Edexcel C3 (Core Mathematics 3)

1. Express as a single fraction in its simplest form

$$\frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 9 } \times \frac { 2 x ^ { 2 } + 6 x } { ( x - 5 ) ^ { 2 } }$$

1. The root of the equation \(\mathrm { f } ( x ) = 0\), where

$$f ( x ) = x + \ln 2 x - 4$$ is to be estimated using the iterative formula \(x _ { n + 1 } = 4 - \ln 2 x _ { n }\), with \(x _ { 0 } = 2.4\).

1. Showing your values of \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\), obtain the value, to 3 decimal places, of the root.
2. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, justify the accuracy of your answer to part (a).

3. The function \(f\) is defined by $$f : x \text { a } | 2 x - a | , \quad x \in ^ { \circ }$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts the axes.
2. On a separate diagram, sketch the graph of \(y = \mathrm { f } ( 2 x )\), showing the coordinates of the points where the graph cuts the axes.
3. Given that a solution of the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\) is \(x = 4\), find the two possible values of \(a\).

4. Prove that $$\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } \equiv \cos 2 \theta$$ \section*{EDEXCEL CORE MATHEMATICS PRACTICE PAPER 1}

1. Express \(\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }\) as a single fraction in its simplest form.
2. The function f , defined for \(x \in ^ { \circ } , x > 0\), is such that

$$\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 2 + \frac { 1 } { x ^ { 2 } }$$

1. Find the value of \(\mathrm { f } ^ { \prime \prime } ( x )\) at \(x = 4\).
2. Given that \(\mathrm { f } ( 3 ) = 0\), find \(\mathrm { f } ( x )\).
3. Prove that f is an increasing function.

7. \(\quad \mathrm { f } ( x ) = \frac { 2 } { x - 1 } - \frac { 6 } { ( x - 1 ) ( 2 x + 1 ) } , x > 1\)

1. Prove that \(\mathrm { f } ( x ) = \frac { 4 } { 2 x + 1 }\).
2. Find the range of f.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
4. Find the range of \(\mathrm { f } ^ { - 1 } ( x )\).

8. The function f is given by $$f : x \text { a } \ln ( 3 x - 6 ) , \quad x \in ^ { \circ } , \quad x > 2$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Write down the domain of \(\mathrm { f } ^ { - 1 }\) and the range of \(\mathrm { f } ^ { - 1 }\).
3. Find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 3\). The function g is given by $$g : x \text { a } \ln | 3 x - 6 | , \quad x \in ^ { \circ } , \quad x \neq 2$$
4. Sketch the graph of \(y = \mathrm { g } ( x )\).
5. Find the exact coordinates of all the points at which the graph of \(y = \mathrm { g } ( x )\) meets the coordinate axes.