Edexcel FP1 (Further Pure Mathematics 1) 2020 June

1. Use l'Hospital's Rule to show that

$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 3 \cos \left( \frac { \pi } { 2 } x ^ { 2 } \right) \quad x \in [ - 1,1 ]$$ A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.
Use Simpson's rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in \(\mathrm { m } ^ { 3 }\) to 4 significant figures.

1. The points \(A , B\) and \(C\), with position vectors \(\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \mathbf { b } = \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }\) and \(\mathbf { c } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) respectively, lie on the plane \(\Pi\)
   1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\)
   2. Find an equation for \(\Pi\) in the form r.n \(= p\)
   The point \(D\) has position vector \(8 \mathbf { i } + 7 \mathbf { j } + 5 \mathbf { k }\)
2. Determine the volume of the tetrahedron \(A B C D\)

4. $$f ( x ) = x ^ { 4 } \sin ( 2 x )$$ Use Leibnitz's theorem to show that the coefficient of \(( x - \pi ) ^ { 8 }\) in the Taylor series expansion of \(\mathrm { f } ( x )\) about \(\pi\) is $$\frac { a \pi + b \pi ^ { 3 } } { 315 }$$ where \(a\) and \(b\) are integers to be determined. The Taylor series expansion of \(\mathrm { f } ( \mathrm { x } )\) about \(\mathrm { x } = \mathrm { k }\) is given by $$f ( x ) = f ( k ) + ( x - k ) f ^ { \prime } ( k ) + \frac { ( x - k ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( k ) + \ldots + \frac { ( x - k ) ^ { r } } { r ! } f ^ { ( r ) } ( k ) + \ldots$$

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 16 } = 1$$ The points \(S\) and \(S ^ { \prime }\) are the foci of \(E\).

1. Find the coordinates of \(S\) and \(S ^ { \prime }\)
2. Show that for any point \(P\) on \(E\), the triangle \(P S S ^ { \prime }\) has constant perimeter and determine its value.

1. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by

$$\begin{aligned} & d _ { A } = | 5 t - 31 |
& d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right| \end{aligned}$$ respectively, where \(t\) is the time in seconds after motion begins.

1. Use algebra to find the range of time for which particle \(A\) is further away from \(O\) than particle \(B\) is from \(O\). It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
2. Use this information to assess the validity of the model.

1. The points \(P \left( 9 p ^ { 2 } , 18 p \right)\) and \(Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q\), lie on the parabola \(C\) with equation

$$y ^ { 2 } = 36 x$$ The line \(l\) passes through the points \(P\) and \(Q\)

1. Show that an equation for the line \(l\) is $$( p + q ) y = 2 ( x + 9 p q )$$ The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(A\).
2. Show that the coordinates of \(A\) are $$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$ Given that the points \(P\) and \(Q\) vary such that \(l\) always passes through the point \(( 12,0 )\)
3. find, in the form \(y ^ { 2 } = \mathrm { f } ( x )\), an equation for the locus of \(A\), giving \(\mathrm { f } ( x )\) in simplest form.

8. $$f ( x ) = \frac { 3 } { 13 + 6 \sin x - 5 \cos x }$$ Using the substitution \(t = \tan \left( \frac { x } { 2 } \right)\)

1. show that \(\mathrm { f } ( x )\) can be written in the form $$\frac { 3 \left( 1 + t ^ { 2 } \right) } { 2 ( 3 t + 1 ) ^ { 2 } + 6 }$$
2. Hence solve, for \(0 < x < 2 \pi\), the equation $$\mathrm { f } ( x ) = \frac { 3 } { 7 }$$ giving your answers to 2 decimal places where appropriate.
3. Use the result of part (a) to show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { 4 \pi } { 3 } } f ( x ) d x = K \left( \arctan \left( \frac { \sqrt { 3 } - 9 } { 3 } \right) - \arctan \left( \frac { \sqrt { 3 } + 3 } { 3 } \right) + \pi \right)$$ where \(K\) is a constant to be determined.