OCR FP2 (Further Pure Mathematics 2) 2008 June

1 It is given that \(\mathrm { f } ( x ) = \frac { 2 a x } { ( x - 2 a ) \left( x ^ { 2 } + a ^ { 2 } \right) }\), where \(a\) is a non-zero constant. Express \(\mathrm { f } ( x )\) in partial fractions.

2
\includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-2_341_1043_466_552} The diagram shows the curve \(y = \mathrm { f } ( x )\). The curve has a maximum point at ( 0,5 ) and crosses the \(x\)-axis at \(( - 2,0 ) , ( 3,0 )\) and \(( 4,0 )\). Sketch the curve \(y ^ { 2 } = \mathrm { f } ( x )\), showing clearly the coordinates of any turning points and of any points where this curve crosses the axes.

3 By using the substitution \(t = \tan \frac { 1 } { 2 } x\), find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 - \cos x } \mathrm {~d} x$$ giving the answer in terms of \(\pi\).

4

1. Sketch, on the same diagram, the curves with equations \(y = \operatorname { sech } x\) and \(y = x ^ { 2 }\).
2. By using the definition of \(\operatorname { sech } x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that the \(x\)-coordinates of the points at which these curves meet are solutions of the equation $$x ^ { 2 } = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 } .$$
3. The iteration $$x _ { n + 1 } = \sqrt { \frac { 2 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { 2 x _ { n } } + 1 } }$$ can be used to find the positive root of the equation in part (ii). With initial value \(x _ { 1 } = 1\), the approximations \(x _ { 2 } = 0.8050 , x _ { 3 } = 0.8633 , x _ { 4 } = 0.8463\) and \(x _ { 5 } = 0.8513\) are obtained, correct to 4 decimal places. State with a reason whether, in this case, the iteration produces a 'staircase' or a ‘cobweb’ diagram.

5 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$

1. By considering \(I _ { n } + I _ { n - 2 }\), or otherwise, show that, for \(n \geqslant 2\), $$( n - 1 ) \left( I _ { n } + I _ { n - 2 } \right) = 1 .$$
2. Find \(I _ { 4 }\) in terms of \(\pi\).

6 It is given that \(\mathrm { f } ( x ) = 1 - \frac { 7 } { x ^ { 2 } }\).

1. Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2.5\), to find the next approximations \(x _ { 2 }\) and \(x _ { 3 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 6 decimal places. [3]
2. The root of \(\mathrm { f } ( x ) = 0\) for which \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) are approximations is denoted by \(\alpha\). Write down the exact value of \(\alpha\).
3. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Find \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\), giving your answers correct to 5 decimal places. Verify that \(e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }\).

7 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\), for \(x > - \frac { 1 } { 2 }\).

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { 1 + 2 x }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\).
2. Show that the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\) can be written as \(\ln a + b x + c x ^ { 2 }\), for constants \(a , b\) and \(c\) to be found.

8 The equation of a curve, in polar coordinates, is $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-3_268_796_1567_717} The diagram shows the part of the curve for which \(0 \leqslant \theta \leqslant \alpha\), where \(\theta = \alpha\) is the equation of the tangent to the curve at \(O\). Find \(\alpha\) in terms of \(\pi\).
2. (a) If \(\mathrm { f } ( \theta ) = 1 - \sin 2 \theta\), show that \(\mathrm { f } \left( \frac { 1 } { 2 } ( 2 k + 1 ) \pi - \theta \right) = \mathrm { f } ( \theta )\) for all \(\theta\), where \(k\) is an integer.
   (b) Hence state the equations of the lines of symmetry of the curve $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
3. Sketch the curve with equation $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$ State the maximum value of \(r\) and the corresponding values of \(\theta\).

9

1. Prove that \(\int _ { 0 } ^ { N } \ln ( 1 + x ) \mathrm { d } x = ( N + 1 ) \ln ( N + 1 ) - N\), where \(N\) is a positive constant.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-4_616_1261_406_482} The diagram shows the curve \(y = \ln ( 1 + x )\), for \(0 \leqslant x \leqslant 70\), together with a set of rectangles of unit width.
   (a) By considering the areas of these rectangles, explain why $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 < \int _ { 0 } ^ { 70 } \ln ( 1 + x ) d x$$ (b) By considering the areas of another set of rectangles, show that $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 > \int _ { 0 } ^ { 69 } \ln ( 1 + x ) d x$$ (c) Hence find bounds between which \(\ln ( 70 ! )\) lies. Give the answers correct to 1 decimal place.