AQA C4 (Core Mathematics 4) 2010 January

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 15 x ^ { 3 } + 19 x ^ { 2 } - 4\).

1. 1. Find \(\mathrm { f } ( - 1 )\).
   2. Show that \(( 5 x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Simplify $$\frac { 15 x ^ { 2 } - 6 x } { f ( x ) }$$ giving your answer in a fully factorised form.

2

1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\), in radians, to three decimal places.
2. 1. Hence write down the minimum value of \(\cos x + 3 \sin x\).
   2. Find the value of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) at which this minimum occurs, giving your answer, in radians, to three decimal places.
3. Solve the equation \(\cos x + 3 \sin x = 2\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving all solutions, in radians, to three decimal places.

3

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence find the binomial expansion of \(\left( 1 + \frac { 3 } { 4 } x \right) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
2. Hence show that \(\sqrt [ 3 ] { \frac { 256 } { 4 + 3 x } } \approx a + b x + c x ^ { 2 }\) for small values of \(x\), stating the values of the constants \(a , b\) and \(c\).

4 The expression \(\frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) }\) can be written in the form \(2 + \frac { A } { x + 1 } + \frac { B } { 5 x - 1 }\), where \(A\) and \(B\) are constants.

1. Find the values of \(A\) and \(B\).
2. Hence find \(\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x\).

5 A curve is defined by the equation $$x ^ { 2 } + x y = \mathrm { e } ^ { y }$$ Find the gradient at the point \(( - 1,0 )\) on this curve.

6

1. 1. Express \(\sin 2 \theta\) and \(\cos 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
   2. Given that \(0 < \theta < \frac { \pi } { 2 }\) and \(\cos \theta = \frac { 3 } { 5 }\), show that \(\sin 2 \theta = \frac { 24 } { 25 }\) and find the value of \(\cos 2 \theta\).
2. A curve has parametric equations $$x = 3 \sin 2 \theta , \quad y = 4 \cos 2 \theta$$
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
   2. At the point \(P\) on the curve, \(\cos \theta = \frac { 3 } { 5 }\) and \(0 < \theta < \frac { \pi } { 2 }\). Find an equation of the tangent to the curve at the point \(P\).

7 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { y } \cos \left( \frac { x } { 3 } \right)\), given that \(y = 1\) when \(x = \frac { \pi } { 2 }\).
Write your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

8 The points \(A , B\) and \(C\) have coordinates \(( 2 , - 1 , - 5 ) , ( 0,5 , - 9 )\) and \(( 9,2,3 )\) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2
- 1
- 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
- 3
2 \end{array} \right]\).

1. Verify that the point \(B\) lies on the line \(l\).
2. Find the vector \(\overrightarrow { B C }\).
3. The point \(D\) is such that \(\overrightarrow { A D } = 2 \overrightarrow { B C }\).
   1. Show that \(D\) has coordinates \(( 20 , - 7,19 )\).
   2. The point \(P\) lies on \(l\) where \(\lambda = p\). The line \(P D\) is perpendicular to \(l\). Find the value of \(p\).

9 A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow. She proposes the model \(h = A \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\), where \(A\) is a constant, for the height \(h\) millimetres of the toadstool, \(t\) hours after it begins to grow.

1. Use this model to:
   1. find the height of the toadstool when \(t = 0\);
   2. show that \(A = 60\), correct to two significant figures.
2. Use the model \(h = 60 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\) to:
   1. show that the time \(T\) hours for the toadstool to grow to a height of 48 millimetres is given by $$T = a \ln b$$ where \(a\) and \(b\) are integers;
   2. show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 15 - \frac { h } { 4 }\);
   3. find the height of the toadstool when it is growing at a rate of 13 millimetres per hour.
      (1 mark)