_r = 12 + 4n^2$$
  1. find the value of \(\sum_{r=1}^{5} a_r\) [2]
  2. Find the value of \(a_6\) [3]
Edexcel M2 2014 January Q6
11 marks Moderate -0.3
\includegraphics{figure_2} The straight line \(l_1\) has equation \(2y = 3x + 7\) The line \(l_1\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.
    1. State the gradient of \(l_1\)
    2. Write down the coordinates of the point \(A\). [2]
Another straight line \(l_2\) intersects \(l_1\) at the point \(B(1, 5)\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle ABC = 90°\),
  1. find an equation of \(l_2\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The rectangle \(ABCD\), shown shaded in Figure 2, has vertices at the points \(A\), \(B\), \(C\) and \(D\).
  1. Find the exact area of rectangle \(ABCD\). [5]
Edexcel M2 2014 January Q7
10 marks Easy -1.2
Shelim starts his new job on a salary of £14000. He will receive a rise of £1500 a year for each full year that he works, so that he will have a salary of £15500 in year 2, a salary of £17000 in year 3 and so on. When Shelim's salary reaches £26000, he will receive no more rises. His salary will remain at £26000.
  1. Show that Shelim will have a salary of £26000 in year 9. [2]
  2. Find the total amount that Shelim will earn in his job in the first 9 years. [2]
Anna starts her new job at the same time as Shelim on a salary of £\(A\). She receives a rise of £1000 a year for each full year that she works, so that she has a salary of £\((A + 1000)\) in year 2, £\((A + 2000)\) in year 3 and so on. The maximum salary for her job, which is reached in year 10, is also £26000.
  1. Find the difference in the total amount earned by Shelim and Anna in the first 10 years. [6]
Edexcel M2 2014 January Q8
7 marks Moderate -0.8
The equation \(2x^2 + 2kx + (k + 2) = 0\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies $$k^2 - 2k - 4 > 0$$ [3]
  2. Find the set of possible values of \(k\). [4]
Edexcel M2 2014 January Q9
12 marks Moderate -0.3
A curve with equation \(y = f(x)\) passes through the point \((3, 6)\). Given that $$f'(x) = (x - 2)(3x + 4)$$
  1. use integration to find \(f(x)\). Give your answer as a polynomial in its simplest form. [5]
  2. Show that \(f(x) = (x - 2)^2(x + p)\), where \(p\) is a positive constant. State the value of \(p\). [3]
  3. Sketch the graph of \(y = f(x)\), showing the coordinates of any points where the curve touches or crosses the coordinate axes. [4]
Edexcel M2 2014 January Q10
10 marks Moderate -0.3
The curve \(C\) has equation \(y = x^3 - 2x^2 - x + 3\) The point \(P\), which lies on \(C\), has coordinates \((2, 1)\).
  1. Show that an equation of the tangent to \(C\) at the point \(P\) is \(y = 3x - 5\) [5]
The point \(Q\) also lies on \(C\). Given that the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\),
  1. find the coordinates of the point \(Q\). [5]
Edexcel M2 2001 June Q1
5 marks Moderate -0.3
At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$\mathbf{r} = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]
Edexcel M2 2001 June Q3
9 marks Standard +0.3
A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5. The other end \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of 30° with the wall. A man of mass \(5m\) stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(ka\). Find the value of \(k\). [9]
Edexcel M2 2001 June Q4
10 marks Moderate -0.3
The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) lie in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) vertical. A ball of mass 0.1 kg is hit by a bat which gives it an impulse of \((3.5\mathbf{i} + 3\mathbf{j})\) Ns. The velocity of the ball immediately after being hit is \((10\mathbf{i} + 25\mathbf{j})\) m s\(^{-1}\).
  1. Find the velocity of the ball immediately before it is hit. [3]
In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.
  1. Find the greatest height of the ball above the ground in the subsequent motion. [3]
The ball is caught when it is again 1 m above the ground.
  1. Find the distance from the point where the ball is hit to the point where it is caught. [4]
Edexcel M2 2001 June Q5
10 marks Standard +0.3
A child is playing with a small model of a fire-engine of mass 0.5 kg and a straight, rigid plank. The plank is inclined at an angle \(\alpha\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(\alpha = 20°\) the fire-engine is projected with an initial speed of 5 m s\(^{-1}\) and first comes to rest after travelling 2 m.
  1. Find, to 3 significant figures, the value of \(R\). [7]
When \(\alpha = 40°\) the fire-engine is again projected with an initial speed of 5 m s\(^{-1}\).
  1. Find how far the fire-engine travels before first coming to rest. [3]
Edexcel M2 2001 June Q6
16 marks Standard +0.3
A particle \(A\) of mass \(2m\) is moving with speed \(2u\) on a smooth horizontal table. The particle collides directly with a particle \(B\) of mass \(4m\) moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).
  1. Show that the speed of \(B\) after the collision is \(\frac{5}{3}u\). [6]
  2. Find the speed of \(A\) after the collision. [2]
Subsequently \(B\) collides directly with a particle \(C\) of mass \(m\) which is at rest on the table. The coefficient of restitution between \(B\) and \(C\) is \(e\). Given that there are no further collisions,
  1. find the range of possible values for \(e\). [8]
Edexcel M2 2001 June Q7
16 marks Standard +0.3
\includegraphics{figure_2} At time \(t = 0\) a small package is projected from a point \(B\) which is 2.4 m above a point \(A\) on horizontal ground. The package is projected with speed 23.75 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{4}{3}\). The package strikes the ground at the point \(C\), as shown in Fig. 2. The package is modelled as a particle moving freely under gravity.
  1. Find the time taken for the package to reach \(C\). [5]
A lorry moves along the line \(AC\), approaching \(A\) with constant speed 18 m s\(^{-1}\). At time \(t = 0\) the rear of the lorry passes \(A\) and the lorry starts to slow down. It comes to rest \(T\) seconds later. The acceleration, \(a\) m s\(^{-2}\) of the lorry at time \(t\) seconds is given by $$a = -\frac{1}{4}t^2, \quad 0 \leq t \leq T.$$
  1. Find the speed of the lorry at time \(t\) seconds. [3]
  2. Hence show that \(T = 6\). [3]
  3. Show that when the package reaches \(C\) it is just under 10 m behind the rear of the moving lorry. [5]
END