OCR MEI C3 (Core Mathematics 3) 2012 June

2 Solve the inequality \(| 2 x + 1 | > 4\).

3 Find the gradient at the point \(( 0 , \ln 2 )\) on the curve with equation \(\mathrm { e } ^ { 2 y } = 5 - \mathrm { e } ^ { - x }\).

4 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h] \end{figure}

1. Find the value of \(a\).
2. Write down the range of \(\mathrm { f } ( x )\).
3. Sketch the curve \(y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1\).

5 A termites' nest has a population of \(P\) million. \(P\) is modelled by the equation \(P = 7 - 2 \mathrm { e } ^ { - k t }\), where \(t\) is in years, and \(k\) is a positive constant.

1. Calculate the population when \(t = 0\), and the long-term population, given by this model.
2. Given that the population when \(t = 1\) is estimated to be 5.5 million, calculate the value of \(k\).

6 Fig. 6 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 2 \arcsin x , - 1 \leqslant x \leqslant 1\).
Fig. 6 also shows the curve \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\).
P is the point on the curve \(y = \mathrm { f } ( x )\) with \(x\)-coordinate \(\frac { 1 } { 2 }\). \begin{figure}[h] \end{figure}

1. Find the \(y\)-coordinate of P , giving your answer in terms of \(\pi\). The point Q is the reflection of P in \(y = x\).
2. Find \(\mathrm { g } ( x )\) and its derivative \(\mathrm { g } ^ { \prime } ( x )\). Hence determine the exact gradient of the curve \(y = \mathrm { g } ( x )\) at the point Q . Write down the exact gradient of \(y = \mathrm { f } ( x )\) at the point P .

7 You are given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are odd functions, defined for \(x \in \mathbb { R }\).

1. Given that \(\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\), prove that \(\mathrm { s } ( x )\) is an odd function.
2. Given that \(\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )\), determine whether \(\mathrm { p } ( x )\) is odd, even or neither.

8 Fig. 8 shows a sketch of part of the curve \(y = x \sin 2 x\), where \(x\) is in radians.
The curve crosses the \(x\)-axis at the point P . The tangent to the curve at P crosses the \(y\)-axis at Q . \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that the \(x\)-coordinates of the turning points of the curve satisfy the equation \(\tan 2 x + 2 x = 0\).
2. Find, in terms of \(\pi\), the \(x\)-coordinate of the point P . Show that the tangent PQ has equation \(2 \pi x + 2 y = \pi ^ { 2 }\).
   Find the exact coordinates of Q .
3. Show that the exact value of the area shaded in Fig. 8 is \(\frac { 1 } { 8 } \pi \left( \pi ^ { 2 } - 2 \right)\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), which has a \(y\)-intercept at \(\mathrm { P } ( 0,3 )\), a minimum point at \(\mathrm { Q } ( 1,2 )\), and an asymptote \(x = - 1\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the images of the points P and Q when the curve \(y = \mathrm { f } ( x )\) is transformed to
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( x + 1 ) + 2\). You are now given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence find the coordinates of the other turning point on the curve \(y = \mathrm { f } ( x )\).
3. Show that \(\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }\).
4. Find \(\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x\) in terms of \(a\) and \(b\). Hence, by choosing suitable values for \(a\) and \(b\), find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).