6
6
\end{tabular}}
\hline 7 &
\hline 8 &
\hline &
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline \multicolumn{2}{|c|}{}
\hline Total &
\hline \end{tabular} \end{center} Turn over

1. A curve \(C\) has the equation

$$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation of the tangent to \(C\) at the point \(( 3 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   2. Given that the binomial expansion of \(( 1 + k x ) ^ { - 4 } , | k x | < 1\), is $$1 - 6 x + A x ^ { 2 } + \ldots$$
3. find the value of the constant \(k\),
4. find the value of the constant \(A\), giving your answer in its simplest form.
   3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-05_659_865_269_550} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 10 } { 2 x + 5 \sqrt { } x } , x > 0\)
   The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the lines with equations \(x = 1\) and \(x = 4\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 10 } { 2 x + 5 \sqrt { } x }\)

   \(x\)	1	2	3	4
   \(y\)	1.42857	0.90326		0.55556

5. Complete the table above by giving the missing value of \(y\) to 5 decimal places.
6. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places.
7. By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of \(R\).
8. Use the substitution \(u = \sqrt { } x\), or otherwise, to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 10 } { 2 x + 5 \sqrt { x } } d x$$ 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-07_618_703_246_625} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is \(h \mathrm {~cm}\), the volume of water \(V \mathrm {~cm} ^ { 3 }\) is given by $$V = 4 \pi h ( h + 4 ) , \quad 0 \leqslant h \leqslant 25$$ Water flows into the vase at a constant rate of \(80 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) Find the rate of change of the depth of the water, in \(\mathrm { cms } ^ { - 1 }\), when \(h = 6\)
   5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9963b13-7db4-4a1d-8c75-829f4aade994-08_675_1262_267_340} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , \quad y = 2 \sin t , \quad 0 \leqslant t < 2 \pi$$
9. Show that $$x + y = 2 \sqrt { 3 } \cos t$$
10. Show that a cartesian equation of \(C\) is $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be determined.
    \includegraphics[max width=\textwidth, alt={}, center]{a9963b13-7db4-4a1d-8c75-829f4aade994-09_104_51_2617_1900}
    6. (i) Find $$\int x \mathrm { e } ^ { 4 x } \mathrm {~d} x$$ (ii) Find $$\int \frac { 8 } { ( 2 x - 1 ) ^ { 3 } } \mathrm {~d} x , \quad x > \frac { 1 } { 2 }$$ (iii) Given that \(y = \frac { \pi } { 6 }\) at \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } \operatorname { cosec } 2 y \operatorname { cosec } y$$