Edexcel FP1 AS (Further Pure 1 AS) 2020 June

1. The variables \(x\) and \(y\) satisfy the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 y ^ { 2 } - x - 1$$ where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and \(y = 0\) at \(x = 0\)
Use the approximations $$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$ with \(h = 0.1\) to find an estimate for the value of \(y\) at \(x = 0.2\)

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

$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$

1. 1. Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to prove that

$$\cot x + \tan \left( \frac { x } { 2 } \right) = \operatorname { cosec } x \quad x \neq n \pi , n \in \mathbb { Z }$$ (ii) \begin{figure}[h] \end{figure} An engineer models the vertical height above the ground of the tip of one blade of a wind turbine, shown in Figure 1. The ground is assumed to be horizontal. The vertical height of the tip of the blade above the ground, \(H\) metres, at time \(x\) seconds after the wind turbine has reached its constant operating speed, is modelled by the equation $$H = 90 - 30 \cos ( 120 x ) ^ { \circ } - 40 \sin ( 120 x ) ^ { \circ }$$

1. Show that \(H = 60\) when \(x = 0\) Using the substitution \(t = \tan ( 60 x ) ^ { \circ }\)
2. show that equation (I) can be rewritten as $$H = \frac { 120 t ^ { 2 } - 80 t + 60 } { 1 + t ^ { 2 } }$$
3. Hence find, according to the model, the value of \(x\) when the tip of the blade is 100 m above the ground for the first time after the wind turbine has reached its constant operating speed.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(P \left( a p ^ { 2 } \right.\), 2ap) lies on \(C\) where \(p > 0\)

1. Write down the coordinates of \(S\).
2. Write down the length of SP in terms of \(a\) and \(p\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p \neq q\), also lies on \(C\).
   The point \(M\) is the midpoint of \(P Q\).
   Given that \(p q = - 1\)
3. prove that, as \(P\) varies, the locus of \(M\) has equation $$y ^ { 2 } = 2 a ( x - a )$$

5. \begin{figure}[h] \end{figure} Figure 3 shows a solid display stand with parallel triangular faces \(A B C\) and \(D E F\). Triangle \(D E F\) is similar to triangle \(A B C\). With respect to a fixed origin \(O\), the points \(A , B\) and \(C\) have coordinates ( \(3 , - 3,1\) ), ( \(- 5,3,3\) ) and ( \(1,7,5\) ) respectively and the points \(D , E\) and \(F\) have coordinates ( \(2 , - 1,8\) ), ( \(- 2,2,9\) ) and ( \(1,4,10\) ) respectively. The units are in centimetres.

1. Show that the area of the triangular face \(D E F\) is \(\frac { 1 } { 2 } \sqrt { 339 } \mathrm {~cm} ^ { 2 }\)
2. Find, in \(\mathrm { cm } ^ { 3 }\), the exact volume of the display stand.