OCR C3 (Core Mathematics 3) 2011 January

1 Solve the equation \(| 3 x + 4 a | = 5 a\), where \(a\) is a positive constant.

2
\includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_538_1061_388_541} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 7 ) = 0\) and that there are stationary points at \(( - 2 , - 6 )\) and \(( 0,0 )\). Sketch the curve with equation \(y = - 4 \mathrm { f } ( x + 3 )\), indicating the coordinates of the stationary points.

3 A giant spherical balloon is being inflated in a theme park. The radius of the balloon is increasing at a rate of 12 cm per hour. Find the rate at which the surface area of the balloon is increasing at the instant when the radius is 150 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per hour correct to 2 significant figures.
[0pt] [Surface area of sphere \(= 4 \pi r ^ { 2 }\).]

4

1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_559_1191_1749_479} The diagram shows the curve with equation \(y = \frac { 6 } { \sqrt { 3 x - 2 } }\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 1 , x = a\) and \(y = 0\), where \(a\) is a constant greater than 1 . It is given that the area of \(R\) is 16 square units. Find the value of \(a\) and hence find the exact volume of the solid formed when \(R\) is rotated completely about the \(x\)-axis.
[0pt] [9]

6 The curve with equation \(y = \frac { 3 x + 4 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }\) has a stationary point at \(P\). It is given that \(P\) is close to the point with coordinates \(( 2.4 , - 1.6 )\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \sqrt [ 3 ] { \frac { 16 } { 3 } x + 1 }$$
2. By first using an iterative process based on the equation in part (i), find the coordinates of \(P\), giving each coordinate correct to 3 decimal places.

7 The function f is defined for \(x > 0\) by \(\mathrm { f } ( x ) = \ln x\) and the function g is defined for all real values of \(x\) by \(\mathrm { g } ( x ) = x ^ { 2 } + 8\).

1. Find the exact, positive value of \(x\) which satisfies the equation \(\operatorname { fg } ( x ) = 8\).
2. State which one of f and g has an inverse and define that inverse function.
3. Find the exact value of the gradient of the curve \(y = \operatorname { gf } ( x )\) at the point with \(x\)-coordinate \(\mathrm { e } ^ { 3 }\).
4. Use Simpson's rule with four strips to find an approximate value of $$\int _ { - 4 } ^ { 4 } \mathrm { fg } ( x ) \mathrm { d } x$$ giving your answer correct to 3 significant figures.

8

1. 1. Sketch the graph of \(y = \operatorname { cosec } x\) for \(0 < x < 4 \pi\).
   2. It is given that \(\operatorname { cosec } \alpha = \operatorname { cosec } \beta\), where \(\frac { 1 } { 2 } \pi < \alpha < \pi\) and \(2 \pi < \beta < \frac { 5 } { 2 } \pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\).
2. 1. Write down the identity giving \(\tan 2 \theta\) in terms of \(\tan \theta\).
   2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2 \phi \tan 4 \phi\), showing all your working.

9

1. The function f is defined for all real values of \(x\) by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$ (a) Show that \(\mathrm { f } ^ { \prime } ( x ) > 0\) for all \(x\).
   (b) Show that the set of values of \(x\) for which \(\mathrm { f } ^ { \prime \prime } ( x ) > 0\) is the same as the set of values of \(x\) for which \(\mathrm { f } ( x ) > 0\), and state what this set of values is.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-04_634_830_641_699} The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x } ,$$ where \(k\) is a constant greater than 1 . The graph of \(y = \mathrm { g } ( x )\) is shown above. Find the range of g , giving your answer in simplified form.

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