Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2015 June

1 Show that \(\frac { 31 } { 6 - \sqrt { 5 } } = 6 + \sqrt { 5 }\).

2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 2\). The curve passes through the point \(( 1,3 )\). Find the equation of the curve.

3 The function f is given by \(\mathrm { f } ( x ) = | x - 2 | + 3\) for \(- 5 \leqslant x \leqslant 5\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Explain why f is not one-one.

4 Find the volume of the solid generated when the region bounded by the \(x\)-axis, \(x = 1 , x = 2\) and the curve given by \(y = x ^ { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

5

1. Show that the equation \(\sin x - x + 1 = 0\) has a root between 1.5 and 2 .
2. Use the iteration \(x _ { n + 1 } = 1 + \sin x _ { n }\), with a suitable starting value, to find that root correct to 2 decimal places.
3. Sketch the graphs of \(y = \sin x\) and \(y = x - 1\), on the same set of axes, for \(0 \leqslant x \leqslant \pi\).
4. Explain why the equation \(\sin x - x + 1 = 0\) has no root other than the one found in part (ii). [1]

6 A cup of tea is served at \(80 ^ { \circ } \mathrm { C }\) in a room which is kept at a constant \(20 ^ { \circ } \mathrm { C }\). The temperature, \(T ^ { \circ } \mathrm { C }\), of the tea after \(t\) minutes can be modelled by assuming that the rate of change of \(T\) is proportional to the difference in temperature between the tea and the room.

1. Explain why the rate of change of the temperature in this model is given by \(\frac { \mathrm { d } T } { \mathrm {~d} t } = - k ( T - 20 )\), where \(k\) is a positive constant.
2. Show by integration that the temperature of the tea after \(t\) minutes is given by \(T = 20 + 60 \mathrm { e } ^ { - k t }\).
3. After 2 minutes the tea has cooled to \(60 ^ { \circ } \mathrm { C }\). Find the value of \(k\).

7 A curve is given parametrically by \(x = 3 t , y = 1 + t ^ { 3 }\) where \(t\) is any real number.

1. Show that a cartesian equation for this curve is given by \(y = 1 + \frac { 1 } { 27 } x ^ { 3 }\). A second curve is given by \(y = x ^ { 2 } + 4 x - 19\).
2. Given that the curves intersect at the point \(( 3,2 )\), find the coordinates of all the other points of intersection between the two curves.

8 The function f is given by \(\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 3 x ^ { 2 } - 1 }\), for \(x > 1\). Show that f is a decreasing function.

9 Find the equations of all the horizontal tangents to the curve with equation \(y ^ { 2 } = x ^ { 4 } - 4 x ^ { 3 } + 36\).

10

1. Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
2. Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
3. Show that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\). Hence, by writing \(\cos 2 \theta - \sin 3 \theta\) in terms of \(\sin \theta\), use your answer to part (ii) to determine the solutions of \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).

11 \includegraphics[max width=\textwidth, alt={}, center]{2f48a6ee-e8ce-47e4-a07f-2c55a6904e7d-3_661_953_767_596} The diagram shows a circle, centre \(O\), radius \(r\). The points \(R\) and \(S\) lie on the circumference of the circle, and the line \(R T\) is a tangent to the circle at \(R\). The angle \(R O S\) is \(\theta\) radians where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Find expressions for the perimeter, \(P\), and the area, \(A\), of the shaded region in terms of \(r\) and \(\theta\).
2. Hence show that \(A \neq r P\).