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 Further Pure Core 2019 June Q14
13 marks Challenging +1.2
14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.
  1. In the case where the planes do not intersect at a unique point,
    1. find \(b\) in terms of \(a\),
    2. find the value of \(c\) for which the planes form a sheaf.
  2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).
OCR MEI Further Pure Core 2019 June Q16
12 marks Challenging +1.2
16
  1. Show that \(\left( 2 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 2 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by \(C = \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \cos n \theta\), \(S = \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \sin n \theta\).
  2. Show that \(C = \frac { 2 ^ { n } ( 2 \cos \theta - 1 ) - 2 \cos ( n + 1 ) \theta + \cos n \theta } { 2 ^ { n } ( 5 - 4 \cos \theta ) }\).
OCR MEI Further Pure Core 2019 June Q17
22 marks Challenging +1.2
17 A cyclist accelerates from rest for 5 seconds then brakes for 5 seconds, coming to rest at the end of the 10 seconds. The total mass of the cycle and rider is \(m \mathrm {~kg}\), and at time \(t\) seconds, for \(0 \leqslant t \leqslant 10\), the cyclist's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance to motion, modelled by a force of magnitude 0.1 mvN , acts on the cyclist during the whole 10 seconds.
  1. Explain why modelling the resistance to motion in this way is likely to be more realistic than assuming this force is constant. During the braking phase of the motion, for \(5 \leqslant t \leqslant 10\), the brakes apply an additional constant resistance force of magnitude \(2 m \mathrm {~N}\) and the cyclist does not provide any driving force.
  2. Show that, for \(5 \leqslant t \leqslant 10 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = - 2\).
    1. Solve the differential equation in part (b).
    2. Hence find the velocity of the cyclist when \(t = 5\). During the acceleration phase ( \(0 \leqslant t \leqslant 5\) ), the cyclist applies a driving force of magnitude directly proportional to \(t\).
  4. Show that, for \(0 \leqslant t \leqslant 5 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = \lambda t\), where \(\lambda\) is a positive constant.
    1. Show by integration that, for \(0 \leqslant t \leqslant 5 , v = 10 \lambda \left( t - 10 + 10 \mathrm { e } ^ { - 0.1 t } \right)\).
    2. Hence find \(\lambda\).
  6. Find the total distance, to the nearest metre, travelled by the cyclist during the motion.
OCR MEI Further Pure Core 2022 June Q1
7 marks Standard +0.3
1
  1. By considering \(( r + 1 ) ^ { 3 } - r ^ { 3 }\), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 3 \mathrm { r } ^ { 2 } + 3 \mathrm { r } + 1 \right)\).
  2. Use this result to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\), expressing your answer in fully factorised form.
OCR MEI Further Pure Core 2022 June Q2
5 marks Challenging +1.2
2 In this question you must show detailed reasoning. Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x\)
OCR MEI Further Pure Core 2022 June Q3
6 marks Standard +0.3
3 In this question you must show detailed reasoning.
Solve the equation \(3 \cosh x = 2 \sinh ^ { 2 } x\), giving your solutions in exact logarithmic form.
OCR MEI Further Pure Core 2022 June Q4
7 marks Standard +0.8
4
  1. A transformation with associated matrix \(\left( \begin{array} { r r r } m & 2 & 1 \\ 0 & 1 & - 2 \\ 2 & 0 & 3 \end{array} \right)\), where \(m\) is a constant, maps the vertices of a cube to points that all lie in a plane. Find \(m\).
  2. The transformations S and T of the plane have associated matrices \(\mathbf { M }\) and \(\mathbf { N }\) respectively, where \(\mathbf { M } = \left( \begin{array} { r r } k & 1 \\ - 3 & 4 \end{array} \right)\) and the determinant of \(\mathbf { N }\) is \(3 k + 1\). The transformation \(U\) is equivalent to the combined transformation consisting of S followed by T . Given that U preserves orientation and has an area scale factor 2, find the possible values of \(k\).
OCR MEI Further Pure Core 2022 June Q5
7 marks Standard +0.3
5
  1. Sketch the polar curve \(\mathrm { r } = \mathrm { a } ( 1 - \cos \theta ) , 0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.
  2. Determine the exact area of the region enclosed by the curve.
OCR MEI Further Pure Core 2022 June Q6
5 marks Standard +0.3
6 Prove by mathematical induction that \(\left( \begin{array} { r l } 2 & 0 \\ - 1 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 1 - 2 ^ { n } & 1 \end{array} \right)\) for all positive integers \(n\). Answer all the questions.
Section B (107 marks)
OCR MEI Further Pure Core 2022 June Q7
9 marks Challenging +1.2
7 In this question you must show detailed reasoning.
Show that \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) } d x = \frac { 1 } { 2 } \ln 2\).
OCR MEI Further Pure Core 2022 June Q8
11 marks Challenging +1.2
8 Two sets of complex numbers are given by \(\left\{ z : \arg ( z - 10 ) = \frac { 3 } { 4 } \pi \right\}\) and \(\{ z : | z - 3 - 6 i | = k \}\), where \(k\) is a positive constant. In an Argand diagram, one of the points of intersection of the two loci representing these sets lies on the imaginary axis.
  1. Sketch the loci on an Argand diagram.
  2. In this question you must show detailed reasoning. Find the complex numbers represented by the points of intersection.
OCR MEI Further Pure Core 2022 June Q9
12 marks Challenging +1.2
9 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \ln ( 1 + \sinh x )\).
  1. Given that \(k\) lies in the domain of this function, explain why \(k\) must be greater than \(\ln ( \sqrt { 2 } - 1 )\).
    1. Find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Show that \(\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = \frac { \mathrm { a } \sinh \mathrm { x } + \mathrm { b } } { ( 1 + \sinh \mathrm { x } ) ^ { 2 } }\), where \(a\) and \(b\) are integers to be determined.
  3. Hence find a quadratic approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
  4. Find the percentage error in this approximation when \(x = 0.1\).
OCR MEI Further Pure Core 2022 June Q10
10 marks Challenging +1.2
10 The equation \(4 x ^ { 4 } + 16 x ^ { 3 } + a x ^ { 2 } + b x + 6 = 0\),
where \(a\) and \(b\) are real, has roots \(\alpha , \frac { 2 } { \alpha } , \beta\) and \(3 \beta\).
  1. Given that \(\beta < 0\), determine all 4 roots.
  2. Determine the values of \(a\) and \(b\).
OCR MEI Further Pure Core 2022 June Q11
8 marks Standard +0.3
11 An Argand diagram with the point A representing a complex number \(z _ { 1 }\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-4_716_778_932_239} The complex numbers \(z _ { 2 }\) and \(z _ { 3 }\) are \(z _ { 1 } \mathrm { e } ^ { \frac { 2 } { 3 } \mathrm { i } \pi }\) and \(z _ { 1 } \mathrm { e } ^ { \frac { 4 } { 3 } \mathrm { i } \pi }\) respectively.
    1. On the copy of the Argand diagram in the Printed Answer Booklet, mark the points B and C representing the complex numbers \(z _ { 2 }\) and \(z _ { 3 }\).
    2. Show that \(z _ { 1 } + z _ { 2 } + z _ { 3 } = 0\).
  2. Given now that \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are roots of the equation \(z ^ { 3 } = 8 \mathrm { i }\), find these three roots, giving your answers in the form \(\mathrm { a } + \mathrm { ib }\), where \(a\) and \(b\) are real and exact.
OCR MEI Further Pure Core 2022 June Q12
9 marks Standard +0.8
12 Solve the differential equation \(\left( 4 - x ^ { 2 } \right) \frac { d y } { d x } - x y = 1\), given that \(y = 1\) when \(x = 0\), giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR MEI Further Pure Core 2022 June Q13
17 marks Standard +0.8
13 The points A and B have coordinates \(( 4,0 , - 1 )\) and \(( 10,4 , - 3 )\) respectively. The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(x - 2 y = 5\) and \(2 x + 3 y - z = - 4\) respectively.
  1. Find the acute angle between the line AB and the plane \(\Pi _ { 1 }\).
  2. Show that the line AB meets \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) at the same point, whose coordinates should be specified.
    1. Find \(( \mathbf { i } - 2 \mathbf { j } ) \times ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )\).
    2. Hence find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
    3. Find the shortest distance between the point A and the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
OCR MEI Further Pure Core 2022 June Q14
8 marks Challenging +1.2
14
  1. Find \(\left( 3 - \mathrm { e } ^ { 2 \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - 2 \mathrm { i } \theta } \right)\) in terms of \(\cos 2 \theta\).
  2. Hence show that the sum of the infinite series \(\sin \theta + \frac { 1 } { 3 } \sin 3 \theta + \frac { 1 } { 9 } \sin 5 \theta + \frac { 1 } { 27 } \sin 7 \theta + \ldots\) can be expressed as \(\frac { 6 \sin \theta } { 5 - 3 \cos 2 \theta }\).
OCR MEI Further Pure Core 2022 June Q15
23 marks Challenging +1.2
15 In an oscillating system, a particle of mass \(m \mathrm {~kg}\) moves in a horizontal line. Its displacement from its equilibrium position O at time \(t\) seconds is \(x\) metres, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), and it is acted on by a force \(2 m x\) newtons acting towards O as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-6_212_914_408_242} Initially, the particle is projected away from O with speed \(1 \mathrm {~ms} ^ { - 1 }\) from a point 2 m from O in the positive direction.
    1. Show that the motion is modelled by the differential equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 x = 0\).
    2. State the type of motion.
    3. Write down the period of the motion.
    4. Find \(x\) in terms of \(t\).
    5. Find the amplitude of the motion.
  2. The motion is now damped by a force \(2 m v\) newtons.
    1. Show that \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 \frac { d x } { d t } + 2 x = 0\).
    2. State, giving a reason, whether the system is under-damped, critically damped or over-damped.
    3. Determine the general solution of this differential equation.
  3. Finally, a variable force \(2 m \cos 2 t\) newtons is added, so that the motion is now modelled by the differential equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } + 2 \frac { d x } { d t } + 2 x = 2 \cos 2 t\).
    1. Find \(x\) in terms of \(t\). In the long term, the particle is seen to perform simple harmonic motion with a period of just over 3 seconds.
    2. Verify that this behaviour is consistent with the answer to part (c)(i).
OCR MEI Further Pure Core 2023 June Q1
7 marks Moderate -0.8
1
  1. The complex number \(\mathrm { a } + \mathrm { ib }\) is denoted by \(z\).
    1. Write down \(z ^ { * }\).
    2. Find \(\operatorname { Re } ( \mathrm { iz } )\).
  2. The complex number \(w\) is given by \(w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }\).
    1. In this question you must show detailed reasoning. Express \(w\) in the form \(\mathrm { x } + \mathrm { iy }\).
    2. Convert \(w\) to modulus-argument form.
OCR MEI Further Pure Core 2023 June Q2
5 marks Moderate -0.3
2 In this question you must show detailed reasoning.
Find the angle between the vector \(3 i + 2 j + \mathbf { k }\) and the plane \(- x + 3 y + 2 z = 8\).
OCR MEI Further Pure Core 2023 June Q3
6 marks Standard +0.8
3
  1. Using partial fractions and the method of differences, show that $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots + \frac { 1 } { \mathrm { n } ( \mathrm { n } + 2 ) } = \frac { 3 } { 4 } - \frac { \mathrm { an } + \mathrm { b } } { 2 ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) }$$ where \(a\) and \(b\) are integers to be determined.
  2. Deduce the sum to infinity of the series. $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$
OCR MEI Further Pure Core 2023 June Q4
6 marks Standard +0.3
4
    1. Given that \(\mathrm { f } ( x ) = \sqrt { 1 + 2 x }\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Hence, find the first three terms of the Maclaurin series for \(\sqrt { 1 + 2 x }\).
  2. Hence, using a suitable value for \(x\), show that \(\sqrt { 5 } \approx \frac { 143 } { 64 }\).
OCR MEI Further Pure Core 2023 June Q5
7 marks Moderate -0.3
5
  1. In this question you must show detailed reasoning.
    Determine the sixth roots of - 64 , expressed in \(r \mathrm { e } ^ { \mathrm { i } \theta }\) form.
  2. Represent the roots on an Argand diagram.
OCR MEI Further Pure Core 2023 June Q6
9 marks Standard +0.3
6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) respectively.
  1. In this question you must show detailed reasoning. Determine whether \(\mathbf { M }\) and \(\mathbf { N }\) commute under matrix multiplication.
  2. Specify the transformation of the plane associated with each of the following matrices.
    1. M
    2. N
  3. State the significance of the result in part (a) for the transformations associated with \(\mathbf { M }\) and \(\mathbf { N }\). [1]
  4. Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with N.
OCR MEI Further Pure Core 2023 June Q7
6 marks Standard +0.8
7 The diagram below shows the curve with polar equation \(r = a ( 1 - 2 \sin \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{76631941-3cd5-4b3e-a7e4-27b8f991975a-4_634_865_486_239} The curve crosses the initial line at A , and the points B and C are the lowest points on the two loops.
  1. Find the values of \(r\) and \(\theta\) at the points A , B and C .
  2. Find the set of values of \(\theta\) for the points on the inner loop (shown in the diagram with a broken line).