OCR MEI C3 (Core Mathematics 3) 2011 June

1 Solve the equation \(| 2 x - 1 | = | x |\).

2 Given that \(\mathrm { f } ( x ) = 2 \ln x\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { x }\), find the composite function \(\mathrm { gf } ( x )\), expressing your answer as simply as possible.

4 The height \(h\) metres of a tree after \(t\) years is modelled by the equation $$h = a - b \mathrm { e } ^ { - k t }$$ where \(a\), \(b\) and \(k\) are positive constants.

1. Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of \(a\) and \(b\).
2. Given also that the tree grows to a height of 6 metres in 8 years, find the value of \(k\), giving your answer correct to 2 decimal places.

5 Given that \(y = x ^ { 2 } \sqrt { 1 + 4 x }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( 5 x + 1 ) } { \sqrt { 1 + 4 x } }\).

6 A curve is defined by the equation \(\sin 2 x + \cos y = \sqrt { 3 }\).

1. Verify that the point \(\mathrm { P } \left( \frac { 1 } { 6 } \pi , \frac { 1 } { 6 } \pi \right)\) lies on the curve.
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point P .
3. Multiply out \(\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)\).
4. Hence prove that if \(n\) is a positive integer then \(3 ^ { 2 n } - 1\) is divisible by 8 .

8 \begin{figure}[h] \end{figure} Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } + 2 }\).

1. Show algebraically that \(\mathrm { f } ( x )\) is an even function, and state how this property relates to the curve \(y = \mathrm { f } ( x )\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }\).
4. Hence, using the substitution \(u = \mathrm { e } ^ { x } + 1\), or otherwise, find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\).
5. Show that there is only one point of intersection of the curves \(y = \mathrm { f } ( x )\) and \(y = \frac { 1 } { 4 } \mathrm { e } ^ { x }\), and find its coordinates.

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\). The endpoints of the curve are \(\mathrm { P } ( - \pi , 1 )\) and \(\mathrm { Q } ( \pi , 3 )\), and \(\mathrm { f } ( x ) = a + \sin b x\), where \(a\) and \(b\) are constants. \begin{figure}[h] \end{figure}

1. Using Fig. 9, show that \(a = 2\) and \(b = \frac { 1 } { 2 }\).
2. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,2 )\). Show that there is no point on the curve at which the gradient is greater than this.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range. Write down the gradient of \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 2,0 )\).
4. Find the area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \pi\).