Edexcel FP1 AS (Further Pure 1 AS) 2023 June

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

$$\frac { 5 x } { x - 2 } \geqslant 12$$ (b) Hence, given that \(x\) is an integer, deduce the value of \(x\).

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

$$3 \cos x - 2 \sin x = 1$$ can be written in the form $$2 t ^ { 2 } + 2 t - 1 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \cos x - 2 \sin x = 1$$ giving your answers to one decimal place.

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The line \(l\) has equation \(x - 2 y = c\)
The points \(P\) and \(Q\) are the points of intersection of \(H\) and \(l\)

1. Determine, in terms of \(c\), the coordinates of \(P\) and the coordinates of \(Q\) The point \(R\) is the midpoint of \(P Q\)
2. Show that, as \(C\) varies, the coordinates of \(R\) satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where \(a\) is a constant to be determined.

1. A teacher made a cup of coffee. The temperature \(\theta ^ { \circ } \mathrm { C }\) of the coffee, \(t\) minutes after it was made, is modelled by the differential equation

$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } + 0.05 ( \theta - 20 ) = 0$$ Given that

• the initial temperature of the coffee was \(95 ^ { \circ } \mathrm { C }\)
• the coffee can only be safely drunk when its temperature is below \(70 ^ { \circ } \mathrm { C }\)
• the teacher made the cup of coffee at 1.15 pm
• the teacher needs to be able to start drinking the coffee by 1.20 pm
  use two iterations of the approximation formula

$$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$$ to estimate whether the teacher will be able to start drinking the coffee at 1.20 pm .

1. The points \(A , B\) and \(C\) are the vertices of a triangle.

Given that

• \(\overrightarrow { A B } = \left( \begin{array} { l } p
  4
  6 \end{array} \right)\) and \(\overrightarrow { A C } = \left( \begin{array} { l } q
  4
  5 \end{array} \right)\) where \(p\) and \(q\) are constants
• \(\overrightarrow { A B } \times \overrightarrow { A C }\) is parallel to \(2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\)
  1. determine the value of \(p\) and the value of \(q\)
  2. Hence, determine the exact area of triangle \(A B C\)

1. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.

The point \(P \left( a t ^ { 2 } , 2 a t \right) , t \neq 0\), lies on \(C\)
The normal to \(C\) at \(P\) is parallel to the line with equation \(y = 2 x\)

1. For the point \(P\), show that \(t = - 2\) The normal to \(C\) at \(P\) intersects \(C\) again when \(x = 9\)
2. Determine the value of \(a\), giving a reason for your answer.