Edexcel AS Paper 1 (AS Paper 1) 2023 June

1. A curve has equation

$$y = \frac { 2 } { 3 } x ^ { 3 } - \frac { 7 } { 2 } x ^ { 2 } - 4 x + 5$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) writing your answer in simplest form.
2. Hence find the range of values of \(x\) for which \(y\) is decreasing.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Using the substitution \(u = \sqrt { x }\) or otherwise, solve $$6 x + 7 \sqrt { x } - 20 = 0$$

3. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing the position of three phone masts, \(A , B\) and \(C\).
The masts are identical and their bases are assumed to lie in the same horizontal plane.
From mast \(C\)

• mast \(A\) is 8.2 km away on a bearing of \(072 ^ { \circ }\)
• mast \(B\) is 15.6 km away on a bearing of \(039 ^ { \circ }\)
  1. Find the distance between masts \(A\) and \(B\), giving your answer in km to one decimal place.

An engineer needs to travel from mast \(A\) to mast \(B\).

Give a reason why the answer to part (a) is unlikely to be an accurate value for the distance the engineer travels.

1. (a) Sketch the curve with equation

$$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The finite region \(R\), shown shaded in Figure 2, is bounded by the curve with equation \(y = 4 x ^ { 2 } + 3\), the \(y\)-axis and the line with equation \(y = 23\) Show that the exact area of \(R\) is \(k \sqrt { 5 }\) where \(k\) is a rational constant to be found.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\). Given that \(C\) does not cut or touch the \(x\)-axis,
2. find the range of possible values for \(k\).

1. The distance a particular car can travel in a journey starting with a full tank of fuel was investigated.

• From a full tank of fuel, 40 litres remained in the car's fuel tank after the car had travelled 80 km
• From a full tank of fuel, 25 litres remained in the car's fuel tank after the car had travelled 200 km

Using a linear model, with \(V\) litres being the volume of fuel remaining in the car's fuel tank and \(d \mathrm {~km}\) being the distance the car had travelled,

1. find an equation linking \(V\) with \(d\). Given that, on a particular journey
   • the fuel tank of the car was initially full
   • the car continued until it ran out of fuel
     find, according to the model,
   • 1. the initial volume of fuel that was in the fuel tank of the car,
     2. the distance that the car travelled on this journey.
   In fact the car travelled 320 km on this journey.
2. Evaluate the model in light of this information.

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\).
Given that

• \(C\) has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression in \(x\)
• \(C\) cuts the \(x\)-axis at 0 and 6
• \(l\) cuts the \(y\)-axis at 60 and intersects \(C\) at the point \(( 10,80 )\)
  use inequalities to define the region \(R\) shown shaded in Figure 3.

1. Using the laws of logarithms, solve the equation

$$2 \log _ { 5 } ( 3 x - 2 ) - \log _ { 5 } x = 2$$

10. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(y = \frac { 3 } { 5 } x + 6\)
The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(B ( 8,0 )\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l _ { 2 }\) is $$5 x + 3 y = 40$$ Given that
   • lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\)
   • line \(l _ { 1 }\) crosses the \(x\)-axis at the point \(A\)
   • find the exact area of triangle \(A B C\), giving your answer as a fully simplified fraction in the form \(\frac { p } { q }\)

1. The height, \(h\) metres, of a plant, \(t\) years after it was first measured, is modelled by the equation

$$h = 2.3 - 1.7 \mathrm { e } ^ { - 0.2 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ Using the model,

1. find the height of the plant when it was first measured,
2. show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year. According to the model, there is a limit to the height to which this plant can grow.
3. Deduce the value of this limit.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$4 \tan x = 5 \cos x$$ can be written as $$5 \sin ^ { 2 } x + 4 \sin x - 5 = 0$$
2. Hence solve, for \(0 < x \leqslant 360 ^ { \circ }\) $$4 \tan x = 5 \cos x$$ giving your answers to one decimal place.
3. Hence find the number of solutions of the equation $$4 \tan 3 x = 5 \cos 3 x$$ in the interval \(0 < x \leqslant 1800 ^ { \circ }\), explaining briefly the reason for your answer.

1. Relative to a fixed origin \(O\)

• point \(A\) has position vector \(10 \mathbf { i } - 3 \mathbf { j }\)
• point \(B\) has position vector \(- 8 \mathbf { i } + 9 \mathbf { j }\)
• point \(C\) has position vector \(- 2 \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant
  1. Find \(\overrightarrow { A B }\)
  2. Find \(| \overrightarrow { A B } |\) giving your answer as a fully simplified surd.

Given that points \(A , B\) and \(C\) lie on a straight line,

1. find the value of \(p\),
2. state the ratio of the area of triangle \(A O C\) to the area of triangle \(A O B\).

1. Find, in simplest form, the coefficient of \(x ^ { 5 }\) in the expansion of

$$\left( 5 + 8 x ^ { 2 } \right) \left( 3 - \frac { 1 } { 2 } x \right) ^ { 6 }$$

1. In this question you must show detailed reasoning.

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation \(y = 8 - 10 x + 6 x ^ { 2 } - x ^ { 3 }\)
The curve \(C _ { 2 }\) has equation \(y = x ^ { 2 } - 12 x + 14\)

1. Verify that when \(x = 1\) the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect. The curves also intersect when \(x = k\).
   Given that \(k < 0\)
2. use algebra to find the exact value of \(k\).

1. A curve has equation \(y = \mathrm { f } ( x ) , x \geqslant 0\)

Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 4 x + a \sqrt { x } + b\), where \(a\) and \(b\) are constants
• the curve has a stationary point at \(( 4,3 )\)
• the curve meets the \(y\)-axis at - 5
  find \(\mathrm { f } ( x )\), giving your answer in simplest form.

1. In this question \(p\) and \(q\) are positive integers with \(q > p\)

Statement 1: \(q ^ { 3 } - p ^ { 3 }\) is never a multiple of 5

1. Show, by means of a counter example, that Statement 1 is not true. Statement 2: When \(p\) and \(q\) are consecutive even integers \(q ^ { 3 } - p ^ { 3 }\) is a multiple of 8
2. Prove, using algebra, that Statement 2 is true.