3 Given that \(y \sin 2 x + \frac { 1 } { x } + y ^ { 2 } = 5\), find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
4 Find the exact value of \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } \ln x \mathrm {~d} x\), giving your answer in the form \(A \ln 2 + B\), where \(A\) and \(B\) are constants to be found.
5 The vector equations of two lines are as follows.
$$L : \mathbf { r } = \left( \begin{array} { l }
1 \\
4 \\
5
\end{array} \right) + s \left( \begin{array} { c }
2 \\
- 1 \\
3
\end{array} \right) \quad M : \mathbf { r } = \left( \begin{array} { c }
3 \\
2 \\
- 5
\end{array} \right) + t \left( \begin{array} { c }
5 \\
- 3 \\
1
\end{array} \right)$$
Show that the lines \(L\) and \(M\) meet, and find the coordinates of the point of intersection.
Show that the line \(L\) can also be represented by the equation \(\mathbf { r } = \left( \begin{array} { c } 7 \\ 1 \\ 14 \end{array} \right) + u \left( \begin{array} { c } - 4 \\ 2 \\ - 6 \end{array} \right)\).
7 Given that the binomial expansion of \(( 1 + k x ) ^ { n }\) is \(1 - 6 x + 30 x ^ { 2 } + \ldots\), find the values of \(n\) and \(k\). State the set of values of \(x\) for which this expansion is valid.
8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by
$$\overrightarrow { O A } = \left( \begin{array} { c }
3 \sin \alpha \\
2 \cos \alpha \\
- 1
\end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c }
2 \cos \alpha \\
4 \sin \alpha \\
3
\end{array} \right)$$
where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.
Calculate the two possible values of \(\alpha\).
Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).
9 A curve has parametric equations \(x = 1 - \cos t , y = \sin t \sin 2 t\), for \(0 \leqslant t \leqslant \pi\).
Find the coordinates of the points where the curve meets the \(x\)-axis.
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos 2 t + 2 \cos ^ { 2 } t\). Hence find, in an exact form, the coordinates of the stationary points.
Find the cartesian equation of the curve. Give your answer in the form \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a polynomial.
Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
It is given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$
and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A \mathrm { e } ^ { n }\).