Edexcel C3 (Core Mathematics 3)

Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]

The function f is given by \(f: x \mapsto 2 + \frac{3}{x + 2}\), \(x \in \mathbb{R}\), \(x \neq -2\).

1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
2. Find an expression for \(f^{-1}(x)\). [3]
3. Write down the domain of \(f^{-1}\). [1]

1. Express as a fraction in its simplest form $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21}.$$ [3]
2. Hence solve $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21} = 1.$$ [3]

1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
2. Find the value of \(x\) for which \(\log_2(x^2 + 4x + 3) - \log_2(x^2 + x) = 4\). [4]

1. Prove, by counter-example, that the statement "\(\sec(A + B) \equiv \sec A + \sec B\), for all \(A\) and \(B\)" is false [2]
2. Prove that $$\tan \theta + \cot \theta = 2\cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]

1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} \equiv \tan \theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [3]
2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]

Given that \(y = \log_a x\), \(x > 0\), where \(a\) is a positive constant,

1. 1. express \(x\) in terms of \(a\) and \(y\), [1]
   2. deduce that \(\ln x = y \ln a\). [1]
2. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x \ln a}\). [2]

The curve \(C\) has equation \(y = \log_{10} x\), \(x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10. Using the result in part (b),

1. find an equation for the tangent to \(C\) at \(A\). [4]

The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).

1. Find the exact \(x\)-coordinate of \(B\). [2]

The curve with equation \(y = \ln 3x\) crosses the \(x\)-axis at the point \(P(p, 0)\).

1. Sketch the graph of \(y = \ln 3x\), showing the exact value of \(p\). [2]

The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.

1. Show that \(x = q\) is a solution of the equation \(x^2 + \ln 3x = 0\). [4]
2. Show that the equation in part (b) can be rearranged in the form \(x = \frac{1}{3}e^{-x^2}\). [2]
3. Use the iteration formula \(x_{n + 1} = \frac{1}{3}e^{-x_n^2}\), with \(x_0 = \frac{1}{3}\), to find \(x_1, x_2, x_3\) and \(x_4\). Hence write down, to 3 decimal places, an approximation for \(q\). [3]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = \text{f}(x)\), \(x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = \text{f}^{-1}(x)\), [2]
2. \(y = 3\text{f}(2x)\). [3]

Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\text{f}: x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

1. state
   1. the value of \(c\),
   2. the range of f.
   [3]
2. Find the value of \(d\), giving your answer to 3 decimal places. [3]

The function g is defined by $$\text{g}: x \mapsto \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find fg(x), giving your answer in its simplest form. [3]