Edexcel FP1 AS (Further Pure 1 AS) 2022 June

1. Use algebra to find the set of values of \(x\) for which

$$x \geqslant \frac { 2 x + 15 } { 2 x + 3 }$$

1. A population of deer was introduced onto an island.

The number of deer, \(P\), on the island at time \(t\) years following their introduction is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { P } { 5000 } \left( 1000 - \frac { P ( t + 1 ) } { 6 t + 5 } \right) \quad t > 0$$ It was estimated that there were 540 deer on the island six months after they were introduced.
Use two applications of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the number of deer on the island 10 months after they were introduced.

1. (a) Use \(t = \tan \frac { \theta } { 2 }\) to show that, where both sides are defined

$$\frac { 29 - 21 \sec \theta } { 20 - 21 \tan \theta } \equiv \frac { 5 t + 2 } { 2 t + 5 }$$ (b) Hence, again using \(t = \tan \frac { \theta } { 2 }\), prove that, where both sides are defined $$\frac { 20 + 21 \tan \theta } { 29 + 21 \sec \theta } \equiv \frac { 29 - 21 \sec \theta } { 20 - 21 \tan \theta }$$

1. The parabola \(C\) has equation \(y ^ { 2 } = 10 x\)

The point \(F\) is the focus of \(C\).

1. Write down the coordinates of \(F\). The point \(P\) on \(C\) has \(y\) coordinate \(q\), where \(q > 0\)
2. Show that an equation for the tangent to \(C\) at \(P\) is given by $$10 x - 2 q y + q ^ { 2 } = 0$$ The tangent to \(C\) at \(P\) intersects the directrix of \(C\) at the point \(A\).
   The point \(B\) lies on the directrix such that \(P B\) is parallel to the \(x\)-axis.
3. Show that the point of intersection of the diagonals of quadrilateral \(P B A F\) always lies on the \(y\)-axis.

5. \begin{figure}[h] \end{figure} The points \(A ( 3,2 , - 4 ) , B ( 9 , - 4,2 ) , C ( - 6 , - 10,8 )\) and \(D ( - 4 , - 5,10 )\) are the vertices of a tetrahedron. The plane with equation \(z = 0\) cuts the tetrahedron into two pieces, one on each side of the plane. The edges \(A B , A C\) and \(A D\) of the tetrahedron intersect the plane at the points \(M , N\) and \(P\) respectively, as shown in Figure 1. Determine

1. the coordinates of the points \(M , N\) and \(P\),
2. the area of triangle \(M N P\),
3. the exact volume of the solid \(B C D P N M\).