Edexcel C3 (Core Mathematics 3) 2008 June

1. The point \(P\) lies on the curve with equation

$$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The \(y\)-coordinate of \(P\) is 8 .

1. Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\).
2. Find the equation of the tangent to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants to be found.

2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\),

1. find the value of \(R\) and the value of \(\alpha\) to 3 decimal places.
2. Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for \(0 \leqslant x < 2 \pi\).
3. 1. Write down the maximum value of \(5 \cos x + 12 \sin x\).
   2. Find the smallest positive value of \(x\) for which this maximum value occurs.

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = f ( x ) , x \in \mathbb { R }\).
The graph consists of two line segments that meet at the point \(P\).
The graph cuts the \(y\)-axis at the point \(Q\) and the \(x\)-axis at the points \(( - 3,0 )\) and \(R\). Sketch, on separate diagrams, the graphs of

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( - x )\). Given that \(\mathrm { f } ( x ) = 2 - | x + 1 |\),
3. find the coordinates of the points \(P , Q\) and \(R\),
4. solve \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\).

4. The function \(f\) is defined by $$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3\).
2. Find the range of f.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\). State the domain of this inverse function. The function \(g\) is defined by $$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R }$$
4. Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 8 }\).

5. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
(b) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$ giving your answers to 1 decimal place.

6. (a) Differentiate with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
   (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
   (c) Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).

7. $$f ( x ) = 3 x ^ { 3 } - 2 x - 6$$

1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between \(x = 1.4\) and \(x = 1.45\)
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { } \left( \frac { 2 } { x } + \frac { 2 } { 3 } \right) , \quad x \neq 0$$
3. Starting with \(x _ { 0 } = 1.43\), use the iteration $$x _ { \mathrm { n } + 1 } = \sqrt { } \left( \frac { 2 } { x _ { \mathrm { n } } } + \frac { 2 } { 3 } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
4. By choosing a suitable interval, show that \(\alpha = 1.435\) is correct to 3 decimal places.