OCR FP2 (Further Pure Mathematics 2) 2012 January

1 Given that \(\mathrm { f } ( x ) = \ln ( \cos 3 x )\), find \(\mathrm { f } ^ { \prime } ( 0 )\) and \(\mathrm { f } ^ { \prime \prime } ( 0 )\). Hence show that the first term in the Maclaurin series for \(\mathrm { f } ( x )\) is \(a x ^ { 2 }\), where the value of \(a\) is to be found.

2 By first completing the square in the denominator, find the exact value of $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { 4 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x$$

3 Express \(\frac { 2 x ^ { 3 } + x + 12 } { ( 2 x - 1 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

4
\includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-2_662_1063_986_484} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { x } }\) for \(0 < x \leqslant 1\). A set of ( \(n - 1\) ) rectangles is drawn under the curve as shown.

1. Explain why a lower bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\) can be expressed as $$\frac { 1 } { n } \left( \mathrm { e } ^ { - n } + \mathrm { e } ^ { - \frac { n } { 2 } } + \mathrm { e } ^ { - \frac { n } { 3 } } + \ldots + \mathrm { e } ^ { - \frac { n } { n - 1 } } \right)$$
2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { - \frac { 1 } { x } } \mathrm {~d} x\).
3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures.
4. When \(n \geqslant N\), the difference between the upper and lower bounds is less than 0.001 . By expressing this difference in terms of \(n\), find the least possible value of \(N\).

5 It is given that \(\mathrm { f } ( x ) = x ^ { 3 } - k\), where \(k > 0\), and that \(\alpha\) is the real root of the equation \(\mathrm { f } ( x ) = 0\). Successive approximations to \(\alpha\), using the Newton-Raphson method, are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots\).

1. Show that \(x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + k } { 3 x _ { n } ^ { 2 } }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for \(\left| \alpha - x _ { 2 } \right|\) to be greater than \(\left| \alpha - x _ { 1 } \right|\). It is now given that \(k = 100\) and \(x _ { 1 } = 5\).
3. Write down the exact value of \(\alpha\) and find \(x _ { 2 }\) and \(x _ { 3 }\) correct to 5 decimal places.
4. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). By finding \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\), verify that \(e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }\).

6

1. Prove that the derivative of \(\cos ^ { - 1 } x\) is \(- \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\). A curve has equation \(y = \cos ^ { - 1 } \left( 1 - x ^ { 2 } \right)\), for \(0 < x < \sqrt { 2 }\).
2. Find and simplify \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and hence show that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = x \frac { \mathrm {~d} y } { \mathrm {~d} x }$$
3. Given that \(y = \sinh ^ { - 1 } x\), prove that \(y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).
4. It is given that \(x\) satisfies the equation \(\sinh ^ { - 1 } x - \cosh ^ { - 1 } x = \ln 2\). Use the logarithmic forms for \(\sinh ^ { - 1 } x\) and \(\cosh ^ { - 1 } x\) to show that $$\sqrt { x ^ { 2 } + 1 } - 2 \sqrt { x ^ { 2 } - 1 } = x$$ Hence, by squaring this equation, find the exact value of \(x\).

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\includegraphics[max width=\textwidth, alt={}, center]{c342b622-a560-46da-9e64-edc4b7b3be93-4_606_915_219_557} The diagram shows two curves, \(C _ { 1 }\) and \(C _ { 2 }\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C _ { 1 }\) is \(r = \sqrt { 2 } \cos \theta\) and the polar equation of \(C _ { 2 }\) is \(r = \sqrt { 2 \sin 2 \theta }\). For both curves, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The value of \(\theta\) at \(P\) is \(\alpha\).

1. Show that \(\tan \alpha = \frac { 1 } { 2 }\).
2. Show that the area of the region common to \(C _ { 1 }\) and \(C _ { 2 }\), shaded in the diagram, is \(\frac { 1 } { 4 } \pi - \frac { 1 } { 2 } \alpha\).

9

1. Show that \(\tanh ( \ln n ) = \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 1 }\). It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { n } u \mathrm {~d} u\).
2. Show that \(I _ { n } - I _ { n - 2 } = - \frac { 1 } { n - 1 } \left( \frac { 3 } { 5 } \right) ^ { n - 1 }\), for \(n \geqslant 2\).
3. Find the value of \(I _ { 3 }\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants.
4. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac { 1 } { 2 } \left( \frac { 3 } { 5 } \right) ^ { 2 } + \frac { 1 } { 4 } \left( \frac { 3 } { 5 } \right) ^ { 4 } + \frac { 1 } { 6 } \left( \frac { 3 } { 5 } \right) ^ { 6 } + \ldots .$$