OCR C4 (Core Mathematics 4) 2011 June

1 Simplify \(\frac { x ^ { 4 } - 10 x ^ { 2 } + 9 } { \left( x ^ { 2 } - 2 x - 3 \right) \left( x ^ { 2 } + 8 x + 15 \right) }\).

2 Find the unit vector in the direction of \(\left( \begin{array} { c } 2
- 3
\sqrt { 12 } \end{array} \right)\).

3

1. Find the quotient when \(3 x ^ { 3 } - x ^ { 2 } + 10 x - 3\) is divided by \(x ^ { 2 } + 3\), and show that the remainder is \(x\).
2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } - x ^ { 2 } + 10 x - 3 } { x ^ { 2 } + 3 } \mathrm {~d} x$$

4 Use the substitution \(x = \frac { 1 } { 3 } \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } } \frac { 1 } { \left( 1 - 9 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 4

6
4 \end{array} \right) + s \left( \begin{array} { l } 3
2
1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 1
0
0 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(A\) lies on \(l _ { 1 }\) and \(O A\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin. Find the position vector of \(A\). 6 Find the coefficient of \(x ^ { 2 }\) in the expansion in ascending powers of \(x\) of $$\sqrt { \frac { 1 + a x } { 4 - x } } ,$$ giving your answer in terms of \(a\).

7 The gradient of a curve at the point \(( x , y )\), where \(x > - 2\), is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 y ^ { 2 } ( x + 2 ) }$$ The points \(( 1,2 )\) and \(( q , 1.5 )\) lie on the curve. Find the value of \(q\), giving your answer correct to 3 significant figures.

8 A curve has parametric equations $$x = \frac { 1 } { t + 1 } , \quad y = t - 1 .$$ The line \(y = 3 x\) intersects the curve at two points.

1. Show that the value of \(t\) at one of these points is - 2 and find the value of \(t\) at the other point.
2. Find the equation of the normal to the curve at the point for which \(t = - 2\).
3. Find the value of \(t\) at the point where this normal meets the curve again.
4. Find a cartesian equation of the curve, giving your answer in the form \(y = \mathrm { f } ( x )\).

9

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x - x ) = \ln x\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{8492b214-aaac-4354-8649-e317bf7b3535-3_481_725_1064_751} In the diagram, \(C\) is the curve \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm { e }\).
   (a) Find the exact volume of the solid of revolution formed by rotating \(R\) completely about the \(x\)-axis.
   (b) The region \(R\) is rotated completely about the \(y\)-axis. Explain why the volume of the solid of revolution formed is given by $$\pi \mathrm { e } ^ { 2 } - \pi \int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 y } \mathrm {~d} y ,$$ and find this volume.