AQA C3 (Core Mathematics 3) 2015 June

1

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1.5 } ^ { 5.5 } \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 ) \mathrm { d } x\), giving your answer to three decimal places.
   [0pt] [4 marks]
2. Find the exact value of the gradient of the curve \(y = \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 )\) at the point on the curve where \(x = 2\).
   [0pt] [4 marks]

2

1. Sketch, on the axes below, the curve with equation \(y = 4 - | 2 x + 1 |\), indicating the coordinates where the curve crosses the axes.
2. Solve the equation \(x = 4 - | 2 x + 1 |\).
3. Solve the inequality \(x < 4 - | 2 x + 1 |\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = | 2 x + 1 |\) onto the graph of \(y = 4 - | 2 x + 1 |\).
   [0pt] [4 marks] \section*{Answer space for question 2}
5. 
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3

1. It is given that the curves with equations \(y = 6 \ln x\) and \(y = 8 x - x ^ { 2 } - 3\) intersect at a single point where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 5 and 6 .
   2. Show that the equation \(x = 4 + \sqrt { 13 - 6 \ln x }\) can be rearranged into the form $$6 \ln x + x ^ { 2 } - 8 x + 3 = 0$$
   3. Use the iterative formula $$x _ { n + 1 } = 4 + \sqrt { 13 - 6 \ln x _ { n } }$$ with \(x _ { 1 } = 5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
2. A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = 6 \ln x + x ^ { 2 } - 8 x + 3\).
   1. Find the exact values of the coordinates of the stationary points of the curve.
   2. Hence, or otherwise, find the exact values of the coordinates of the stationary points of the curve with equation $$y = 2 \mathrm { f } ( x - 4 )$$

4 The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 5 - \mathrm { e } ^ { 3 x } , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for } x \neq 1.5 \end{array}$$

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 \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
3. Find an expression for \(\operatorname { gg } ( x )\), giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
   [0pt] [3 marks]

5

1. By writing \(\tan x\) as \(\frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \tan x ) = \sec ^ { 2 } x\).
   [0pt] [2 marks]
2. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
   [0pt] [4 marks]
3. The region bounded by the curve \(y = ( 5 \sqrt { x } ) \sec x\), the \(x\)-axis from 0 to 1 and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the value of the volume of the solid generated, giving your answer to two significant figures.
   [0pt] [3 marks]

6

1. Sketch, on the axes below, the curve with equation \(y = \sin ^ { - 1 } ( 3 x )\), where \(y\) is in radians. State the exact values of the coordinates of the end points of the graph.
2. Given that \(x = \frac { 1 } { 3 } \sin y\), write down \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) and hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). \section*{Answer space for question 6}
3. 
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7 Use the substitution \(u = 6 - x ^ { 2 }\) to find the value of \(\int _ { 1 } ^ { 2 } \frac { x ^ { 3 } } { \sqrt { 6 - x ^ { 2 } } } \mathrm {~d} x\), giving your answer in the form \(p \sqrt { 5 } + q \sqrt { 2 }\), where \(p\) and \(q\) are rational numbers.
[0pt] [7 marks]

8

1. Show that the equation \(4 \operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta = k\), where \(k \neq 4\), can be written in the form $$\sec ^ { 2 } \theta = \frac { k - 1 } { k - 4 }$$
2. Hence, or otherwise, solve the equation $$4 \operatorname { cosec } ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) - \cot ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) = 5$$ giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
   [0pt] [5 marks]
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