CAIE P3 (Pure Mathematics 3) 2011 June

1 Expand \(\sqrt [ 3 ] { } ( 1 - 6 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.

2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \ln ( 1 + \sin 2 x )\),
2. \(y = \frac { \tan x } { x }\).

3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).

1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between \(p\) and the \(y\)-axis.

4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$

1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.

5 The curve with equation $$6 \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { y } + \mathrm { e } ^ { 2 y } = c$$ where \(k\) and \(c\) are constants, passes through the point \(P\) with coordinates \(( \ln 3 , \ln 2 )\).

1. Show that \(58 + 2 k = c\).
2. Given also that the gradient of the curve at \(P\) is - 6 , find the values of \(k\) and \(c\).

6
\includegraphics[max width=\textwidth, alt={}, center]{cc85b13a-7f15-4025-a545-373cda454de8-3_456_495_255_824} The diagram shows a circle with centre \(O\) and radius 10 cm . The chord \(A B\) divides the circle into two regions whose areas are in the ratio \(1 : 4\) and it is required to find the length of \(A B\). The angle \(A O B\) is \(\theta\) radians.

1. Show that \(\theta = \frac { 2 } { 5 } \pi + \sin \theta\).
2. Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1 , to find \(\theta\) correct to 2 decimal places. Hence find the length of \(A B\) in centimetres correct to 1 decimal place.

7 The integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } 4 t ^ { 3 } \ln \left( t ^ { 2 } + 1 \right) \mathrm { d } t\).

1. Use the substitution \(x = t ^ { 2 } + 1\) to show that \(I = \int _ { 1 } ^ { 5 } ( 2 x - 2 ) \ln x \mathrm {~d} x\).
2. Hence find the exact value of \(I\).

8 The complex number \(u\) is defined by \(u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }\).

1. Showing all your working, find the modulus of \(u\) and show that the argument of \(u\) is \(- \frac { 1 } { 2 } \pi\).
2. For complex numbers \(z\) satisfying \(\arg ( z - u ) = \frac { 1 } { 4 } \pi\), find the least possible value of \(| z |\).
3. For complex numbers \(z\) satisfying \(| z - ( 1 + \mathrm { i } ) u | = 1\), find the greatest possible value of \(| z |\).

9

1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
2. Hence
   (a) solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
   (b) find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).

10 The number of birds of a certain species in a forested region is recorded over several years. At time \(t\) years, the number of birds is \(N\), where \(N\) is treated as a continuous variable. The variation in the number of birds is modelled by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 1800 - N ) } { 3600 }$$ It is given that \(N = 300\) when \(t = 0\).

1. Find an expression for \(N\) in terms of \(t\).
2. According to the model, how many birds will there be after a long time?