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
Edexcel M3 2021 October Q7
14 marks Challenging +1.2
\hspace{0pt} [You may assume that the volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).] A uniform solid right circular cone \(C\), with vertex \(V\), has base radius \(r\) and height \(h\).
  1. Show that the centre of mass of \(C\) is \(\frac { 3 } { 4 } h\) from \(V\) A solid \(F\), shown below in Figure 4, is formed by removing the solid right circular cone \(C ^ { \prime }\) from \(C\), where cone \(C ^ { \prime }\) has height \(\frac { 1 } { 3 } h\) and vertex \(V\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-24_666_670_854_639} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure}
  2. Show that the distance of the centre of mass of \(F\) from its larger plane face is \(\frac { 3 } { 13 } h\) The solid \(F\) rests in equilibrium with its curved surface in contact with a horizontal plane.
  3. Show that \(13 r ^ { 2 } \leqslant 17 h ^ { 2 }\)
    \includegraphics[max width=\textwidth, alt={}]{9777abb8-a564-40d5-8d96-d5649913737b-28_2642_1844_116_114}
Edexcel M3 2018 Specimen Q1
8 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-02_397_526_561_715} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl, of internal radius \(r\), is fixed with its circular rim upwards and horizontal. A particle \(P\) of mass \(m\) moves on the smooth inner surface of the bowl. The particle moves with constant angular speed in a horizontal circle. The centre of the circle is at a distance \(\frac { 1 } { 2 } r\) vertically below the centre of the bowl, as shown in Figure 1.
The time taken by \(P\) to complete one revolution of its circular path is \(T\).
Show that \(T = \pi \sqrt { \frac { 2 r } { g } }\).
Edexcel M3 2018 Specimen Q2
9 marks Standard +0.3
2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.
  1. Show that \(K = m g R ^ { 2 }\) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
  2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHM ION OOVI4V SIHI NI JIIIM I ON OO
Edexcel M3 2018 Specimen Q3
12 marks Standard +0.3
3. At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) has magnitude \(2 ( t + 4 ) ^ { - \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\).
  1. Show that, at time \(t\) seconds, the velocity of \(P\) is \(16 - 4 ( t + 4 ) ^ { \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 1 }\)
  2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHM ION OOVI4V SIHI NI JIIIM I ON OO
Edexcel M3 2018 Specimen Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-12_403_497_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass \(3 m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A\) is horizontal and \(O A = a\). The particle is projected vertically downwards from \(A\) with speed \(u\), as shown in Figure 2. The particle moves in complete vertical circles.
  1. Show that \(u ^ { 2 } \geqslant 3 a g\). Given that the greatest tension in the string is three times the least tension in the string, (b) show that \(u ^ { 2 } = 6 a g\).
    VIIIV SIHI NI JIIYM ION OCVIIVV SIHI NI JIIIAM ION OOVEYV SIHIL NI JIIIM ION OO
Edexcel M3 2018 Specimen Q5
17 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-16_193_931_269_520} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two fixed points \(A\) and \(B\) are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\). A second light elastic string, of natural length 1.2 m and modulus of elasticity 15 N , has one end attached to \(P\) and the other end attached to \(B\). Initially \(P\) rests in equilibrium at the point \(O\), as shown in Figure 3.
  1. Show that \(A O = 3 \mathrm {~m}\). The particle is now pulled towards \(A\) and released from rest at the point \(C\), where \(A C B\) is a straight line and \(O C = 1 \mathrm {~m}\).
  2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion.
  3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. The particle first comes to instantaneous rest at the point \(D\).
  4. Find the distance \(D B\).
Edexcel M3 2018 Specimen Q6
17 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_442_723_237_605} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,
  1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
  2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_483_469_1402_767} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
    The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
  3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
  4. Find the size of the angle between \(V A\) and the vertical.
    Leave
    blank
    Q6
    VIIIV SIHI NI JAIIM ION OCVIIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM IONOO
OCR MEI C3 2006 June Q1
3 marks Moderate -0.8
1 Solve the equation \(| 3 x - 2 | = x\).
OCR MEI C3 2006 June Q2
6 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } x \sin 2 x \mathrm {~d} x = \frac { 3 \sqrt { 3 } - \pi } { 24 }\).
OCR MEI C3 2006 June Q3
7 marks Moderate -0.3
3 Fig. 3 shows the curve defined by the equation \(y = \arcsin ( x - 1 )\), for \(0 \leqslant x \leqslant 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72de4365-c3f0-4106-8daf-3047afeab723-2_684_551_829_758} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find \(x\) in terms of \(y\), and show that \(\frac { \mathrm { d } x } { \mathrm {~d} y } = \cos y\).
  2. Hence find the exact gradient of the curve at the point where \(x = 1.5\).
OCR MEI C3 2006 June Q4
6 marks Moderate -0.3
4 Fig. 4 is a diagram of a garden pond. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72de4365-c3f0-4106-8daf-3047afeab723-3_296_746_351_657} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The volume \(V \mathrm {~m} ^ { 3 }\) of water in the pond when the depth is \(h\) metres is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 3 - h )$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} h }\). Water is poured into the pond at the rate of \(0.02 \mathrm {~m} ^ { 3 }\) per minute.
  2. Find the value of \(\frac { \mathrm { d } h } { \mathrm {~d} t }\) when \(h = 0.4\).
OCR MEI C3 2006 June Q5
6 marks Standard +0.3
5 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
  1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).
OCR MEI C3 2006 June Q6
8 marks Moderate -0.8
6 The mass \(M \mathrm {~kg}\) of a radioactive material is modelled by the equation $$M = M _ { 0 } \mathrm { e } ^ { - k t } ,$$ where \(M _ { 0 }\) is the initial mass, \(t\) is the time in years, and \(k\) is a constant which measures the rate of radioactive decay.
  1. Sketch the graph of \(M\) against \(t\).
  2. For Carbon \(14 , k = 0.000121\). Verify that after 5730 years the mass \(M\) has reduced to approximately half the initial mass. The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the initial mass.
  3. Show that, in general, the half-life \(T\) is given by \(T = \frac { \ln 2 } { k }\).
  4. Hence find the half-life of Plutonium 239, given that for this material \(k = 2.88 \times 10 ^ { - 5 }\).
OCR MEI C3 2007 June Q1
7 marks Moderate -0.3
1
  1. Differentiate \(\sqrt { 1 + 2 x }\).
  2. Show that the derivative of \(\ln \left( 1 - \mathrm { e } ^ { - x } \right)\) is \(\frac { 1 } { \mathrm { e } ^ { x } - 1 }\).
OCR MEI C3 2007 June Q2
3 marks Easy -1.2
2 Given that \(\mathrm { f } ( x ) = 1 - x\) and \(\mathrm { g } ( x ) = | x |\), write down the composite function \(\mathrm { gf } ( x )\).
On separate diagrams, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { gf } ( x )\).
OCR MEI C3 2007 June Q3
8 marks Standard +0.3
3 A curve has equation \(2 y ^ { 2 } + y = 9 x ^ { 2 } + 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point \(\mathrm { A } ( 1,2 )\).
  2. Find the coordinates of the points on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
OCR MEI C3 2007 June Q4
8 marks Moderate -0.3
4 A cup of water is cooling. Its initial temperature is \(100 ^ { \circ } \mathrm { C }\). After 3 minutes, its temperature is \(80 ^ { \circ } \mathrm { C }\).
  1. Given that \(T = 25 + a \mathrm { e } ^ { - k t }\), where \(T\) is the temperature in \({ } ^ { \circ } \mathrm { C } , t\) is the time in minutes and \(a\) and \(k\) are constants, find the values of \(a\) and \(k\).
  2. What is the temperature of the water
    (A) after 5 minutes,
    (B) in the long term?
OCR MEI C3 2007 June Q5
2 marks Easy -1.2
5 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.
OCR MEI C3 2007 June Q6
8 marks Moderate -0.3
6 Fig. 6 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \arctan x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-3_378_725_367_669} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Find the range of the function \(\mathrm { f } ( x )\), giving your answer in terms of \(\pi\).
  2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Find the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the origin.
  3. Hence write down the gradient of \(y = \frac { 1 } { 2 } \arctan x\) at the origin.
OCR MEI C3 2007 June Q7
16 marks Standard +0.3
7 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-3_654_1034_1505_497} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
  3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).
OCR MEI C3 2007 June Q8
20 marks Standard +0.3
8 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-4_421_965_349_550} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the exact coordinates of P .
  2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
  3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
  5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
  6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.
OCR MEI C3 2008 June Q1
4 marks Easy -1.2
1 Solve the inequality \(| 2 x - 1 | \leqslant 3\).
OCR MEI C3 2008 June Q2
4 marks Moderate -0.5
2 Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
OCR MEI C3 2008 June Q3
5 marks Moderate -0.8
3
  1. State the algebraic condition for the function \(\mathrm { f } ( x )\) to be an even function.
    What geometrical property does the graph of an even function have?
  2. State whether the following functions are odd, even or neither.
    (A) \(\mathrm { f } ( x ) = x ^ { 2 } - 3\) (B) \(\mathrm { g } ( x ) = \sin x + \cos x\) (C) \(\mathrm { h } ( x ) = \frac { 1 } { x + x ^ { 3 } }\)
OCR MEI C3 2008 June Q4
4 marks Moderate -0.3
4 Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 6\).