Questions — Edexcel (10514 questions)

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Edexcel C4 2013 June Q2
9 marks Moderate -0.3
  1. Use the binomial expansion to show that $$\left. \sqrt { ( } \frac { 1 + x } { 1 - x } \right) \approx 1 + x + \frac { 1 } { 2 } x ^ { 2 } , \quad | x | < 1$$
  2. Substitute \(x = \frac { 1 } { 26 }\) into $$\sqrt { \left( \frac { 1 + x } { 1 - x } \right) = 1 + x + \frac { 1 } { 2 } x ^ { 2 } }$$ to obtain an approximation to \(\sqrt { } 3\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
Edexcel C4 Specimen Q2
6 marks Standard +0.3
The curve \(C\) has equation $$13 x ^ { 2 } + 13 y ^ { 2 } - 10 x y = 52$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) and \(y\), simplifying your answer.
(6)
Edexcel F1 2017 January Q2
7 marks Standard +0.3
The quadratic equation $$2 x ^ { 2 } - x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
  3. find a quadratic equation which has roots $$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
Edexcel FP2 2006 January Q2
13 marks Standard +0.3
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
  2. Given that \(x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\) at \(t = 0\), find the particular solution of the differential equation, giving your answer in the form \(x = \mathrm { f } ( t )\).
  3. Sketch the curve with equation \(x = \mathrm { f } ( t ) , 0 \leq t \leq \pi\), showing the coordinates, as multiples of \(\pi\), of the points where the curve cuts the \(x\)-axis.
    (4)(Total 13 marks)
Edexcel FP2 2002 June Q3
13 marks Standard +0.8
  1. Show that \(y = \frac { 1 } { 2 } x ^ { 2 } \mathrm { e } ^ { x }\) is a solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$
  2. Solve the differential equation \(\quad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }\).
    given that at \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\).
Edexcel FP2 2004 June Q2
10 marks Standard +0.3
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \left( 1 + \frac { 3 } { x } \right) = \frac { 1 } { x ^ { 2 } } , \quad x > 0$$
  1. Verify that \(x ^ { 3 } \mathrm { e } ^ { x }\) is an integrating factor for the differential equation.
  2. Find the general solution of the differential equation.
  3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\).
    (3)(Total 10 marks)
Edexcel FP2 2005 June Q2
7 marks Standard +0.8
Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \cot 2 x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(Total 7 marks)
Edexcel FP2 2009 June Q2
6 marks Standard +0.8
Solve the equation $$z ^ { 3 } = 4 \sqrt { } 2 - 4 \sqrt { } 2 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(- \pi < \theta \leqslant \pi\).
Edexcel M1 2011 January Q1
5 marks Moderate -0.8
  1. Two particles \(B\) and \(C\) have mass \(m \mathrm {~kg}\) and 3 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. The two particles collide directly. Immediately before the collision, the speed of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(C\) is reversed and the direction of motion of \(B\) is unchanged. Immediately after the collision, the speed of \(B\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find
  1. the value of \(m\),
  2. the magnitude of the impulse received by \(C\).
Edexcel M1 2011 January Q2
8 marks Moderate -0.3
2. A ball is thrown vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) at height \(h\) metres above the ground. The ball hits the ground 0.75 s later. The speed of the ball immediately before it hits the ground is \(6.45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle.
  1. Show that \(u = 0.9\)
  2. Find the height above \(P\) to which the ball rises before it starts to fall towards the ground again.
  3. Find the value of \(h\).
Edexcel M1 2011 January Q3
10 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-04_245_860_260_543} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform beam \(A B\) has mass 20 kg and length 6 m . The beam rests in equilibrium in a horizontal position on two smooth supports. One support is at \(C\), where \(A C = 1 \mathrm {~m}\), and the other is at the end \(B\), as shown in Figure 1. The beam is modelled as a rod.
  1. Find the magnitudes of the reactions on the beam at \(B\) and at \(C\). A boy of mass 30 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The magnitudes of the reactions on the beam at \(B\) and at \(C\) are now equal. The boy is modelled as a particle.
  2. Find the distance \(A D\).
Edexcel M1 2011 January Q4
11 marks Moderate -0.5
  1. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. The velocity of \(P\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 0\), and \(( 7 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 5 \mathrm {~s}\).
Find
  1. the speed of \(P\) at \(t = 0\),
  2. the vector \(\mathbf { F }\) in the form \(a \mathbf { i } + b \mathbf { j }\),
  3. the value of \(t\) when \(P\) is moving parallel to \(\mathbf { i }\).
Edexcel M1 2011 January Q5
10 marks Moderate -0.8
A car accelerates uniformly from rest for 20 seconds. It moves at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 40 seconds and then decelerates uniformly for 10 seconds until it comes to rest.
  1. For the motion of the car, sketch
    1. a speed-time graph,
    2. an acceleration-time graph. Given that the total distance moved by the car is 880 m ,
  2. find the value of \(v\).
Edexcel M1 2011 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_426_768_239_653} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.
  1. Show that the normal reaction between the particle and the plane has magnitude 114 N . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_433_774_1464_604} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.
  2. Find the greatest possible value of \(P\).
  3. Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).
Edexcel M1 2011 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-12_581_1211_235_370} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane.
  1. Find the magnitude of the acceleration of \(B\) immediately after release.
  2. Find the speed of \(B\) when it has moved 1 m up the plane. When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
  3. find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.
Edexcel M1 2012 January Q1
5 marks Moderate -0.8
  1. A railway truck \(P\), of mass \(m \mathrm {~kg}\), is moving along a straight horizontal track with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Truck \(P\) collides with a truck \(Q\) of mass 3000 kg , which is at rest on the same track. Immediately after the collision the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is reversed by the collision.
Modelling the trucks as particles, find
  1. the magnitude of the impulse exerted by \(P\) on \(Q\),
  2. the value of \(m\).
Edexcel M1 2012 January Q2
6 marks Moderate -0.8
2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N . The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and \(R\) newtons respectively. Given that the acceleration of the car and the caravan is \(0.88 \mathrm {~ms} ^ { - 2 }\),
  1. show that \(R = 860\),
  2. find the tension in the tow-bar.
Edexcel M1 2012 January Q3
8 marks Moderate -0.8
3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,
  1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
  2. the magnitude of \(\mathbf { R }\),
  3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).
Edexcel M1 2012 January Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{724254f3-3a6a-4820-b3a1-979458e24437-05_241_794_219_575} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(5 d\), rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = D B = d\), as shown in Figure 1. The centre of mass of the rod is at the point \(G\). A particle of mass \(\frac { 5 } { 2 } m\) is placed on the rod at \(B\) and the rod is on the point of tipping about \(D\).
  1. Show that \(G D = \frac { 5 } { 2 } d\). The particle is moved from \(B\) to the mid-point of the rod and the rod remains in equilibrium.
  2. Find the magnitude of the normal reaction between the support at \(D\) and the rod.
Edexcel M1 2012 January Q5
11 marks Moderate -0.8
  1. A stone is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After projection the stone moves freely under gravity until it returns to \(A\). The time between the instant that the stone is projected and the instant that it returns to \(A\) is \(3 \frac { 4 } { 7 }\) seconds.
Modelling the stone as a particle,
  1. show that \(u = 17 \frac { 1 } { 2 }\),
  2. find the greatest height above \(A\) reached by the stone,
  3. find the length of time for which the stone is at least \(6 \frac { 3 } { 5 } \mathrm {~m}\) above \(A\).
Edexcel M1 2012 January Q6
13 marks Moderate -0.3
A car moves along a straight horizontal road from a point \(A\) to a point \(B\), where \(A B = 885 \mathrm {~m}\). The car accelerates from rest at \(A\) to a speed of \(15 \mathrm {~ms} ^ { - 1 }\) at a constant rate \(a \mathrm {~ms} ^ { - 2 }\). The time for which the car accelerates is \(\frac { 1 } { 3 } T\) seconds. The car maintains the speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds. The car then decelerates at a constant rate of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) stopping at \(B\).
  1. Find the time for which the car decelerates.
  2. Sketch a speed-time graph for the motion of the car.
  3. Find the value of \(T\).
  4. Find the value of \(a\).
  5. Sketch an acceleration-time graph for the motion of the car.
Edexcel M1 2012 January Q7
9 marks Moderate -0.8
7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. Position vectors are relative to a fixed origin \(O\).] A boat \(P\) is moving with constant velocity \(( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Calculate the speed of \(P\). When \(t = 0\), the boat \(P\) has position vector \(( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p ~ k m }\).
  2. Write down \(\mathbf { p }\) in terms of \(t\). A second boat \(Q\) is also moving with constant velocity. At time \(t\) hours, the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$ Find
  3. the value of \(t\) when \(P\) is due west of \(Q\),
  4. the distance between \(P\) and \(Q\) when \(P\) is due west of \(Q\).
Edexcel M1 2012 January Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{724254f3-3a6a-4820-b3a1-979458e24437-13_334_538_219_703} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find
  1. the magnitude of the normal reaction between \(P\) and the plane,
  2. the value of \(\mu\). The force of magnitude 36 N is removed.
  3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.
Edexcel M1 2001 June Q1
6 marks Moderate -0.8
  1. Two small balls \(A\) and \(B\) have masses 0.5 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) immediately after the collision is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of the motion of \(A\) is unchanged as a result of the collision.
By modelling the balls as particles, find
  1. the speed of \(B\) immediately after the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision.
Edexcel M1 2001 June Q2
8 marks Moderate -0.8
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-2_272_592_1239_648}
\end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle. The force \(\mathbf { P }\) has magnitude 5 N and the force \(\mathbf { Q }\) has magnitude 3 N . The angle between the directions of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(40 ^ { \circ }\), as shown in Fig. 1. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(\mathbf { F }\).
  1. Find, to 3 significant figures, the magnitude of \(\mathbf { F }\).
  2. Find, in degrees to 1 decimal place, the angle between the directions of \(\mathbf { F }\) and \(\mathbf { P }\).