Edexcel FP1 AS (Further Pure 1 AS) 2024 June

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Sketch the graph of the curve with equation
   $$y = \frac { 1 } { x ^ { 2 } }$$
2. Solve, using algebra, the inequality $$3 - 2 x ^ { 2 } > \frac { 1 } { x ^ { 2 } }$$

1. An area of woodland contains a mixture of blue and yellow flowers.

A study found that the proportion, \(x\), of blue flowers in the woodland area satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x t ( 0.8 - x ) } { x ^ { 2 } + 5 t } \quad t > 0$$ where \(t\) is the number of years since the start of the study.
Given that exactly 3 years after the start of the study half of the flowers in the woodland area were blue,

1. use one application 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 proportion of blue flowers in the woodland area half a year later.
2. Deduce from the differential equation the proportion of flowers that will be blue in the long term.

1. Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by

$$\mathbf { u } = 5 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \mathbf { v } = a \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k }$$ where \(a\) is a constant.

1. Determine, in terms of \(a\), the vector product \(\mathbf { u } \times \mathbf { v }\) Given that
   • \(\overrightarrow { A B } = 2 \mathbf { u }\)
   • \(\overrightarrow { A C } = \mathbf { v }\)
   • the area of triangle \(A B C\) is 15
   • determine the possible values of \(a\).

1. (a) Given that \(t = \tan \frac { X } { 2 }\) prove that

$$\cos x \equiv \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (b) Show that the equation $$3 \tan x - 10 \cos x = 10$$ can be written in the form $$( t + 2 ) \left( a t ^ { 2 } + b t + c \right) = 0$$ where \(t = \tan \frac { X } { 2 }\) and \(a , b\) and \(c\) are integers to be determined.
(c) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \tan x - 10 \cos x = 10$$

1. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\)

The point \(P\) on \(C\) has \(y\) coordinate \(p\), where \(p\) is a positive constant.

1. Show that an equation of the tangent to \(C\) at \(P\) is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
2. Write down the equation of the directrix of \(C\). The line \(l\) is the reflection of the tangent to \(C\) at \(P\) in the directrix of \(C\).
   Given that \(l\) passes through the focus of \(C\),
3. determine the exact value of \(p\).