5
\end{array} \right) + \mu \left( \begin{array} { r }
5
- 4
- 3
\end{array} \right)$$
where \(\mu\) is a scalar parameter.
The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).
- Write down the coordinates of \(A\).
The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
- Find the value of \(\cos \theta\).
The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).
The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\). - Find a vector equation of \(l _ { 2 }\).
- Find the length of \(O B\), giving your answer as a simplified surd.
The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
- find the length of \(O X\), giving your answer to 3 significant figures.
5. The curve \(C\) has the equation
$$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$ - Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).
The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\). - Find the exact value of the \(x\)-coordinate of \(A\).