Edexcel C3 (Core Mathematics 3) 2018 June

1. Given \(y = 2 x ( 3 x - 1 ) ^ { 5 }\),
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer as a single fully factorised expression.
   2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } \leqslant 0\)
      

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { 6 } { 2 x + 5 } + \frac { 2 } { 2 x - 5 } + \frac { 60 } { 4 x ^ { 2 } - 25 } , \quad x > 4$$

1. Show that \(\mathrm { f } ( x ) = \frac { A } { B x + C }\) where \(A , B\) and \(C\) are constants to be found.
2. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

1. The value of a car is modelled by the formula

$$V = 16000 \mathrm { e } ^ { - k t } + A , \quad t \geqslant 0 , t \in \mathbb { R }$$ where \(V\) is the value of the car in pounds, \(t\) is the age of the car in years, and \(k\) and \(A\) are positive constants. Given that the value of the car is \(\pounds 17500\) when new and \(\pounds 13500\) two years later,

1. find the value of \(A\),
2. show that \(k = \ln \left( \frac { 2 } { \sqrt { 3 } } \right)\)
3. Find the age of the car, in years, when the value of the car is \(\pounds 6000\) Give your answer to 2 decimal places.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { - 2 x } + x ^ { 2 } - 3$$ The curve \(C\) crosses the \(y\)-axis at the point \(A\). The line \(l\) is the normal to \(C\) at the point \(A\).

1. Find the equation of \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The line \(l\) meets \(C\) again at the point \(B\), as shown in Figure 1 .
2. Show that the \(x\) coordinate of \(B\) is a solution of $$x = \sqrt { 1 + \frac { 1 } { 2 } x - \mathrm { e } ^ { - 2 x } }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 1 } { 2 } x _ { n } - \mathrm { e } ^ { - 2 x _ { n } } }$$ with \(x _ { 1 } = 1\)
3. find \(x _ { 2 }\) and \(x _ { 3 }\) to 3 decimal places.

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 5 - x | + 3 , \quad x \geqslant 0$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly one root,

1. state the set of possible values of \(k\).
2. Solve the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x + 10\) The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\). The vertex on the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\) has coordinates \(( p , q )\).
3. State the value of \(p\) and the value of \(q\).

1. (i) Using the identity for \(\tan ( A \pm B )\), solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\),

$$\frac { \tan 2 x + \tan 32 ^ { \circ } } { 1 - \tan 2 x \tan 32 ^ { \circ } } = 5$$ Give your answers, in degrees, to 2 decimal places.
(ii) (a) Using the identity for \(\tan ( A \pm B )\), show that $$\tan \left( 3 \theta - 45 ^ { \circ } \right) \equiv \frac { \tan 3 \theta - 1 } { 1 + \tan 3 \theta } , \quad \theta \neq ( 60 n + 45 ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 < \theta < 180 ^ { \circ }\), $$( 1 + \tan 3 \theta ) \tan \left( \theta + 28 ^ { \circ } \right) = \tan 3 \theta - 1$$

1. The curve \(C\) has equation \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 } , \quad x \in \mathbb { R }\)
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a single fraction, simplifying your answer.
   2. Hence find the exact coordinates of the stationary points of \(C\).
      

1. (a) By writing \(\sec \theta = \frac { 1 } { \cos \theta }\), show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \sec \theta ) = \sec \theta \tan \theta\)
   (b) Given that

$$x = \mathrm { e } ^ { \sec y } \quad x > \mathrm { e } , \quad 0 < y < \frac { \pi } { 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \sqrt { \mathrm {~g} ( x ) } } , \quad x > \mathrm { e }$$ where \(\mathrm { g } ( x )\) is a function of \(\ln x\).

1. (a) Express \(\sin \theta - 2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and the value of \(\alpha\), in radians, to 3 decimal places. $$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$ (b) Find

1. the maximum value of \(\mathrm { M } ( \theta )\),
2. the smallest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { M } ( \theta )\) occurs. $$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$ (c) Find
3. the maximum value of \(\mathrm { N } ( \theta )\),
4. the largest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { N } ( \theta )\) occurs.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

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