12
0
5
\end{array} \right) + s \left( \begin{array} { r }
1
- 4
- 2
\end{array} \right) .$$
- Show that the lines intersect.
- Find the angle between the lines.
- Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
- Solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$
given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\).
8
- Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
- Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
- Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\).
9 The parametric equations of a curve are
$$x = 2 \theta + \sin 2 \theta , \quad y = 4 \sin \theta$$
and part of its graph is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{b8ba126f-c5fa-4828-9439-e5162a03ca5b-3_646_1150_1050_500}
- Find the value of \(\theta\) at \(A\) and the value of \(\theta\) at \(B\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec \theta\).
- At the point \(C\) on the curve, the gradient is 2 . Find the coordinates of \(C\), giving your answer in an exact form.