14
2
\end{array} \right) ,
\end{aligned}$$
and
where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.
Given that the two lines intersect,
- find the position vector of their point of intersection,
- find the value of \(a\).
Given also that \(\theta\) is the acute angle between the lines,
- find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.
4. continued
5. A curve has the equation
$$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$ - Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
- Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation
$$3 x - 2 y + 1 = 0$$
The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
- Find the coordinates of \(Q\).
5. continued
6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\). - Write down a differential equation connecting \(N\) and \(t\).
Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
- Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
Given also that the number of bacteria present doubles every six hours,
- find the value of \(k\),
- find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.
6. continued
7. A curve has parametric equations
$$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$ - Show that \(x + \frac { 1 } { x } = 2 \sec \theta\).
Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
- find a cartesian equation for the curve.
- Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
7. continued
7. continued