AQA C4 (Core Mathematics 4) 2008 June

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(3 x + 1\).
2. 1. Show that f \(\left( - \frac { 2 } { 3 } \right) = 0\).
   2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
   3. Simplify $$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$

2 A curve is defined, for \(t \neq 0\), by the parametric equations $$x = 4 t + 3 , \quad y = \frac { 1 } { 2 t } - 1$$ At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).

1. Find the gradient of the curve at the point \(P\).
2. Find an equation of the normal to the curve at the point \(P\).
3. Find a cartesian equation of the curve.

3

1. By writing \(\sin 3 x\) as \(\sin ( x + 2 x )\), show that \(\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x\) for all values of \(x\).
2. Hence, or otherwise, find \(\int \sin ^ { 3 } x \mathrm {~d} x\).

4

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence show that \(( 81 - 16 x ) ^ { \frac { 1 } { 4 } } \approx 3 - \frac { 4 } { 27 } x - \frac { 8 } { 729 } x ^ { 2 }\) for small values of \(x\).
2. Use the result from part (a)(ii) to find an approximation for \(\sqrt [ 4 ] { 80 }\), giving your answer to seven decimal places.

5

1. The angle \(\alpha\) is acute and \(\sin \alpha = \frac { 4 } { 5 }\).
   1. Find the value of \(\cos \alpha\).
   2. Express \(\cos ( \alpha - \beta )\) in terms of \(\sin \beta\) and \(\cos \beta\).
   3. Given also that the angle \(\beta\) is acute and \(\cos \beta = \frac { 5 } { 13 }\), find the exact value of \(\cos ( \alpha - \beta )\).
2. 1. Given that \(\tan 2 x = 1\), show that \(\tan ^ { 2 } x + 2 \tan x - 1 = 0\).
   2. Hence, given that \(\tan 45 ^ { \circ } = 1\), show that \(\tan 22 \frac { 1 } { 2 } ^ { \circ } = \sqrt { 2 } - 1\).

6

1. Express \(\frac { 2 } { x ^ { 2 } - 1 }\) in the form \(\frac { A } { x - 1 } + \frac { B } { x + 1 }\).
2. Hence find \(\int \frac { 2 } { x ^ { 2 } - 1 } \mathrm {~d} x\).
3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y } { 3 \left( x ^ { 2 } - 1 \right) }\), given that \(y = 1\) when \(x = 3\). Show that the solution can be written as \(y ^ { 3 } = \frac { 2 ( x - 1 ) } { x + 1 }\).

7 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2,1\) ) and ( \(5,3,0\) ) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 5
3
0 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
0
- 3 \end{array} \right]\).

1. Find the distance between \(A\) and \(B\).
2. Find the acute angle between the lines \(A B\) and \(l\). Give your answer to the nearest degree.
3. The points \(B\) and \(C\) lie on \(l\) such that the distance \(A C\) is equal to the distance \(A B\). Find the coordinates of \(C\).

8

1. The number of fish in a lake is decreasing. After \(t\) years, there are \(x\) fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
   1. Formulate a differential equation, in the variables \(x\) and \(t\) and a constant of proportionality \(k\), where \(k > 0\), to model the rate at which the number of fish in the lake is decreasing.
   2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of \(k\).
2. The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, \(P\), in another lake, where \(t\) is the time in years and \(A\) is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
   1. Taking 1 January 2008 as \(t = 0\), find the value of \(A\).
   2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.