Edexcel C3 (Core Mathematics 3)

1. Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec ^ { 2 } x\).
2. The function f is given by \(\mathrm { f } : x \propto 2 + \frac { 3 } { x + 2 } , x \in \mathbb { R } , x \neq - 2\).
   1. Express \(2 + \frac { 3 } { x + 2 }\) as a single fraction.
   2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   3. Write down the domain of \(\mathrm { f } ^ { - 1 }\).
   4. (a) Express as a fraction in its simplest form
   $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 }$$
3. Hence solve $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 } = 1$$

1. (a) Simplify \(\frac { x ^ { 2 } + 4 x + 3 } { x ^ { 2 } + x }\).
   (b) Find the value of \(x\) for which \(\log _ { 2 } \left( x ^ { 2 } + 4 x + 3 \right) - \log _ { 2 } \left( x ^ { 2 } + x \right) = 4\).
2. (i) Prove, by counter-example, that the statement

$$\text { " } \sec ( A + B ) \equiv \sec A + \sec B , \text { for all } A \text { and } B \text { " }$$ is false
(ii) Prove that $$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$

1. (a) Prove that

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

1. Given that \(y = \log _ { a } x , x > 0\), where \(a\) is a positive constant,
   1. (i) express \(x\) in terms of \(a\) and \(y\),
      (ii) deduce that \(\ln x = y \ln a\).
   2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln a }\).
   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),
2. find an equation for the tangent to \(C\) at \(A\). The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
3. Find the exact \(x\)-coordinate of \(B\).

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

1. Sketch the graph of \(y = \ln 3 x\), showing the exact value of \(p\). The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.
2. Show that \(x = q\) is a solution of the equation \(x ^ { 2 } + \ln 3 x = 0\).
3. Show that the equation in part (b) can be rearranged in the form \(x = \frac { 1 } { 3 } \mathrm { e } ^ { - x ^ { 2 } }\).
4. Use the iteration formula \(x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } ^ { 2 } }\), 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 \(q\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \geq 0\). The curve meets the coordinate axes at the points \(( 0 , c )\) and \(( d , 0 )\). In separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ^ { - 1 } ( x )\),
2. \(y = 3 \mathrm { f } ( 2 x )\).
   (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 $$\mathrm { f } : x \mapsto 3 \left( 2 ^ { - x } \right) - 1 , x \in \mathbb { R } , x \geq 0 ,$$
3. state
   1. the value of \(c\),
   2. the range of \(f\).
4. Find the value of \(d\), giving your answer to 3 decimal places. The function g is defined by $$\mathrm { g } : x \mapsto \log _ { 2 } x , x \in \mathbb { R } , x \geq 1 .$$
5. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.