OCR Further Pure Core 1 (Further Pure Core 1) Specimen

1 Show that \(\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }\).

2 In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).

1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).

3 In this question you must show detailed reasoning. The diagram below shows the curve \(r = 2 \cos 4 \theta\) for \(- k \pi \leq \theta \leq k \pi\) where \(k\) is a constant to be determined. Calculate the exact area enclosed by the curve.

4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).

5

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }\).
2. Find \(\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x\).

6 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
Find the values of \(p , q\) and \(r\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 3 } { 1 } = \frac { y - 5 } { 2 } = \frac { z + 2 } { - 3 }\) and \(\frac { x - 4 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z - 7 } { 4 }\).

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

8

1. Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3
   2 x + 4 y + 5 z = & 9
   7 x + 11 y + 12 z = & 20 \end{array}$$
2. Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r l } x + & y + & z = & 3
   2 x + & 4 y + & 5 z = & 9
   7 x + & 11 y + & p z = & k \end{array}$$

9 Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

1. Justify each of the following statements.
   (a) \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   (b) \(A B C D E\) is a regular pentagon.
   (c) \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   (d) \(b ^ { * } = e\)
   (e) \(a + b + c + d + e = 0\)
2. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).

11 A company is required to weigh any goods before exporting them overseas. When a crate is placed on a set of weighing scales, the mass displayed takes time to settle down to its final value. The company wishes to model the mass, \(m \mathrm {~kg}\), which is displayed \(t\) seconds after a crate X is placed on the scales.
For the displayed mass it is assumed that the rate of change of the quantity \(\left( 0.5 \frac { \mathrm {~d} m } { \mathrm {~d} t } + m \right)\) with respect to time is proportional to \(( 80 - m )\).

1. Show that \(\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 2 \mathrm {~km} = 160 \mathrm { k }\), where \(k\) is a real constant. It is given that the complementary function for the differential equation in part (i) is \(\mathrm { e } ^ { \lambda t } ( A \cos 2 t + B \sin 2 t )\), where \(A\) and \(B\) are arbitrary constants.
2. Show that \(k = \frac { 5 } { 2 }\) and state the value of the constant \(\lambda\). When X is initially placed on the scales the displayed mass is zero and the rate of increase of the displayed mass is \(160 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\).
3. Find \(m\) in terms of \(t\).
4. Describe the long term behaviour of \(m\).
5. With reference to your answer to part (iv), comment on a limitation of the model.
6. (a) Find the value of \(m\) that corresponds to the stationary point on the curve \(m = \mathrm { f } ( t )\) with the smallest positive value of \(t\).
   (b) Interpret this value of \(m\) in the context of the model.
7. Adapt the differential equation \(\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 5 m = 400\) to model the mass displayed \(t\) seconds after a crate Y , of mass 100 kg , is placed on the scales. \section*{END OF QUESTION PAPER} \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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