Edexcel C3 (Core Mathematics 3)

1. Find the exact value of \(x\) such that $$3 \arctan (x - 2) + \pi = 0.$$ [3]
2. Solve, for \(-\pi < \theta < \pi\), the equation $$\cos 2\theta - \sin \theta - 1 = 0,$$ giving your answers in terms of \(\pi\). [5]

1. Express $$\frac{4x}{x^2 - 9} - \frac{2}{x + 3}$$ as a single fraction in its simplest form. [4]
2. Simplify $$\frac{x^3 - 8}{3x^2 - 8x + 4}.$$ [5]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(x^2 e^{-x}\) [3]
3. \(\frac{\sin x}{3 + 2\cos x}\) [4]

1. Find, as natural logarithms, the solutions of the equation $$e^{2x} - 8e^x + 15 = 0.$$ [4]
2. Use proof by contradiction to prove that \(\log_5 3\) is irrational. [6]

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

1. State the range of f. [1]
2. Find an expression for \(f^{-1}(x)\) and state its domain. [4]

The function g is defined by $$g : x \to 5x - 2, \quad x \in \mathbb{R}.$$ Find, in terms of e,

1. the value of gf(ln 2), [3]
2. the solution of the equation $$f^{-1}g(x) = 4.$$ [4]

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

1. Show that the equation \(f(x) = 0\) can be written in the form $$x = 2 - e^{kx^2},$$ where \(k\) is a constant to be found. [3]

The root, \(\alpha\), of the equation \(f(x) = 0\) is \(1.9\) correct to \(1\) decimal place.

1. Use the iteration formula $$x_{n+1} = 2 - e^{kx_n^2},$$ with \(x_0 = 1.9\) and your value of \(k\), to find \(\alpha\) to \(3\) decimal places and justify the accuracy of your answer. [5]
2. Solve the equation \(f'(x) = 0\). [5]

\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) 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. \(y = f(|x|)\),
   2. \(y = 1 + 2f(x)\). [6]

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,

1. show that f(x) can be expressed in the form $$f(x) = A + R \cos (x + \alpha)^{\circ},$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
2. state the value of \(A\), [1]
3. find, to \(1\) decimal place, the \(x\)-coordinates of the points where the curve \(y = f(x)\) crosses the \(x\)-axis. [4]