CAIE P3 (Pure Mathematics 3) 2009 November

1 Solve the equation $$\ln ( 5 - x ) = \ln 5 - \ln x$$ giving your answers correct to 3 significant figures.

2 The equation \(x ^ { 3 } - 8 x - 13 = 0\) has one real root.

1. Find the two consecutive integers between which this root lies.
2. Use the iterative formula $$x _ { n + 1 } = \left( 8 x _ { n } + 13 \right) ^ { \frac { 1 } { 3 } }$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the equation of the tangent to the curve at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\).

4 The angles \(\alpha\) and \(\beta\) lie in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\), and are such that $$\tan \alpha = 2 \tan \beta \quad \text { and } \quad \tan ( \alpha + \beta ) = 3 .$$ Find the possible values of \(\alpha\) and \(\beta\).

5 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). The result of differentiating \(\mathrm { p } ( x )\) with respect to \(x\) is denoted by \(\mathrm { p } ^ { \prime } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and of \(\mathrm { p } ^ { \prime } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

6

1. Use the substitution \(x = 2 \tan \theta\) to show that $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$$
2. Hence find the exact value of $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$$

7 The complex numbers \(- 2 + \mathrm { i }\) and \(3 + \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.

1. Find, in the form \(x + \mathrm { i } y\), the complex numbers
   (a) \(u + v\),
   (b) \(\frac { u } { v }\), showing all your working.
2. State the argument of \(\frac { u } { v }\). In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u + v\) respectively.
3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
4. State fully the geometrical relationship between the line segments \(O A\) and \(B C\).

8

1. Express \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

9 The temperature of a quantity of liquid at time \(t\) is \(\theta\). The liquid is cooling in an atmosphere whose temperature is constant and equal to \(A\). The rate of decrease of \(\theta\) is proportional to the temperature difference \(( \theta - A )\). Thus \(\theta\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - A )$$ where \(k\) is a positive constant.

1. Find, in any form, the solution of this differential equation, given that \(\theta = 4 A\) when \(t = 0\).
2. Given also that \(\theta = 3 A\) when \(t = 1\), show that \(k = \ln \frac { 3 } { 2 }\).
3. Find \(\theta\) in terms of \(A\) when \(t = 2\), expressing your answer in its simplest form.

10 The plane \(p\) has equation \(2 x - 3 y + 6 z = 16\). The plane \(q\) is parallel to \(p\) and contains the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\).

1. Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the perpendicular distance between \(p\) and \(q\).
3. The line \(l\) is parallel to the plane \(p\) and also parallel to the plane with equation \(x - 2 y + 2 z = 5\). Given that \(l\) passes through the origin, find a vector equation for \(l\).