CAIE P3 (Pure Mathematics 3) 2011 November

1 Expand \(\frac { 16 } { ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 1 + \mathrm { e } ^ { 2 x } }\). Show that the gradient of the curve at the point for which \(x = \ln 3\) is \(\frac { 9 } { 50 }\).

3

1. Express \(8 \cos \theta + 15 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(8 \cos \theta + 15 \sin \theta = 12\), giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

4 During an experiment, the number of organisms present at time \(t\) days is denoted by \(N\), where \(N\) is treated as a continuous variable. It is given that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = 1.2 \mathrm { e } ^ { - 0.02 t } N ^ { 0.5 }$$ When \(t = 0\), the number of organisms present is 100 .

1. Find an expression for \(N\) in terms of \(t\).
2. State what happens to the number of organisms present after a long time.

5 It is given that \(\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.

6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).

1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - x ^ { 2 } + 4 x - a$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
2. When \(a\) has the value found in part (i), express \(\frac { 8 x - 13 } { \mathrm { p } ( x ) }\) in partial fractions.

8 \includegraphics[max width=\textwidth, alt={}, center]{6025cf1d-525e-4f12-9517-f20ef5fff2fa-3_698_1006_758_571} The diagram shows the curve with parametric equations $$x = \sin t + \cos t , \quad y = \sin ^ { 3 } t + \cos ^ { 3 } t$$ for \(\frac { 1 } { 4 } \pi < t < \frac { 5 } { 4 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 3 \sin t \cos t\).
2. Find the gradient of the curve at the origin.
3. Find the values of \(t\) for which the gradient of the curve is 1 , giving your answers correct to 2 significant figures.

9 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } a \\ 1 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)\), where \(a\) is a constant. The plane \(p\) has equation \(2 x - 2 y + z = 10\).

1. Given that \(l\) does not lie in \(p\), show that \(l\) is parallel to \(p\).
2. Find the value of \(a\) for which \(l\) lies in \(p\).
3. It is now given that the distance between \(l\) and \(p\) is 6 . Find the possible values of \(a\).

10

1. Use the substitution \(u = \tan x\) to show that, for \(n \neq - 1\), $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
2. Hence find the exact value of
   1. \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x\),
   2. \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).