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OCR MEI Further Pure Core 2020 November Q7
6 marks Standard +0.3
7 Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
OCR MEI Further Pure Core 2020 November Q8
9 marks Standard +0.3
8
  1. Given that the lines \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 2 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 1 \\ 3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ k \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)\) meet, determine \(k\).
  2. In this question you must show detailed reasoning. Find the acute angle between the two lines.
OCR MEI Further Pure Core 2020 November Q9
8 marks Standard +0.8
9 A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2 \\ \lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant. constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.
OCR MEI Further Pure Core 2020 November Q12
8 marks Challenging +1.2
12
  1. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
  2. By considering \(\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 }\), find constants \(A\) and \(B\) such that \(\sin ^ { 3 } \theta \cos ^ { 3 } \theta = A \sin 6 \theta + B \sin 2 \theta\).
OCR MEI Further Pure Core 2020 November Q13
9 marks Challenging +1.3
13
  1. Using exponentials, prove that \(\sinh 2 x = 2 \cosh x \sinh x\).
  2. Hence show that if \(\mathrm { f } ( x ) = \sinh ^ { 2 } x\), then \(\mathrm { f } ^ { \prime \prime } ( x ) = 2 \cosh 2 x\).
  3. Explain why the coefficients of odd powers in the Maclaurin series for \(\sinh ^ { 2 } x\) are all zero.
  4. Find the coefficient of \(x ^ { n }\) in this series when \(n\) is a positive even number.
OCR MEI Further Pure Core 2020 November Q14
11 marks Challenging +1.3
14 Solve the simultaneous differential equations \(\frac { \mathrm { d } x } { \mathrm {~d} t } + 2 x = 4 y , \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 x = 5 y\),
given that when \(t = 0 , x = 0\) and \(y = 1\).
OCR MEI Further Pure Core 2020 November Q15
17 marks Standard +0.3
15
  1. Show that the three planes with equations $$\begin{aligned} x + \lambda y + 3 z & = - 12 \\ 2 x + y + 5 z & = - 11 \\ x - 2 y + 2 z & = - 9 \end{aligned}$$ where \(\lambda\) is a constant, meet at a unique point except for one value of \(\lambda\) which is to be determined.
  2. In the case \(\lambda = - 2\), use matrices to find the point of intersection P of the planes, showing your method clearly. The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z + 2 } { - 2 }\).
  3. Find a vector equation of \(l\).
  4. Find the shortest distance between the point P and \(l\).
    1. Show that \(l\) is parallel to the plane \(x - 2 y + 2 z = - 9\).
    2. Find the distance between \(l\) and the plane \(x - 2 y + 2 z = - 9\).
OCR MEI Further Pure Core 2020 November Q16
25 marks Challenging +1.2
16 The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is \(A\).
  1. One simple model is to assume that the rate of change of population density is directly proportional to \(A - P\).
    1. Formulate a differential equation for this model.
    2. Verify that \(P = A \left( 1 - \mathrm { e } ^ { - k t } \right)\), where \(k\) is a positive constant, satisfies
      • this differential equation,
  2. the initial condition,
  3. the long-term condition.
    4. An alternative model uses the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } - \frac { P } { t \left( 1 + t ^ { 2 } \right) } = \mathrm { Q } ( t )$$ where \(\mathrm { Q } ( t )\) is a function of \(t\).
  4. Find the integrating factor for this differential equation, showing that it can be written in the $$\text { form } \frac { \sqrt { 1 + t ^ { 2 } } } { t } \text {. }$$
  5. Suppose that \(\mathrm { Q } ( t ) = 0\). $$\text { (i) Show that } P = \frac { A t } { \sqrt { 1 + t ^ { 2 } } } \text {. }$$ (ii) Find the time predicted by this model for the population density to reach half its longterm value. Give your answer correct to the nearest minute.
  6. Now suppose that \(\mathrm { Q } ( t ) = \frac { t \mathrm { e } ^ { - t } } { \sqrt { 1 + t ^ { 2 } } }\). $$\text { Show that } \left. P = \frac { A t - t e ^ { - t } } { \sqrt { 1 + t ^ { 2 } } } \text {. [You may assume that } \lim _ { t \rightarrow \infty } t e ^ { - t } = 0 . \right]$$ It is found that the long-term value of \(P\) is 10, and \(P\) reaches half this value after 37 minutes.
  7. Determine which of the models proposed in parts (c) and (d) is more consistent with these data.
OCR MEI Further Pure Core 2021 November Q1
7 marks Standard +0.3
1
  1. Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
  2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.
OCR MEI Further Pure Core 2021 November Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning.
Find the gradient of the curve \(y = 6 \arcsin ( 2 x )\) at the point with \(x\)-coordinate \(\frac { 1 } { 4 }\). Express the result in the form \(\mathrm { m } \sqrt { \mathrm { n } }\), where \(m\) and \(n\) are integers.
OCR MEI Further Pure Core 2021 November Q3
6 marks
3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = - 2 + 2 i\) and \(z _ { 2 } = 2 \left( \cos \frac { 1 } { 6 } \pi + i \sin \frac { 1 } { 6 } \pi \right)\).
  1. Find the modulus and argument of \(z _ { 1 }\).
  2. Hence express \(\frac { z _ { 1 } } { z _ { 2 } }\) in exact modulus-argument form.
OCR MEI Further Pure Core 2021 November Q4
4 marks Standard +0.3
4 In this question you must show detailed reasoning.
Determine the mean value of \(\frac { 1 } { 1 + 4 x ^ { 2 } }\) between \(x = - 1\) and \(x = 1\). Give your answer to 3 significant
figures. figures.
OCR MEI Further Pure Core 2021 November Q5
6 marks Standard +0.8
5
  1. Use a Maclaurin series to find a quadratic approximation for \(\ln ( 1 + 2 x )\).
  2. Find the percentage error in using the approximation in part (a) to calculate \(\ln ( 1.2 )\).
  3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid.
OCR MEI Further Pure Core 2021 November Q6
4 marks Standard +0.3
6 Given that \(y = m x\) is an invariant line of the transformation with matrix \(\left( \begin{array} { r r } 1 & 2 \\ 2 & - 2 \end{array} \right)\), determine the possible values of \(m\). Section B (113 marks)
Answer all the questions.
OCR MEI Further Pure Core 2021 November Q7
6 marks Challenging +1.2
7 Prove that \(\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }\) for all \(n \geqslant 1\).
OCR MEI Further Pure Core 2021 November Q8
9 marks Challenging +1.2
8 The equation \(4 \mathrm { x } ^ { 4 } - 4 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , - \alpha , \beta\) and \(\frac { 1 } { \beta }\).
  1. Determine the exact roots of the equation.
  2. Determine the values of \(p\) and \(q\).
OCR MEI Further Pure Core 2021 November Q9
11 marks Standard +0.3
9 The transformation Too the plane has associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } - 1 & 0 \\ - 2 & 1 \end{array} \right)\).
  1. On the grid in the Printed Answer Booklet, plot the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) of the unit square OABC under the transformation T.
    1. Calculate the value of \(\operatorname { det } \mathbf { M }\).
    2. Explain the significance of the value of \(\operatorname { det } \mathbf { M }\) in relation to the image \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\).
  3. T is equivalent to a sequence of two transformations of the plane.
    1. Specify fully two transformations equivalent to T .
    2. Use matrices to verify your answer.
OCR MEI Further Pure Core 2021 November Q10
13 marks
10
  1. Show on an Argand diagram the points representing the three cube roots of unity.
    1. Find the exact roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\), expressing them in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta < \pi\).
    2. The points representing the cube roots of unity form a triangle \(\Delta _ { 1 }\). The points representing the roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\) form a triangle \(\Delta _ { 2 }\). State a sequence of two transformations that maps \(\Delta _ { 1 }\) onto \(\Delta _ { 2 }\).
    3. The three roots in part (b)(i) are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\). By simplifying \(z _ { 1 } + z _ { 2 } + z _ { 3 }\), verify that the sum of these roots is zero.
    4. Hence show that \(\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }\).
OCR MEI Further Pure Core 2021 November Q11
9 marks Standard +0.8
11
  1. Given that \(\mathbf { u } = \lambda \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(\mathbf { v } = \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\), find the following, giving your answers in terms of \(\lambda\).
    1. u.v
    2. \(\mathbf { u } \times \mathbf { v }\)
  2. Hence determine
    1. the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x + 2 y - 2 z = 10\),
    2. the shortest distance between the lines \(\frac { x - 3 } { 3 } = \frac { y } { 1 } = \frac { z - 2 } { - 3 }\) and \(\frac { x } { 1 } = \frac { y - 4 } { 2 } = \frac { z + 2 } { - 2 }\), giving your answer as a multiple of \(\sqrt { 2 }\).
OCR MEI Further Pure Core 2021 November Q12
4 marks Challenging +1.2
12 Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers \(z\) and \(w\) respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82808722-0abc-411a-9aa3-c0f368a4c95e-4_641_659_1201_242} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Prove that \(\arg ( z + w ) = \frac { 1 } { 2 } ( \arg z + \arg w )\).
[0pt] [A copy of Fig. 12 is provided in the Printed Answer Booklet.]
OCR MEI Further Pure Core 2021 November Q13
7 marks Standard +0.3
13 Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = 2 e ^ { x }\).
OCR MEI Further Pure Core 2021 November Q14
14 marks Standard +0.8
14 A curve has polar equation \(\mathrm { r } = \mathrm { a } ( \cos \theta + 2 \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \pi\).
  1. Determine the polar coordinates of the point on the curve which is furthest from the pole.
    1. Show that the curve is a circle whose radius should be specified.
    2. Write down the polar coordinates of the centre of the circle.
OCR MEI Further Pure Core 2021 November Q15
6 marks Standard +0.8
15 The equations of three planes are $$\begin{aligned} - 4 x + k y + 7 z & = 4 \\ x - 2 y + 5 z & = 1 \\ 2 x + 3 y + z & = 2 \end{aligned}$$ Given that the planes form a sheaf, determine the values of \(k\) and \(l\).
OCR MEI Further Pure Core 2021 November Q16
14 marks Challenging +1.2
16
  1. Show using exponentials that \(\cosh 2 u = 1 + 2 \sinh ^ { 2 } u\).
  2. Show that \(\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x = 2 \sqrt { 2 } - 2 \ln ( 1 + \sqrt { 2 } )\).
OCR MEI Further Pure Core 2021 November Q17
20 marks Challenging +1.2
17 In a chemical process, a vessel contains 1 litre of pure water. A liquid chemical is then passed into the top of the vessel at a constant rate of \(a\) litres per minute and thoroughly mixed with the water. At the same time, the resulting mixture is drawn from the bottom of the vessel at a constant rate of \(b\) litres per minute. You may assume that the chemical mixes instantly and uniformly with the water. After \(t\) minutes, the mixture in the vessel contains \(x\) litres of the chemical.
    1. Show that the proportion of chemical present in the vessel after \(t\) minutes is $$\frac { x } { 1 + ( a - b ) t } .$$
    2. Hence show that \(\frac { d x } { d t } + \frac { b x } { 1 + ( a - b ) t } = a\).
  2. First, consider the case where \(\mathbf { b } = \mathbf { a }\).
    1. Solve the differential equation to find \(x\) in terms of \(a\) and \(t\).
    2. Given that after 1 minute the vessel contains equal amounts of water and chemical, find the rate of inflow of chemical.
  3. Now consider the case where \(\mathrm { b } = 2 \mathrm { a }\).
    1. Explain why the differential equation in part (a)(ii) is now invalid for \(\mathrm { t } \geqslant \frac { 1 } { \mathrm { a } }\).
    2. Find the maximum amount of chemical in the vessel.