OCR Further Pure Core 1 (Further Pure Core 1) 2018 September

1 In this question you must show detailed reasoning.
For the complex number \(z\) it is given that \(| z | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
Find the following in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact numbers.

1. \(z\)
2. \(z ^ { 2 }\)
3. \(\frac { z } { z ^ { * } }\)

2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 1 | = 5\) and \(\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) respectively.

1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate by shading on your Argand diagram the following set of points. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$

3 A sequence is defined by \(a _ { 1 } = 6\) and \(a _ { n + 1 } = 5 a _ { n } - 2\) for \(n \geqslant 1\).
Prove by induction that for all integers \(n \geqslant 1 , a _ { n } = \frac { 11 \times 5 ^ { n - 1 } + 1 } { 2 }\).

4 In this question you must show detailed reasoning.
Find the exact value of each of the following.

1. \(\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x + 10 } \mathrm {~d} x\)
2. The mean value of \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) in the interval \([ 0,0.5 ]\)

5 Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), have equations \(3 x + 2 y + z = 4\) and \(2 x + y + z = 3\) respectively.

1. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(L\) has equation \(x = 1 - y = 2 - z\).
2. Show that \(L\) lies in both planes.

6

1. Find as a single algebraic fraction an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
2. Determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).

7 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 } \mathrm {~d} x\), expressing your answer in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers.

8

1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\sinh 2 x = 2 \sinh x \cosh x\). You are given the function \(\mathrm { f } ( x ) = a \cosh x - \cosh 2 x\), where \(a\) is a positive constant.
2. Verify that, for any value of \(a\), the curve \(y = \mathrm { f } ( x )\) has a stationary point on the \(y\)-axis.
3. Find the coordinates of the stationary point found in part (ii).
4. Determine the maximum value of \(a\) for which the stationary point found in part (ii) is the only stationary point on the curve \(y = \mathrm { f } ( x )\). You are given that for any value of \(a\) greater than the value found in part (iv) there are three stationary points, the one found in part (ii) and two others, one of which satisfies \(x > 0\).
5. Find the coordinates of this point when \(a = 6\). Give your answer in the form \(\left( \cosh ^ { - 1 } p , q \right)\).

9 The diagram below shows the curve \(r = 4 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{c03cae53-eb00-496b-948f-ccff676bc03c-3_311_775_1713_644}

1. On the diagram in your Printed Answer Booklet, shade the region \(R\) for which $$r \leqslant 4 \sin 3 \theta \text { and } 0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi .$$ In this question you must show detailed reasoning.
2. Find the exact area of the region \(R\).

10

1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
2. Find the range of \(x\) for which the expansion is valid.
3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$

11 A particular radioactive substance decays over time.
A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t .$$

1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). Initially there was 10 g of the substance.
2. Find the particular solution of the differential equation.
3. Find to 6 significant figures the amount of substance that would be predicted by the model at
   (a) 6 hours,
   (b) 6.25 hours.
4. Comment on the appropriateness of the model for predicting the amount of substance over time. \section*{END OF QUESTION PAPER}