AQA C3 (Core Mathematics 3) 2011 January

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 3 } - 1 \right) ^ { 6 }\).
2. A curve has equation \(y = x \ln x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln x\) at the point on the curve where \(x = \mathrm { e }\).

2 A curve is defined by the equation \(y = \left( x ^ { 2 } - 4 \right) \ln ( x + 2 )\) for \(x \geqslant 3\).
The curve intersects the line \(y = 15\) at a single point, where \(x = \alpha\).

1. Show that \(\alpha\) lies between 3.5 and 3.6.
2. Show that the equation \(\left( x ^ { 2 } - 4 \right) \ln ( x + 2 ) = 15\) can be arranged into the form $$x = \pm \sqrt { 4 + \frac { 15 } { \ln ( x + 2 ) } }$$ (2 marks)
3. Use the iteration $$x _ { n + 1 } = \sqrt { 4 + \frac { 15 } { \ln \left( x _ { n } + 2 \right) } }$$ with \(x _ { 1 } = 3.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
   (2 marks)

3

1. Given that \(x = \tan ( 3 y + 1 )\) :
   1. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\);
   2. find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = - \frac { 1 } { 3 }\).
2. Sketch the graph of \(y = \tan ^ { - 1 } x\).

4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 \cos \frac { 1 } { 2 } x , & \text { for } 0 \leqslant x \leqslant 2 \pi
\mathrm {~g} ( x ) = | x | , & \text { for all real values of } x \end{array}$$

1. Find the range of f .
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 1\), giving your answer in an exact form.
3. 1. Write down an expression for \(\mathrm { gf } ( x )\).
   2. Sketch the graph of \(y = \operatorname { gf } ( x )\) for \(0 \leqslant x \leqslant 2 \pi\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos \frac { 1 } { 2 } x\).

5

1. Find \(\int \frac { 1 } { 3 + 2 x } \mathrm {~d} x\).
2. By using integration by parts, find \(\int x \sin \frac { x } { 2 } \mathrm {~d} x\).

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 0.4 } \cos \sqrt { 3 x + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Use the substitution \(u = 3 x + 1\) to find the exact value of \(\int _ { 0 } ^ { 1 } x \sqrt { 3 x + 1 } \mathrm {~d} x\).
   (6 marks)

7

1. Solve the equation \(\sec x = - 5\), giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
2. Show that the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ can be written in the form $$\sec ^ { 2 } x = 25$$
3. Hence, or otherwise, solve the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
   (3 marks)

8

1. Given that \(\mathrm { e } ^ { - 2 x } = 4\), find the exact value of \(x\).
2. The diagram shows the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\).
   \includegraphics[max width=\textwidth, alt={}, center]{6761e676-48ae-47e9-9617-153342cdf5c4-9_490_1185_463_440} The curve crosses the \(y\)-axis at the point \(A\), the \(x\)-axis at the point \(B\), and has a stationary point at \(M\).
   1. State the \(y\)-coordinate of \(A\).
   2. Find the \(x\)-coordinate of \(B\), giving your answer in an exact form.
   3. Find the \(x\)-coordinate of the stationary point, \(M\), giving your answer in an exact form.
   4. The shaded region \(R\) is bounded by the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\), the lines \(x = 0\) and \(x = \ln 2\) and the \(x\)-axis. Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\frac { p } { q } \pi\), where \(p\) and \(q\) are integers.
      (7 marks)