OCR C4 (Core Mathematics 4) 2011 January

1

1. Expand \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
2. Hence expand \(\left( 1 - 2 y + 4 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).

2

1. Express \(\frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\), where \(A\) and \(B\) are constants.
2. Hence find the exact value of \(\int _ { 4 } ^ { 5 } \frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\).

3

1. Show that the derivative of \(\sec x\) can be written as \(\sec x \tan x\).
2. Find \(\int \frac { \tan x } { \sqrt { 1 + \cos 2 x } } \mathrm {~d} x\).

4 A curve has parametric equations $$x = 2 + t ^ { 2 } , \quad y = 4 t$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the normal at the point where \(t = 4\), giving your answer in the form \(y = m x + c\).
3. Find a cartesian equation of the curve.

5 In this question, \(I\) denotes the definite integral \(\int _ { 2 } ^ { 5 } \frac { 5 - x } { 2 + \sqrt { x - 1 } } \mathrm {~d} x\). The value of \(I\) is to be found using two different methods.

1. Show that the substitution \(u = \sqrt { x - 1 }\) transforms \(I\) to \(\int _ { 1 } ^ { 2 } \left( 4 u - 2 u ^ { 2 } \right) \mathrm { d } u\) and hence find the exact value of \(I\).
2. (a) Simplify \(( 2 + \sqrt { x - 1 } ) ( 2 - \sqrt { x - 1 } )\).
   (b) By first multiplying the numerator and denominator of \(\frac { 5 - x } { 2 + \sqrt { x - 1 } }\) by \(2 - \sqrt { x - 1 }\), find the exact value of \(I\).

6 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 3
0
- 2 \end{array} \right) + s \left( \begin{array} { r } 2
3
- 4 \end{array} \right)\). The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 5
3
2 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 2 \end{array} \right)\).

1. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
3. One of the numbers in the equation of line \(l _ { 1 }\) is changed so that the equation becomes \(\mathbf { r } = \left( \begin{array} { l } 3
   0
   a \end{array} \right) + s \left( \begin{array} { r } 2
   3
   - 4 \end{array} \right)\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) now intersect, find \(a\).

7 Show that \(\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10\).

8 The points \(P\) and \(Q\) lie on the curve with equation $$2 x ^ { 2 } - 5 x y + y ^ { 2 } + 9 = 0$$ The tangents to the curve at \(P\) and \(Q\) are parallel, each having gradient \(\frac { 3 } { 8 }\).

1. Show that the \(x\) - and \(y\)-coordinates of \(P\) and \(Q\) are such that \(x = 2 y\).
2. Hence find the coordinates of \(P\) and \(Q\).

9 Paraffin is stored in a tank with a horizontal base. At time \(t\) minutes, the depth of paraffin in the tank is \(x \mathrm {~cm}\). When \(t = 0 , x = 72\). There is a tap in the side of the tank through which the paraffin can flow. When the tap is opened, the flow of the paraffin is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 4 ( x - 8 ) ^ { \frac { 1 } { 3 } }$$

1. How long does it take for the level of paraffin to fall from a depth of 72 cm to a depth of 35 cm ?
2. The tank is filled again to its original depth of 72 cm of paraffin and the tap is then opened. The paraffin flows out until it stops. How long does this take?