OCR MEI Further Pure Core (Further Pure Core) 2021 November

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.

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.

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.

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.

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.

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.

7 Prove that \(\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }\) for all \(n \geqslant 1\).

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 \(q\).

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.
2. 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.

10

1. Show on an Argand diagram the points representing the three cube roots of unity.
2. 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 \(\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }\).

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 \(\sqrt { 2 }\).

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] \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.]

13 Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = 2 e ^ { x }\).

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.
2. 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.

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 \(l\).

16

1. Show using exponentials that \(\cosh 2 u = 1 + 2 \sinh ^ { 2 } u\).
2. Show that \(\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x = 2 \sqrt { 2 } - 2 \ln ( 1 + \sqrt { 2 } )\).

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. 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.