AQA C3 (Core Mathematics 3) 2013 June

1 The diagram below shows the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-02_579_1150_351_482}

1. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
   (3 marks)
2. Hence, or otherwise, solve the inequality $$| 2 x - 3 | \geqslant | x |$$ (2 marks)

2

1. Given that \(y = x ^ { 4 } \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. Find the gradient of the curve with equation \(y = \frac { x ^ { 2 } } { x - 1 }\) at the point where \(x = 3\).
   (3 marks)

3

1. The equation \(\mathrm { e } ^ { - x } - 2 + \sqrt { x } = 0\) has a single root, \(\alpha\).
   Show that \(\alpha\) lies between 3 and 4 .
2. Use the recurrence relation \(x _ { n + 1 } = \left( 2 - e ^ { - x _ { n } } \right) ^ { 2 }\), with \(x _ { 1 } = 3.5\), to find \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
3. The diagram below shows parts of the graphs of \(y = \left( 2 - \mathrm { e } ^ { - x } \right) ^ { 2 }\) and \(y = x\), and a position of \(x _ { 1 }\). On the diagram, draw a staircase or cobweb diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
   \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-03_1100_1402_881_367}

4 By forming and solving a quadratic equation, solve the equation $$8 \sec x - 2 \sec ^ { 2 } x = \tan ^ { 2 } x - 2$$ in the interval \(0 < x < 2 \pi\), giving the values of \(x\) in radians to three significant figures.

5 The diagram shows a sketch of the graph of \(y = \sqrt { 27 + x ^ { 3 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-04_762_988_365_534}

1. The area of the shaded region, bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\), is given by \(\int _ { 0 } ^ { 4 } \sqrt { 27 + x ^ { 3 } } \mathrm {~d} x\). Use the mid-ordinate rule with five strips to find an estimate for this area. Give your answer to three significant figures.
2. With the aid of a diagram, explain whether the mid-ordinate rule applied in part (a) gives an estimate which is smaller than or greater than the area of the shaded region.
   (2 marks)

6

1. Sketch the graph of \(y = \cos ^ { - 1 } x\), where \(y\) is in radians. State the coordinates of the end points of the graph.
2. Sketch the graph of \(y = \pi - \cos ^ { - 1 } x\), where \(y\) is in radians. State the coordinates of the end points of the graph.
   (2 marks)
   \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-05_759_1258_678_431}
   \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-05_751_1241_1564_443}

7 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-06_620_1216_356_422}

1. On Figure 1, below, sketch the curve with equation \(y = - \mathrm { f } ( 3 x )\), indicating the values where the curve cuts the coordinate axes.
2. On Figure 2, on page 7, sketch the curve with equation \(y = \mathrm { f } ( | x | )\), indicating the values where the curve cuts the coordinate axes.
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } \left( - \frac { 1 } { 2 } x \right)\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-06_732_1237_1649_443}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-07_727_1211_340_466}
   \end{figure}

8 The curve with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }\), is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}

1. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. State the range of \(\mathrm { f } ^ { - 1 }\).
   3. Sketch, on the axes given on page 9 , the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating the value of the \(y\)-coordinate of the point where the curve intersects the \(y\)-axis.
      (2 marks)
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
   1. Find \(\mathrm { gf } ( x )\), giving your answer in the form \(( a x - b ) ^ { 2 } - c\), where \(a , b\) and \(c\) are integers.
   2. Write down an expression for \(\mathrm { fg } ( x )\), and hence find the exact solution of the equation \(\operatorname { fg } ( x ) = \ln 5\).
      \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_1079_1422_233_358}

9 The shape of a vase can be modelled by rotating the curve with equation \(16 x ^ { 2 } - ( y - 8 ) ^ { 2 } = 32\) between \(y = 0\) and \(y = 16\) completely about the \(\boldsymbol { y }\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_890_1210_1555_424} The vase has a base.
Find the volume of water needed to fill the vase, giving your answer as an exact value.

10

1. 1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
   2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
      (4 marks)
2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)