5. Points \(A , B\) and \(C\) have coordinates \(( 4,2,0 ) , ( 1,5,3 )\) and \(( 1,4 , - 2 )\) respectively. The line \(l\) passes through \(A\) and \(B\).
- Find a cartesian equation for \(l\).
\(M\) is the point on \(l\) that is closest to \(C\). - Find the coordinates of \(M\).
- Find the exact area of the triangle \(A B C\).
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\section*{6. The curve \(C\) has equation}
$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$
where \(a\) and \(p\) are positive constants and \(p > 2\)
There are exactly four points on \(C\) where the tangent is perpendicular to the initial line. - Show that the range possible values for \(p\) is
- Sketch the curve with equation
$$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$
John digs a hole in his garden in order to make a pond.
The pond has a uniform horizontal cross section that is modelled by the curve with equation
$$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$
where \(r\) is measured in centimetres.
The depth of the pond is 90 centimetres.
Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
Given that the pond is initially empty, - determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
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[TURN OVER FOR QUESTION 7]