Questions (33218 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI C3 2007 January Q3
7 marks Moderate -0.8
3 The value \(\pounds V\) of a car is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(t\) is the age of the car in years and \(A\) and \(k\) are constants. Its value when new is \(\pounds 10000\), and after 3 years its value is \(\pounds 6000\).
  1. Find the values of \(A\) and \(k\).
  2. Find the age of the car when its value is \(\pounds 2000\).
OCR MEI C3 2007 January Q4
3 marks Moderate -0.8
4 Use the method of exhaustion to prove the following result.
No 1 - or 2 -digit perfect square ends in \(2,3,7\) or 8
State a generalisation of this result.
OCR MEI C3 2007 January Q5
8 marks Standard +0.3
5 The equation of a curve is \(y = \frac { x ^ { 2 } } { 2 x + 1 }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( x + 1 ) } { ( 2 x + 1 ) ^ { 2 } }\).
  2. Find the coordinates of the stationary points of the curve. You need not determine their nature.
OCR MEI C3 2007 January Q6
8 marks Moderate -0.3
6 Fig. 6 shows the triangle OAP , where O is the origin and A is the point \(( 0,3 )\). The point \(\mathrm { P } ( x , 0 )\) moves on the positive \(x\)-axis. The point \(\mathrm { Q } ( 0 , y )\) moves between O and A in such a way that \(\mathrm { AQ } + \mathrm { AP } = 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-3_490_839_438_612} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Write down the length AQ in terms of \(y\). Hence find AP in terms of \(y\), and show that $$( y + 3 ) ^ { 2 } = x ^ { 2 } + 9 .$$
  2. Use this result to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { y + 3 }\).
  3. When \(x = 4\) and \(y = 2 , \frac { \mathrm {~d} x } { \mathrm {~d} t } = 2\). Calculate \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) at this time. Section B (36 marks)
OCR MEI C3 2007 January Q7
18 marks Standard +0.3
7 Fig. 7 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x \sqrt { 1 + x }\). The curve meets the \(x\)-axis at the origin and at the point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-4_491_881_476_588} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Verify that the point P has coordinates \(( - 1,0 )\). Hence state the domain of the function \(\mathrm { f } ( x )\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + 3 x } { 2 \sqrt { 1 + x } }\).
  3. Find the exact coordinates of the turning point of the curve. Hence write down the range of the function.
  4. Use the substitution \(u = 1 + x\) to show that $$\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$ Hence find the area of the region enclosed by the curve and the \(x\)-axis.
OCR MEI C3 2007 January Q8
18 marks Moderate -0.3
8 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \text { for } x \geqslant 0 .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-5_707_876_440_593} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the origin and at the point \(( \ln 2,1 )\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \ln ( 1 + \sqrt { x } )\) for \(x \geqslant 0\).
  2. Show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are inverse functions. Hence sketch the graph of \(y = \mathrm { g } ( x )\). Write down the gradient of the curve \(y = \mathrm { g } ( x )\) at the point \(( 1 , \ln 2 )\).
  3. Show that \(\int \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { x } + x + c\). Hence evaluate \(\int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \mathrm {~d} x\), giving your answer in an exact form.
  4. Using your answer to part (iii), calculate the area of the region enclosed by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis and the line \(x = 1\).
Edexcel M3 2014 January Q1
5 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force of magnitude \(F\) newtons. The force acts along the \(x\)-axis in the direction of \(x\) increasing. When \(P\) is \(x\) metres from the origin \(O\), it is moving away from \(O\) with speed \(\sqrt { \left( 8 x ^ { \frac { 3 } { 2 } } - 4 \right) } \mathrm { ms } ^ { - 1 }\).
Find \(F\) when \(P\) is 4 m from \(O\).
Edexcel M3 2014 January Q2
9 marks Standard +0.8
2. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring, of natural length \(l\) and modulus of elasticity \(2 m g\). The other end of the spring is attached to a fixed point \(A\) on a rough horizontal plane. The particle is held at rest on the plane at a point \(B\), where \(A B = \frac { 1 } { 2 } l\), and released from rest. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\) Find the distance of \(P\) from \(B\) when \(P\) first comes to rest.
Edexcel M3 2014 January Q3
8 marks Challenging +1.2
3. A light rod \(A B\) of length \(2 a\) has a particle \(P\) of mass \(m\) attached to \(B\). The rod is rotating in a vertical plane about a fixed smooth horizontal axis through \(A\). Given that the greatest tension in the rod is \(\frac { 9 m g } { 8 }\), find, to the nearest degree, the angle between the rod and the downward vertical when the speed of \(P\) is \(\sqrt { \left( \frac { a g } { 20 } \right) }\).
Edexcel M3 2014 January Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-05_568_620_269_653} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the region \(R\) bounded by the curve with equation \(y = \mathrm { e } ^ { - x }\), the line \(x = 1\), the \(x\)-axis and the \(y\)-axis. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the volume of \(S\) is \(\frac { \pi } { 2 } \left( 1 - \mathrm { e } ^ { - 2 } \right)\).
  2. Find, in terms of e, the distance of the centre of mass of \(S\) from \(O\).
Edexcel M3 2014 January Q5
12 marks Standard +0.8
5. A solid \(S\) consists of a uniform solid hemisphere of radius \(r\) and a uniform solid circular cylinder of radius \(r\) and height \(3 r\). The circular face of the hemisphere is joined to one of the circular faces of the cylinder, so that the centres of the two faces coincide. The other circular face of the cylinder has centre \(O\). The mass per unit volume of the hemisphere is \(3 k\) and the mass per unit volume of the cylinder is \(k\).
  1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 9 r } { 4 }\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-07_501_1082_653_422} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The solid \(S\) is held in equilibrium by a horizontal force of magnitude \(P\). The circular face of \(S\) has one point in contact with a fixed rough horizontal plane and is inclined at an angle \(\alpha\) to the horizontal. The force acts through the highest point of the circular face of \(S\) and in the vertical plane through the axis of the cylinder, as shown in Figure 2. The coefficient of friction between \(S\) and the plane is \(\mu\). Given that \(S\) is on the point of slipping along the plane in the same direction as \(P\),
  2. show that \(\mu = \frac { 1 } { 8 } ( 9 - 4 \cot \alpha )\).
Edexcel M3 2014 January Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-09_1089_1072_278_466} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A light inextensible string of length \(14 a\) has its ends attached to two fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = 10 a\). A particle of mass \(m\) is attached to the string at the point \(P\), where \(A P = 8 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\) and with both parts of the string taut, as shown in Figure 3.
  1. Show that angle \(A P B = 90 ^ { \circ }\).
  2. Show that the time for the particle to make one complete revolution is less than $$2 \pi \sqrt { \left( \frac { 32 a } { 5 g } \right) } .$$
Edexcel M3 2014 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-11_517_254_278_845} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A smooth hollow narrow tube of length \(l\) has one open end and one closed end. The tube is fixed in a vertical position with the closed end at the bottom. A light elastic spring has natural length \(l\) and modulus of elasticity \(8 m g\). The spring is inside the tube and has one end attached to a fixed point \(O\) on the closed end of the tube. The other end of the spring is attached to a particle \(P\) of mass \(m\). The particle rests in equilibrium at a distance \(e\) below the top of the tube, as shown in Figure 4.
  1. Find \(e\) in terms of \(l\). The particle \(P\) is now held inside the tube at a distance \(\frac { 1 } { 2 } l\) below the top of the tube and released from rest at time \(t = 0\)
  2. Prove that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \left( \frac { l } { 8 g } \right) }\). The particle \(P\) passes through the open top of the tube with speed \(u\).
  3. Find \(u\) in terms of \(g\) and \(l\).
  4. Find the time taken for \(P\) to first attain a speed of \(\sqrt { \left( \frac { 9 g l } { 32 } \right) }\).
Edexcel M3 2015 January Q1
6 marks Standard +0.8
  1. A particle \(P\) of mass 3 kg is moving along the horizontal \(x\)-axis. At time \(t = 0 , P\) passes through the origin \(O\) moving in the positive \(x\) direction. At time \(t\) seconds, \(O P = x\) metres and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds, the resultant force acting on \(P\) is \(\frac { 9 } { 2 } ( 26 - x ) \mathrm { N }\), measured in the positive \(x\) direction. For \(t > 0\) the maximum speed of \(P\) is \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find \(v ^ { 2 }\) in terms of \(x\).
Edexcel M3 2015 January Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-03_547_671_260_648} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina is in the shape of the region \(R\) which is bounded by the curve with equation \(y = \frac { 3 } { x ^ { 2 } }\), the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis, as shown in Figure 1. The centre of mass of the lamina has coordinates \(( \bar { x } , \bar { y } )\).
Use algebraic integration to find
  1. the value of \(\bar { x }\),
  2. the value of \(\bar { y }\).
Edexcel M3 2015 January Q3
8 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-05_828_624_264_676} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). An identical string has one end attached to the fixed point \(B\), where \(B\) is vertically below \(A\) and \(A B = 4 a\), and the other end attached to \(P\), as shown in Figure 2. The particle is moving in a horizontal circle with constant angular speed \(\omega\), with both strings taut and inclined at \(30 ^ { \circ }\) to the vertical. The tension in the upper string is twice the tension in the lower string. Find \(\omega\) in terms of \(a\) and \(g\).
Edexcel M3 2015 January Q4
11 marks Standard +0.8
A light elastic string has natural length 5 m and modulus of elasticity 20 N . The ends of the string are attached to two fixed points \(A\) and \(B\), which are 6 m apart on a horizontal ceiling. A particle \(P\) is attached to the midpoint of the string and hangs in equilibrium at a point which is 4 m below \(A B\).
  1. Calculate the weight of \(P\). The particle is now raised to the midpoint of \(A B\) and released from rest.
  2. Calculate the speed of \(P\) when it has fallen 4 m .
Edexcel M3 2015 January Q5
10 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-09_270_919_267_557} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a uniform solid \(S\) formed by joining the plane faces of two solid right circular cones, of base radius \(r\), so that the centres of their bases coincide at \(O\). One cone, with vertex \(V\), has height \(4 r\) and the other cone has height \(k r\), where \(k > 4\)
  1. Find the distance of the centre of mass of \(S\) from \(O\).
    (4) The point \(A\) lies on the circumference of the common base of the cones. The solid is placed on a horizontal surface with VA in contact with the surface. Given that \(S\) rests in equilibrium,
  2. find the greatest possible value of \(k\). When \(S\) is suspended from \(A\) and hangs freely in equilibrium, \(O A\) makes an angle of \(12 ^ { \circ }\) with the downward vertical.
  3. Find the value of \(k\).
Edexcel M3 2015 January Q6
15 marks Standard +0.8
6. A smooth sphere, with centre \(O\) and radius \(a\), is fixed with its lowest point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\sqrt { \frac { a g } { 5 } }\) and moves along the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the sphere, the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = \frac { a g } { 5 } ( 11 - 10 \cos \theta )\). The particle leaves the surface of the sphere at the point \(C\).
    Find
  2. the speed of \(P\) at \(C\) in terms of \(a\) and \(g\),
  3. the size of the angle between the floor and the direction of motion of \(P\) at the instant immediately before \(P\) hits the floor.
Edexcel M3 2015 January Q7
16 marks Challenging +1.2
7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane. The particle rests in equilibrium at the point \(B\), where \(B\) is lower than \(A\) and \(A B = \frac { 6 } { 5 } a\).
  1. Show that \(\lambda = \frac { 5 } { 2 } m g\). The particle is now pulled down a line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 5 } a\), and released from rest.
  2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { 2 a } { 5 g } }\)
  3. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(P\) while the string is taut. The midpoint of \(B C\) is \(D\) and the string becomes slack for the first time at the point \(E\).
  4. Find, in terms of \(a\) and \(g\), the time taken by \(P\) to travel directly from \(D\) to \(E\).
Edexcel M3 2016 January Q1
6 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-02_503_524_121_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl of internal radius \(2 r\) is fixed with its circular rim horizontal. A particle \(P\) is moving in a horizontal circle of radius \(r\) on the smooth inner surface of the bowl, as shown in Figure 1. Particle \(P\) is moving with constant angular speed \(\omega\). Show that \(\omega = \sqrt { \frac { g \sqrt { 3 } } { 3 r } }\)
Edexcel M3 2016 January Q2
8 marks Standard +0.3
2. A particle \(P\) is moving in a straight line. At time \(t\) seconds, the distance of \(P\) from a fixed point \(O\) on the line is \(x\) metres and the acceleration of \(P\) is \(( 6 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the velocity of \(P\) in terms of \(t\).
  2. Find the total distance travelled by \(P\) in the first 4 seconds.
Edexcel M3 2016 January Q3
9 marks Standard +0.8
3. A car of mass 800 kg is driven at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) round a bend in a race track. Around the bend, the track is banked at \(20 ^ { \circ }\) to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5 Given that the tyres do not slip sideways on the track, find the maximum value of \(v\).
Edexcel M3 2016 January Q4
10 marks Standard +0.8
4. Fixed points \(A\) and \(B\) are on a horizontal ceiling, where \(A B = 4 a\). A light elastic string has natural length \(3 a\) and modulus of elasticity \(\lambda\). One end of the string is attached to \(A\) and the other end is attached to \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at the point \(C\), where \(C\) is at a distance \(\frac { 3 } { 2 } a\) vertically below the ceiling.
  1. Show that \(\lambda = \frac { 5 m g } { 4 }\) (5) The point \(D\) is the midpoint of \(A B\). The particle is now raised vertically upwards to \(D\), and released from rest.
  2. Find the speed of \(P\) as it passes through \(C\).
    \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-05_542_51_2026_1982}VIIV SIHI NI JIIIM IONOOVI4V SIHI NI JIIYM ION OO
Edexcel M3 2016 January Q5
13 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-07_371_800_262_573} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The particle rests in equilibrium on the plane at the point \(B\) with the string lying along a line of greatest slope of the plane, as shown in Figure 2. Given that \(A B = \frac { 6 } { 5 } l\)
  1. show that \(\lambda = 3 \mathrm { mg }\) The particle is pulled down the line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 2 } l\), and released from rest.
  2. Show that, while the string remains taut, \(P\) moves with simple harmonic motion about centre \(B\).
  3. Find the greatest magnitude of the acceleration of \(P\) while the string remains taut. The point \(D\) is the midpoint of \(B C\). The time taken by \(P\) to move directly from \(D\) to the point where the string becomes slack for the first time is \(k \sqrt { \frac { l } { g } }\), where \(k\) is a constant.
  4. Find, to 2 significant figures, the value of \(k\).