OCR Further Pure Core 2 (Further Pure Core 2) Specimen

1 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 )\). Give your answer in a fully factorised form.

2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]

1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.

4 It is given that \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + k x } \equiv \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + k }\) where \(k , A , B\) and \(C\) are positive integers.
Determine the set of possible values of \(k\).

5 In this question you must show detailed reasoning.
Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]

6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).

1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3
   2
   - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5
   3
   2 \end{array} \right)\) and \(\Pi\).
2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1
   4
   2 \end{array} \right) + \mu \left( \begin{array} { r } 3
   - 1
   5 \end{array} \right)\) and \(\Pi\).

1. Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2 x \sin 4 x\) up to and including the term in \(x ^ { 3 }\).
2. Hence find, in exact surd form, an approximation to the least positive root of the equation \(2 \sin x \sin 2 x \sin 4 x = x\).

8 The equation of a curve is \(y = \cosh ^ { 2 } x - 3 \sinh x\). Show that \(\left( \ln \left( \frac { 3 + \sqrt { 13 } } { 2 } \right) , - \frac { 5 } { 4 } \right)\) is the only stationary point on the curve.

9 A curve has equation \(x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }\), where \(x\) and \(y\) are not both zero.

1. Show that the equation of the curve in polar coordinates is \(r ^ { 2 } = \frac { 2 } { 2 - \sin ^ { 2 } 2 \theta }\).
2. Deduce that no point on the curve \(x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }\) is further than \(\sqrt { 2 }\) from the origin.

10 Let \(C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos r \theta\). Show that \(C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos 10 \theta\).

11 During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 0.3 x - 0.2 y \text { and } \frac { \mathrm { d } z } { \mathrm {~d} t } = 0.2 y + 0.1 x$$ where \(x , y\) and \(z\) are the amounts in kg of \(X , Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substances \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.

1. Show that \(x = A \mathrm { e } ^ { - 0.4 t }\), stating the value of \(A\).
2. (a) Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } z } { \mathrm {~d} t } = 0\).
   (b) Comment on this result in the context of the industrial process.
3. Express \(y\) in terms of \(t\).
4. Determine the maximum amount of substance \(Y\) present during the process.
5. How long does it take to produce 9 kg of substance \(Z\) ? \section*{END OF QUESTION PAPER} {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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