Edexcel C4 (Core Mathematics 4)

1. A curve has the equation

$$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.

2. Use the substitution \(x = 2 \tan u\) to show that $$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 1 } { 2 } ( 4 - \pi )$$

1. (a) Show that \(\left( 1 \frac { 1 } { 24 } \right) ^ { - \frac { 1 } { 2 } } = k \sqrt { 6 }\), where \(k\) is rational.
   (b) Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { - \frac { 1 } { 2 } } , | x | < 2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   (c) Use your answer to part (b) with \(x = \frac { 1 } { 12 }\) to find an approximate value for \(\sqrt { 6 }\), giving your answer to 5 decimal places.
2. continued
3. Relative to a fixed origin, two lines have the equations

$$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where \(s\) and \(t\) are scalar parameters.
(a) Show that the two lines intersect and find the position vector of the point where they meet.
(b) Find, in degrees to 1 decimal place, the acute angle between the lines.

5. A curve has parametric equations $$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }\).
2. Find an equation for the normal to the curve at the point where \(t = 1\).
3. Show that the cartesian equation of the curve can be written in the form $$y = \frac { 1 + x } { 1 + 3 x }$$
   1. continued
   2. (a) Find \(\int \tan ^ { 2 } x d x\).
   3. Show that
   $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-10_566_789_648_504} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
   The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
4. Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a hemispherical bowl of radius 5 cm .
The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h \mathrm {~cm}\) and the volume of water in the bowl is \(V \mathrm {~cm} ^ { 3 }\), where $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 15 - h ) .$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).

1. Show that $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { k h ( 15 - h ) } { 3 ( 10 - h ) } ,$$ where \(k\) is a positive constant.
2. Express \(\frac { 3 ( 10 - h ) } { h ( 15 - h ) }\) in partial fractions. Given that when \(t = 0 , h = 5\),
3. show that $$h ^ { 2 } ( 15 - h ) = 250 \mathrm { e } ^ { - k t }$$ Given also that when \(t = 2 , h = 4\),
4. find the value of \(k\) to 3 significant figures.
   7. continued
   7. continued