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
CAIE P3 2015 June Q8
10 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [5]
CAIE P3 2016 June Q9
9 marks Challenging +1.2
  1. Sketch this diagram and state fully the geometrical relationship between \(O B\) and \(A C\).
  2. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex number \(\frac { u } { v }\).
  3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
CAIE P3 2017 June Q5
8 marks Moderate -0.3
  1. Show that \(x\) satisfies the equation \(x = \frac { 1 } { 3 } ( \pi + \sin x )\).
  2. Verify by calculation that \(x\) lies between 1 and 1.5.
  3. Use an iterative formula based on the equation in part (i) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2017 March Q8
10 marks Standard +0.3
  1. Showing all your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( z ) = 0\).
  2. Find the other three roots of the equation \(\mathrm { p } ( z ) = 0\).
CAIE P3 2013 November Q7
10 marks Standard +0.3
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
CAIE P3 2016 November Q9
10 marks Standard +0.3
  1. Solve the equation \(( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 1 - \mathrm { i } | \leqslant 2\) and \(- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
CAIE P3 2016 November Q7
9 marks Standard +0.3
  1. Find the modulus and argument of \(z\).
  2. Express each of the following in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact:
    1. \(z + 2 z ^ { * }\);
    2. \(\frac { z ^ { * } } { \mathrm { i } z }\).
    3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z ^ { * }\) and \(\mathrm { i } z\) respectively. Prove that angle \(A O B\) is equal to \(\frac { 1 } { 6 } \pi\).
CAIE P3 2017 November Q8
10 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2019 November Q7
9 marks Standard +0.8
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the equation of the plane containing \(l\) and \(m\).
CAIE P3 2019 November Q6
7 marks Standard +0.3
  1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
  2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
  3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
CAIE FP1 2017 November Q8
11 marks Challenging +1.8
  1. Find the value of \(I _ { 2 }\).
  2. Show that, for \(n > 2\), $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
  3. The curve \(C\) has equation \(y = \sec ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2019 November Q5
9 marks Challenging +1.8
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
CAIE Further Paper 2 2020 June Q4
8 marks Challenging +1.2
  1. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x\).
CAIE Further Paper 2 2020 November Q4
8 marks Challenging +1.2
  1. By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) d x \leqslant \frac { 3 n ^ { 2 } + 2 n - 1 } { 4 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) \mathrm { dx }\).
CAIE Further Paper 2 2021 November Q4
10 marks Challenging +1.8
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
  2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }\).
CAIE FP2 2018 June Q5
12 marks Challenging +1.2
  1. Show that the moment of inertia of the object about the axis \(l\) is \(180 M a ^ { 2 }\).
  2. Show that small oscillations of the object about the axis \(l\) are approximately simple harmonic, and state the period.
CAIE FP2 2019 June Q4
11 marks Challenging +1.3
  1. Find the moment of inertia of the object, consisting of the rod and two spheres, about \(L\).
    The object is pivoted at \(A\) so that it can rotate freely about \(L\). The object is released from rest with the rod making an angle of \(60 ^ { \circ }\) to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is \(\frac { 9 } { 20 } \sqrt { } \left( \frac { g } { a } \right)\).
  2. Find the value of \(k\).
CAIE FP2 2017 November Q5
12 marks Challenging +1.3
  1. Show that the moment of inertia of the system, consisting of frame and small object, about an axis through \(O\) perpendicular to the plane of the frame, is \(\frac { 169 } { 3 } m a ^ { 2 }\).
  2. Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.
CAIE FP2 2017 Specimen Q3
11 marks Standard +0.8
  1. Find the value of \(k\).
  2. The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion.
  3. In the case where \(P\) is released from rest at a distance \(0.2 a \mathrm {~m}\) from \(M\), the speed of \(P\) is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(P\) is \(0.05 a \mathrm {~m}\) from \(M\). Find the value of \(a\).
CAIE M1 2022 June Q4
8 marks Standard +0.3
  1. In the case where \(F = 20\), find the tensions in each of the strings.
  2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.
CAIE M1 2022 June Q6
10 marks Standard +0.3
  1. It is given that the plane \(B C\) is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.
  2. It is given instead that the plane \(B C\) is rough. A force of magnitude 3 N is applied to \(Q\) directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between \(Q\) and the plane \(B C\) for which the particles remain at rest.
CAIE M1 2011 June Q4
7 marks Standard +0.3
  1. Make a rough copy of the diagram and shade the region whose area represents the displacement of \(P\) from \(X\) at the instant when \(Q\) starts. It is given that \(P\) has travelled 70 m at the instant when \(Q\) starts.
  2. Find the value of \(T\).
  3. Find the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Sketch a single diagram showing the displacement-time graphs for both \(P\) and \(Q\), with values shown on the \(t\)-axis at which the speed of either particle changes.
CAIE M1 2015 June Q6
11 marks Challenging +1.2
  1. Find the value of \(h\).
  2. Find the value of \(m\), and find also the tension in the string while \(Q\) is moving.
  3. The string is slack while \(Q\) is at rest on the ground. Find the total time from the instant that \(P\) is released until the string becomes taut again.
CAIE M1 2019 June Q4
10 marks Standard +0.8
  1. Show that, before the string breaks, the magnitude of the acceleration of each particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the difference in the times that it takes the particles to hit the ground.
CAIE M1 2011 November Q5
8 marks Standard +0.8
  1. Show that \(\mu \geqslant \frac { 6 } { 17 }\). When the applied force acts upwards as in Fig. 2 the block slides along the floor.
  2. Find another inequality for \(\mu\).