Edexcel C3 (Core Mathematics 3)

1. (a) Find the exact value of \(x\) such that

$$3 \arctan ( x - 2 ) + \pi = 0$$ (b) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).

2. (a) Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
(b) Simplify $$\frac { x ^ { 3 } - 8 } { 3 x ^ { 2 } - 8 x + 4 } .$$

1. Differentiate each of the following with respect to \(x\) and simplify your answers.
   1. \(\cot x ^ { 2 }\)
   2. \(x ^ { 2 } \mathrm { e } ^ { - x }\)
   3. \(\frac { \sin x } { 3 + 2 \cos x }\)
   4. (a) Find, as natural logarithms, the solutions of the equation
   $$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$
2. Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.

5. The function f is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R } .$$

1. State the range of f.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R } .$$ Find, in terms of e,
3. the value of \(\mathrm { gf } ( \ln 2 )\),
4. the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4 .$$

6. $$f ( x ) = 2 x ^ { 2 } + 3 \ln ( 2 - x ) , \quad x \in \mathbb { R } , \quad x < 2 .$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = 2 - \mathrm { e } ^ { k x ^ { 2 } } ,$$ where \(k\) is a constant to be found. The root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) is 1.9 correct to 1 decimal place.
2. Use the iteration formula $$x _ { n + 1 } = 2 - \mathrm { e } ^ { k x _ { n } ^ { 2 } } ,$$ with \(x _ { 0 } = 1.9\) and your value of \(k\), to find \(\alpha\) to 3 decimal places and justify the accuracy of your answer.
3. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\).

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).

1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
   1. \(y = \mathrm { f } ( | x | )\),
   2. \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ } ,$$ where \(R > 0\) and \(0 < \alpha < 90\),
3. state the value of \(A\),
4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.