Questions — Edexcel (10514 questions)

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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
Edexcel M2 2002 June Q1
8 marks Moderate -0.3
The velocity \(v\) m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$
  1. Show that the acceleration of \(P\) is constant. [2]
At \(t = 0\), the position vector of \(P\) relative to a fixed origin O is \(3\mathbf{i}\) m.
  1. Find the distance of \(P\) from O when \(t = 2\). [6]
Edexcel M2 2002 June Q2
8 marks Standard +0.3
A particle \(P\) moves in a straight line so that, at time \(t\) seconds, its acceleration \(a\) m s\(^{-2}\) is given by $$a = \begin{cases} 4t - t^2, & 0 \leq t \leq 3, \\ \frac{27}{t^2}, & t > 3. \end{cases}$$ At \(t = 0\), \(P\) is at rest. Find the speed of \(P\) when
  1. \(t = 3\), [3]
  2. \(t = 6\). [5]
Edexcel M2 2002 June Q3
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at 8 m s\(^{-1}\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m. When she reaches \(B\), her speed is 5 m s\(^{-1}\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m. By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N,
  1. find the work done by the cyclist in moving from \(A\) to \(B\). [5]
At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at 0.5 m s\(^{-2}\),
  1. find the power generated by the cyclist at \(B\). [4]
Edexcel M2 2002 June Q4
11 marks Standard +0.3
\includegraphics{figure_1} A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(ABCD\), of side 10 cm, and removing the semi-circle with diameter \(AB\) from the square, as shown in Fig. 2.
  1. Find, in cm to 2 decimal places, the distance of the centre of mass of the lamina \(L\) from the mid-point of \(AB\). [7]
[The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac{4a}{3\pi}\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.
  1. Find, in degrees to one decimal place, the angle between \(CD\) and the vertical. [4]
Edexcel M2 2002 June Q5
12 marks Standard +0.3
A particle is projected from a point with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When it has moved a horizontal distance \(x\), its height above the point of projection is \(y\).
  1. Show that $$y = x \tan \alpha - \frac{gx^2}{2u^2}(1 + \tan^2 \alpha).$$ [5]
A shot-putter puts a shot from a point \(A\) at a height of 2 m above horizontal ground. The shot is projected at an angle of elevation of 45° with a speed of 14 m s\(^{-1}\). By modelling the shot as a particle moving freely under gravity,
  1. find, to 3 significant figures, the horizontal distance of the shot from \(A\) when the shot hits the ground, [5]
  2. find, to 2 significant figures, the time taken by the shot in moving from \(A\) to reach the ground. [2]
Edexcel M2 2002 June Q6
13 marks Standard +0.8
A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(u\) when it collides directly with another small smooth ball \(B\) of mass \(3m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.
  1. Find, in terms of \(e\) and \(u\), the speeds of \(A\) and \(B\) immediately after the collision. [7]
In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac{1}{3}\). Given that there is a second collision between \(A\) and \(B\),
  1. find the range of values of \(e\) for which the motion described is possible. [6]
Edexcel M2 2002 June Q7
14 marks Standard +0.8
\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6. The log is modelled as a uniform rod and the cable as light.
  1. Show that the normal reaction on the log at \(A\) is \(\frac{5}{8}W\). [6]
  2. Find the value of \(\beta\). [6]
The tension in the cable is \(kW\).
  1. Find the value of \(k\). [2]
Edexcel M2 2003 June Q1
5 marks Moderate -0.3
A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\) m s\(^{-1}\) in the direction of \(x\) increasing, where \(v = 6t - 2t^2\). When \(t = 0\), \(P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest after leaving \(O\). [5]
Edexcel M2 2003 June Q2
8 marks Moderate -0.3
A tennis ball of mass \(0.2\) kg is moving with velocity \((-10\mathbf{i})\) m s\(^{-1}\) when it is struck by a tennis racket. Immediately after being struck, the ball has velocity \((15\mathbf{i}+ 15\mathbf{j})\) m s\(^{-1}\). Find
  1. the magnitude of the impulse exerted by the racket on the ball, [4]
  2. the angle, to the nearest degree, between the vector \(\mathbf{i}\) and the impulse exerted by the racket, [2]
  3. the kinetic energy gained by the ball as a result of being struck. [2]
Edexcel M2 2003 June Q3
9 marks Standard +0.3
\includegraphics{figure_1} A uniform lamina \(ABCD\) is made by taking a uniform sheet of metal in the form of a rectangle \(ABED\), with \(AB = 3a\) and \(AD = 2a\), and removing the triangle \(BCE\), where \(C\) lies on \(DE\) and \(CE = a\), as shown in Fig. 1.
  1. Find the distance of the centre of mass of the lamina from \(AD\). [5]
The lamina has mass \(M\). A particle of mass \(m\) is attached to the lamina at \(B\). When the loaded lamina is freely suspended from the mid-point of \(AB\), it hangs in equilibrium with \(AB\) horizontal.
  1. Find \(m\) in terms of \(M\). [4]
Edexcel M2 2003 June Q4
12 marks Moderate -0.3
\includegraphics{figure_2} A uniform steel girder \(AB\), of mass 40 kg and length 3 m, is freely hinged at \(A\) to a vertical wall. The girder is supported in a horizontal position by a steel cable attached to the girder at \(B\). The other end of the cable is attached to the point \(C\) vertically above \(A\) on the wall, with \(\angle ABC = \alpha\), where \(\tan \alpha = \frac{4}{3}\). A load of mass 60 kg is suspended by another cable from the girder at the point \(D\), where \(AD = 2\) m, as shown in Fig. 2. The girder remains horizontal and in equilibrium. The girder is modelled as a rod, and the cables as light inextensible strings.
  1. Show that the tension in the cable \(BC\) is 980 N. [5]
  2. Find the magnitude of the reaction on the girder at \(A\). [6]
  3. Explain how you have used the modelling assumption that the cable at \(D\) is light. [1]
Edexcel M2 2003 June Q5
12 marks Standard +0.3
\includegraphics{figure_3} A ball is thrown from a point 4 m above horizontal ground. The ball is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac{4}{3}\). The ball hits the ground at a point which is a horizontal distance 8 m from its point of projection, as shown in Fig. 3. The initial speed of the ball is \(u\) m s\(^{-1}\) and the time of flight is \(T\) seconds.
  1. Prove that \(uT = 10\). [2]
  2. Find the value of \(u\). [5]
As the ball hits the ground, its direction of motion makes an angle \(\phi\) with the horizontal.
  1. Find \(\tan \phi\). [5]
Edexcel M2 2003 June Q6
14 marks Standard +0.3
A girl and her bicycle have a combined mass of 64 kg. She cycles up a straight stretch of road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{14}\). She cycles at a constant speed of 5 m s\(^{-1}\). When she is cycling at this speed, the resistance to motion from non-gravitational forces has magnitude 20 N.
  1. Find the rate at which the cyclist is working. [4]
She now turns round and comes down the same road. Her initial speed is 5 m s\(^{-1}\), and the resistance to motion is modelled as remaining constant with magnitude 20 N. She free-wheels down the road for a distance of 80 m. Using this model,
  1. find the speed of the cyclist when she has travelled a distance of 80 m. [5]
The cyclist again moves down the same road, but this time she pedals down the road. The resistance is now modelled as having magnitude proportional to the speed of the cyclist. Her initial speed is again 5 m s\(^{-1}\) when the resistance to motion has magnitude 20 N.
  1. Find the magnitude of the resistance to motion when the speed of the cyclist is 8 m s\(^{-1}\). [1]
The cyclist works at a constant rate of 200 W.
  1. Find the magnitude of her acceleration when her speed is 8 m s\(^{-1}\). [4]