AQA C3 (Core Mathematics 3) 2011 June

1 The diagram shows the curve with equation \(y = \ln ( 6 x )\).
\includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-2_448_501_370_790}

1. State the \(x\)-coordinate of the point of intersection of the curve with the \(x\)-axis.
   (1 mark)
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (2 marks)
3. Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for \(\int _ { 1 } ^ { 7 } \ln ( 6 x ) \mathrm { d } x\), giving your answer to three significant figures.

2

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \mathrm { e } ^ { 2 x }\).
   2. Find an equation of the tangent to the curve \(y = x \mathrm { e } ^ { 2 x }\) at the point \(\left( 1 , \mathrm { e } ^ { 2 } \right)\).
2. Given that \(y = \frac { 2 \sin 3 x } { 1 + \cos 3 x }\), use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { 1 + \cos 3 x }$$ where \(k\) is an integer.

3 The curve \(y = \cos ^ { - 1 } ( 2 x - 1 )\) intersects the curve \(y = \mathrm { e } ^ { x }\) at a single point where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.4 and 0.5 .
2. Show that the equation \(\cos ^ { - 1 } ( 2 x - 1 ) = \mathrm { e } ^ { x }\) can be written as \(x = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x } \right)\).
3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.

4

1. 1. Solve the equation \(\operatorname { cosec } \theta = - 4\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).
   2. Solve the equation $$2 \cot ^ { 2 } \left( 2 x + 30 ^ { \circ } \right) = 2 - 7 \operatorname { cosec } \left( 2 x + 30 ^ { \circ } \right)$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \operatorname { cosec } x\) onto the graph of \(y = \operatorname { cosec } \left( 2 x + 30 ^ { \circ } \right)\).

5 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 2 x + 1 } & \text { for real values of } x , \quad x \neq - 0.5 \end{array}$$

1. Explain why f does not have an inverse.
2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
3. State the range of \(\mathrm { g } ^ { - 1 }\).
4. Solve the equation \(\mathrm { fg } ( x ) = \mathrm { g } ( x )\).

6

1. Given that \(3 \ln x = 4\), find the exact value of \(x\).
2. By forming a quadratic equation in \(\ln x\), solve \(3 \ln x + \frac { 20 } { \ln x } = 19\), giving your answers for \(x\) in an exact form.

7

1. On separate diagrams:
   1. sketch the curve with equation \(y = | 3 x + 3 |\);
   2. sketch the curve with equation \(y = \left| x ^ { 2 } - 1 \right|\).
2. 1. Solve the equation \(| 3 x + 3 | = \left| x ^ { 2 } - 1 \right|\).
   2. Hence solve the inequality \(| 3 x + 3 | < \left| x ^ { 2 } - 1 \right|\).
      \(8 \quad\) Use the substitution \(u = 1 + 2 \tan x\) to find $$\int \frac { 1 } { ( 1 + 2 \tan x ) ^ { 2 } \cos ^ { 2 } x } d x$$

9

1. Use integration by parts to find \(\int x \ln x \mathrm {~d} x\).
2. Given that \(y = ( \ln x ) ^ { 2 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (2 marks)
3. The diagram shows part of the curve with equation \(y = \sqrt { x } \ln x\).
   \includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-5_406_645_696_719} The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \ln x\), the line \(x = \mathrm { e }\) and the \(x\)-axis from \(x = 1\) to \(x = \mathrm { e }\). Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in an exact form.
   (6 marks)