OCR C3 (Core Mathematics 3) 2006 June

1 Find the equation of the tangent to the curve \(y = \sqrt { 4 x + 1 }\) at the point ( 2,3 ).

2 Solve the inequality \(| 2 x - 3 | < | x + 1 |\).

3 The equation \(2 x ^ { 3 } + 4 x - 35 = 0\) has one real root.

1. Show by calculation that this real root lies between 2 and 3 .
2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { 17.5 - 2 x _ { n } }$$ with a suitable starting value, to find the real root of the equation \(2 x ^ { 3 } + 4 x - 35 = 0\) correct to 2 decimal places. You should show the result of each iteration.

4 It is given that \(y = 5 ^ { x - 1 }\).

1. Show that \(x = 1 + \frac { \ln y } { \ln 5 }\).
2. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
3. Hence find the exact value of the gradient of the curve \(y = 5 ^ { x - 1 }\) at the point (3, 25).

5

1. Write down the identity expressing \(\sin 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
2. Given that \(\sin \alpha = \frac { 1 } { 4 }\) and \(\alpha\) is acute, show that \(\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }\).
3. Solve, for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\), the equation \(5 \sin 2 \beta \sec \beta = 3\).

6
\includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-3_563_583_267_781} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$

1. Evaluate \(\mathrm { ff } ( - 3 )\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). Indicate the coordinates of the points where the graph meets the axes.

7

1. Find the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 } { ( 4 x - 1 ) ^ { 2 } } \mathrm {~d} x\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-3_563_753_1681_735} The diagram shows part of the curve \(y = \frac { 1 } { x }\). The point \(P\) has coordinates \(\left( a , \frac { 1 } { a } \right)\) and the point \(Q\) has coordinates \(\left( 2 a , \frac { 1 } { 2 a } \right)\), where \(a\) is a positive constant. The point \(R\) is such that \(P R\) is parallel to the \(x\)-axis and \(Q R\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(P R\) and \(Q R\). Show that the area of this shaded region is \(\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\).

8

1. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\).
3. Solve, for \(0 ^ { \circ } < x < 360 ^ { \circ }\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1 ^ { \circ }\).

9
\includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-4_556_720_676_715} The diagram shows the curve with equation \(y = 2 \ln ( x - 1 )\). The point \(P\) has coordinates ( \(0 , p\) ). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0 , y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(\boldsymbol { y }\)-axis to form a solid.

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the solid is given by $$V = \pi \left( \mathrm { e } ^ { p } + 4 \mathrm { e } ^ { \frac { 1 } { 2 } p } + p - 5 \right) .$$
2. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \mathrm {~cm} \mathrm {~min} ^ { - 1 }\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures.