CAIE P3 (Pure Mathematics 3) 2010 November

1 Expand \(( 1 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2 The parametric equations of a curve are $$x = \frac { t } { 2 t + 3 } , \quad y = \mathrm { e } ^ { - 2 t }$$ Find the gradient of the curve at the point for which \(t = 0\).

3 The complex number \(w\) is defined by \(w = 2 + \mathrm { i }\).

1. Showing your working, express \(w ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real. Find the modulus of \(w ^ { 2 }\).
2. Shade on an Argand diagram the region whose points represent the complex numbers \(z\) which satisfy $$\left| z - w ^ { 2 } \right| \leqslant \left| w ^ { 2 } \right|$$

4 It is given that \(\mathrm { f } ( x ) = 4 \cos ^ { 2 } 3 x\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 9 } \pi \right)\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).

5 Show that \(\int _ { 0 } ^ { 7 } \frac { 2 x + 7 } { ( 2 x + 1 ) ( x + 2 ) } \mathrm { d } x = \ln 50\).

6 The straight line \(l\) passes through the points with coordinates \(( - 5,3,6 )\) and \(( 5,8,1 )\). The plane \(p\) has equation \(2 x - y + 4 z = 9\).

1. Find the coordinates of the point of intersection of \(l\) and \(p\).
2. Find the acute angle between \(l\) and \(p\).

7

1. Given that \(\int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = \frac { 2 } { 5 }\), show that \(a = \frac { 5 } { 3 } ( 1 + \ln a )\).
2. Use an iteration formula based on the equation \(a = \frac { 5 } { 3 } ( 1 + \ln a )\) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.

8

1. Express \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta\) 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, in each of the following cases, find the smallest positive angle \(\theta\) which satisfies the equation
   (a) \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta = - 4\),
   (b) \(( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3\).

9 A biologist is investigating the spread of a weed in a particular region. At time \(t\) weeks after the start of the investigation, the area covered by the weed is \(A \mathrm {~m} ^ { 2 }\). The biologist claims that the rate of increase of \(A\) is proportional to \(\sqrt { } ( 2 A - 5 )\).

1. Write down a differential equation representing the biologist's claim.
2. At the start of the investigation, the area covered by the weed was \(7 \mathrm {~m} ^ { 2 }\) and, 10 weeks later, the area covered was \(27 \mathrm {~m} ^ { 2 }\). Assuming that the biologist's claim is correct, find the area covered 20 weeks after the start of the investigation.

10 The polynomial \(\mathrm { p } ( z )\) is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where \(m\) is a constant. It is given that \(( z + 2 )\) is a factor of \(\mathrm { p } ( z )\).

1. Find the value of \(m\).
2. Hence, showing all your working, find
   (a) the three roots of the equation \(\mathrm { p } ( z ) = 0\),
   (b) the six roots of the equation \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).