Edexcel C3 (Core Mathematics 3)

Express as a single fraction in its simplest form $$\frac{x^2 - 8x + 15}{x^2 - 9} \times \frac{2x^2 + 6x}{(x - 5)^2}$$ [4]

The root of the equation f(x) = 0, where $$f(x) = x + \ln 2x - 4$$ is to be estimated using the iterative formula \(x_{n+1} = 4 - \ln 2x_n\), 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. [4]
2. By considering the change of sign of f(x) in a suitable interval, justify the accuracy of your answer to part (a). [2]

The function f is defined by $$f: x \mapsto |2x - a|, \quad x \in \mathbb{R}$$ where \(a\) is a positive constant.

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

Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta$$ [6]

Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]

The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}$$

1. Find the value of f''(x) at \(x = 4\). [3]
2. Given that f(3) = 0, find f(x). [4]
3. Prove that f is an increasing function. [3]

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

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

The function f is given by $$f: x \mapsto \ln(3x - 6), \quad x \in \mathbb{R}, \quad x > 2$$

1. Find \(f^{-1}(x)\). [3]
2. Write down the domain of \(f^{-1}\) and the range of \(f^{-1}\). [2]
3. Find, to 3 significant figures, the value of \(x\) for which f(x) = 3. [2]

The function g is given by $$g: x \mapsto \ln|3x - 6|, \quad x \in \mathbb{R}, \quad x \neq 2$$

1. Sketch the graph of \(y = g(x)\). [3]
2. Find the exact coordinates of all the points at which the graph of \(y = g(x)\) meets the coordinate axes. [3]