OCR MEI C3 (Core Mathematics 3) 2008 June

1 Solve the inequality \(| 2 x - 1 | \leqslant 3\).

2 Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).

3

1. State the algebraic condition for the function \(\mathrm { f } ( x )\) to be an even function.
   What geometrical property does the graph of an even function have?
2. State whether the following functions are odd, even or neither.
   (A) \(\mathrm { f } ( x ) = x ^ { 2 } - 3\) (B) \(\mathrm { g } ( x ) = \sin x + \cos x\) (C) \(\mathrm { h } ( x ) = \frac { 1 } { x + x ^ { 3 } }\)

4 Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 6\).

5 Show that the curve \(y = x ^ { 2 } \ln x\) has a stationary point when \(x = \frac { 1 } { \sqrt { \mathrm { e } } }\).

6 In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
2. Find the time when the mass is 30 grams.
3. Sketch the graph of \(m\) against \(t\).

7 Given that \(x ^ { 2 } + x y + y ^ { 2 } = 12\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 1 + \cos x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
P is the point on the curve with \(x\)-coordinate \(\frac { 1 } { 3 } \pi\). \begin{figure}[h] \end{figure}

1. Find the \(y\)-coordinate of P .
2. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the gradient of the curve at the point P .
3. Show that the derivative of \(\frac { \sin x } { 1 + \cos x }\) is \(\frac { 1 } { 1 + \cos x }\). Hence find the exact area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 3 } \pi\).
4. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arccos \left( \frac { 1 } { x } - 1 \right)\). State the domain of this inverse function, and add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.

9 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt { 4 - x ^ { 2 } }\) for \(- 2 \leqslant x \leqslant 2\).

1. Show that the curve \(y = \sqrt { 4 - x ^ { 2 } }\) is a semicircle of radius 2 , and explain why it is not the whole of this circle. Fig. 9 shows a point \(\mathrm { P } ( a , b )\) on the semicircle. The tangent at P is shown. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8feffafd-4eba-4968-b4d2-88fa364d6170-4_625_933_589_607} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
2. (A) Use the gradient of OP to find the gradient of the tangent at P in terms of \(a\) and \(b\).
   (B) Differentiate \(\sqrt { 4 - x ^ { 2 } }\) and deduce the value of \(\mathrm { f } ^ { \prime } ( a )\).
   (C) Show that your answers to parts ( \(A\) ) and ( \(B\) ) are equivalent. The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } ( x - 2 )\), for \(0 \leqslant x \leqslant 4\).
3. Describe a sequence of two transformations that would map the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\). Hence sketch the curve \(y = \mathrm { g } ( x )\).
4. Show that if \(y = \mathrm { g } ( x )\) then \(9 x ^ { 2 } + y ^ { 2 } = 36 x\).