OCR C4 (Core Mathematics 4) 2009 January

1 Simplify \(\frac { 20 - 5 x } { 6 x ^ { 2 } - 24 x }\).

2 Find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).

3

1. Expand \(( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. Hence find the expansion of \(\frac { ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } } { ( 1 + x ) ^ { 3 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in part (ii) is valid.

4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 1 + \sin x ) ^ { 2 } \mathrm {~d} x\).

5

1. Show that the substitution \(u = \sqrt { x }\) transforms \(\int \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x\) to \(\int \frac { 2 } { u ( 1 + u ) } \mathrm { d } u\).
2. Hence find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x\).

6 A curve has parametric equations $$x = t ^ { 2 } - 6 t + 4 , \quad y = t - 3 .$$ Find

1. the coordinates of the point where the curve meets the \(x\)-axis,
2. the equation of the curve in cartesian form, giving your answer in a simple form without brackets,
3. the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7

1. Show that the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2
   - 3
   5 \end{array} \right) + t \left( \begin{array} { r } 1
   4
   - 2 \end{array} \right)\) meets the line passing through ( \(9,7,5\) ) and ( \(7,8,2\) ), and find the point of intersection of these lines.
2. Find the acute angle between these lines.

8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } = 6 x y\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Show that the point \(\left( 2 ^ { \frac { 4 } { 3 } } , 2 ^ { \frac { 5 } { 3 } } \right)\) lies on the curve and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at this point.
3. The point \(( a , a )\), where \(a > 0\), lies on the curve. Find the value of \(a\) and the gradient of the curve at this point.

9 A liquid is being heated in an oven maintained at a constant temperature of \(160 ^ { \circ } \mathrm { C }\). It may be assumed that the rate of increase of the temperature of the liquid at any particular time \(t\) minutes is proportional to \(160 - \theta\), where \(\theta ^ { \circ } \mathrm { C }\) is the temperature of the liquid at that time.

1. Write down a differential equation connecting \(\theta\) and \(t\). When the liquid was placed in the oven, its temperature was \(20 ^ { \circ } \mathrm { C }\) and 5 minutes later its temperature had risen to \(65 ^ { \circ } \mathrm { C }\).
2. Find the temperature of the liquid, correct to the nearest degree, after another 5 minutes. 4