OCR FP2 (Further Pure Mathematics 2) 2006 June

1 Find the first three non-zero terms of the Maclaurin series for $$( 1 + x ) \sin x$$ simplifying the coefficients.

2

1. Given that \(y = \tan ^ { - 1 } x\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
2. Verify that \(y = \tan ^ { - 1 } x\) satisfies the equation $$\left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0$$

3 The equation of a curve is \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. State the equation of the asymptote of the curve.
2. Show that \(- \frac { 1 } { 6 } \leqslant y \leqslant \frac { 1 } { 2 }\).

4

1. Using the definition of \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), prove that $$\cosh 2 x = 2 \cosh ^ { 2 } x - 1$$
2. Hence solve the equation $$\cosh 2 x - 7 \cosh x = 3$$ giving your answer in logarithmic form.

5

1. Express \(t ^ { 2 } + t + 1\) in the form \(( t + a ) ^ { 2 } + b\).
2. By using the substitution \(\tan \frac { 1 } { 2 } x = t\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$

6 \includegraphics[max width=\textwidth, alt={}, center]{52b43f20-e0e6-4ddd-9518-bea9782982bf-3_623_1354_262_392} The diagram shows the curve with equation \(y = 3 ^ { x }\) for \(0 \leqslant x \leqslant 1\). The area \(A\) under the curve between these limits is divided into \(n\) strips, each of width \(h\) where \(n h = 1\).

1. By using the set of rectangles indicated on the diagram, show that \(A > \frac { 2 h } { 3 ^ { h } - 1 }\).
2. By considering another set of rectangles, show that \(A < \frac { ( 2 h ) 3 ^ { h } } { 3 ^ { h } - 1 }\).
3. Given that \(h = 0.001\), use these inequalities to find values between which \(A\) lies.

7 The equation of a curve, in polar coordinates, is $$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$

1. Find the equation of the tangent at the pole.
2. State the greatest value of \(r\) and the corresponding value of \(\theta\).
3. Sketch the curve.
4. Find the exact area of the region enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 4 } \pi\).

8 The curve with equation \(y = \frac { \sinh x } { x ^ { 2 } }\), for \(x > 0\), has one turning point.

1. Show that the \(x\)-coordinate of the turning point satisfies the equation \(x - 2 \tanh x = 0\).
2. Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next two approximations, \(x _ { 2 }\) and \(x _ { 3 }\), to the positive root of \(x - 2 \tanh x = 0\).
3. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\). (You are not expected to evaluate \(x _ { 4 }\).)

9

1. Given that \(y = \sinh ^ { - 1 } x\), prove that \(y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).
2. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \alpha } \sinh ^ { n } \theta \mathrm {~d} \theta$$ where \(\alpha = \sinh ^ { - 1 } 1\). Show that $$n I _ { n } = \sqrt { 2 } - ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2 .$$
3. Evaluate \(I _ { 4 }\), giving your answer in terms of \(\sqrt { 2 }\) and logarithms.