Edexcel C3 (Core Mathematics 3) 2017 June

1. Express \(\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }\) as a single fraction in its simplest form.
   

2. Find the exact solutions, in their simplest form, to the equations

1. \(\mathrm { e } ^ { 3 x - 9 } = 8\)
2. \(\ln ( 2 y + 5 ) = 2 + \ln ( 4 - y )\)
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3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph of \(y = \mathrm { g } ( x )\), where $$g ( x ) = 3 + \sqrt { x + 2 } , \quad x \geqslant - 2$$

1. State the range of g .
2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
3. Find the exact value of \(x\) for which $$\mathrm { g } ( x ) = x$$
4. Hence state the value of \(a\) for which $$\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )$$

1. (a) Write \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 \leqslant \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
(b) Show that the equation $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ can be rewritten in the form $$5 \cos 2 x - 2 \sin 2 x = c$$ where \(c\) is a positive constant to be determined.
(c) Hence or otherwise, solve, for \(0 \leqslant x < \pi\), $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ giving your answers to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 2 \ln ( 2 x + 5 ) - \frac { 3 x } { 2 } , \quad x > - 2.5$$ The point \(P\) with \(x\) coordinate - 2 lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The normal to \(C\) at \(P\) cuts the curve again at the point \(Q\), as shown in Figure 2
2. Show that the \(x\) coordinate of \(Q\) is a solution of the equation $$x = \frac { 20 } { 11 } \ln ( 2 x + 5 ) - 2$$ The iteration formula $$x _ { n + 1 } = \frac { 20 } { 11 } \ln \left( 2 x _ { n } + 5 \right) - 2$$ can be used to find an approximation for the \(x\) coordinate of \(Q\).
3. Taking \(x _ { 1 } = 2\), find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving each answer to 4 decimal places.

1. Given that \(a\) and \(b\) are positive constants,
   1. on separate diagrams, sketch the graph with equation
      1. \(y = | 2 x - a |\)
      2. \(y = | 2 x - a | + b\)
   Show, on each sketch, the coordinates of each point at which the graph crosses or meets the axes. Given that the equation $$| 2 x - a | + b = \frac { 3 } { 2 } x + 8$$ has a solution at \(x = 0\) and a solution at \(x = c\),
2. find \(c\) in terms of \(a\).

1. 1. Given \(y = 2 x \left( x ^ { 2 } - 1 \right) ^ { 5 }\), show that
      1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { g } ( x ) \left( x ^ { 2 } - 1 \right) ^ { 4 }\) where \(\mathrm { g } ( x )\) is a function to be determined.
   2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } \geqslant 0\)
      (ii) Given
   $$x = \ln ( \sec 2 y ) , \quad 0 < y < \frac { \pi } { 4 }$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) in its simplest form.

8. \begin{figure}[h] \end{figure} The number of rabbits on an island is modelled by the equation $$P = \frac { 100 \mathrm { e } ^ { - 0.1 t } } { 1 + 3 \mathrm { e } ^ { - 0.9 t } } + 40 , \quad t \in \mathbb { R } , t \geqslant 0$$ where \(P\) is the number of rabbits, \(t\) years after they were introduced onto the island.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 3.

1. Calculate the number of rabbits that were introduced onto the island.
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) The number of rabbits initially increases, reaching a maximum value \(P _ { T }\) when \(t = T\)
3. Using your answer from part (b), calculate
   1. the value of \(T\) to 2 decimal places,
   2. the value of \(P _ { T }\) to the nearest integer.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) For \(t > T\), the number of rabbits decreases, as shown in Figure 3, but never falls below \(k\), where \(k\) is a positive constant.
4. Use the model to state the maximum value of \(k\).

1. (a) Prove that

$$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Given that \(x \neq 90 ^ { \circ }\) and \(x \neq 270 ^ { \circ }\), solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\sin 2 x - \tan x = 3 \tan x \sin x$$ Give your answers in degrees to one decimal place where appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)