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Edexcel AS Paper 1 2023 June Q1
  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.
Edexcel AS Paper 1 2023 June Q2
  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 $$6 x + 7 \sqrt { x } - 20 = 0$$
Edexcel AS Paper 1 2023 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce4f8375-0d88-4e48-85de-35f7e90b014d-06_478_513_283_776} \captionsetup{labelformat=empty} \caption{Figure 1}
\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.
    •
    Edexcel AS Paper 1 2023 June Q4
    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 $$\frac { 16 } { x } \leqslant 2$$
    Edexcel AS Paper 1 2023 June Q5
    1. In this question you must show all stages of your working.
    Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce4f8375-0d88-4e48-85de-35f7e90b014d-10_488_519_365_772} \captionsetup{labelformat=empty} \caption{Figure 2}
    \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.
    Edexcel AS Paper 1 2023 June Q6
    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 \(k\).
    Edexcel AS Paper 1 2023 June Q7
    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.
    Edexcel AS Paper 1 2023 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce4f8375-0d88-4e48-85de-35f7e90b014d-16_661_855_283_605} \captionsetup{labelformat=empty} \caption{Figure 3}
    \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.
    Edexcel AS Paper 1 2023 June Q9
    1. Using the laws of logarithms, solve the equation
    $$2 \log _ { 5 } ( 3 x - 2 ) - \log _ { 5 } x = 2$$
    Edexcel AS Paper 1 2023 June Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ce4f8375-0d88-4e48-85de-35f7e90b014d-20_643_767_276_648} \captionsetup{labelformat=empty} \caption{Figure 4}
    \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 \(\frac { p } { q }\)
    Edexcel AS Paper 1 2023 June Q11
    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.
    Edexcel AS Paper 1 2023 June Q12
    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.
    Edexcel AS Paper 1 2023 June Q13
    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 \(A O B\).
    •
    Edexcel AS Paper 1 2023 June Q14
    1. Find, in simplest form, the coefficient of \(x ^ { 5 }\) in the expansion of
    $$\left( 5 + 8 x ^ { 2 } \right) \left( 3 - \frac { 1 } { 2 } x \right) ^ { 6 }$$
    Edexcel AS Paper 1 2023 June Q15
    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 \(k\).
    Edexcel AS Paper 1 2023 June Q16
    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.
    Edexcel AS Paper 1 2023 June Q17
    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.
    Edexcel AS Paper 1 2024 June Q1
    1. Find
    $$\int \frac { 2 \sqrt { x } - 3 } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
    Edexcel AS Paper 1 2024 June Q2
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$ where \(a\) and \(b\) are constants.
    Given that ( \(x - 4\) ) is a factor of \(\mathrm { f } ( x )\),
    1. use the factor theorem to show that $$10 a = 32 + b$$ Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
    2. express \(\mathrm { f } ( x )\) in the form $$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$ where \(k\) is a constant to be found.
    3. Hence,
      1. state the number of real roots of the equation \(\mathrm { f } ( x ) = 0\)
      2. write down the largest root of the equation \(\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0\)
    Edexcel AS Paper 1 2024 June Q3
    1. Relative to a fixed origin \(O\),
    • point \(P\) has position vector \(9 \mathbf { i } - 8 \mathbf { j }\)
    • point \(Q\) has position vector \(3 \mathbf { i } - 5 \mathbf { j }\)
      1. Find \(\overrightarrow { P Q }\)
    Given that \(R\) is the point such that \(\overrightarrow { Q R } = 9 \mathbf { i } + 18 \mathbf { j }\)
  • show that angle \(P Q R = 90 ^ { \circ }\) Given also that \(S\) is the point such that \(\overrightarrow { P S } = 3 \overrightarrow { Q R }\)
  • find the exact area of \(P Q R S\)
    •
    Edexcel AS Paper 1 2024 June Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-10_547_1475_306_294} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of triangle \(A B D\) and triangle \(B C D\)
    Given that
    • \(A D C\) is a straight line
    • \(B D = ( x + 3 ) \mathrm { cm }\)
    • \(B C = x \mathrm {~cm}\)
    • angle \(B D C = 30 ^ { \circ }\)
    • angle \(B C D = 140 ^ { \circ }\)
      1. show that \(x = 10.5\) correct to 3 significant figures.
    Given also that \(A D = ( x - 2 ) \mathrm { cm }\)
  • find the length of \(A B\), giving your answer to 3 significant figures.
    •
    Edexcel AS Paper 1 2024 June Q5
    1. The curve \(C _ { 1 }\) has equation
    $$y = \frac { 6 } { x } + 3$$
      1. Sketch \(C _ { 1 }\) stating the coordinates of any points where the curve cuts the coordinate axes.
      2. State the equations of any asymptotes to the curve \(C _ { 1 }\) The curve \(C _ { 2 }\) has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
    2. Show that \(C _ { 1 }\) and \(C _ { 2 }\) intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the \(x\) coordinate of one of the points of intersection is \(- \frac { 2 } { 3 }\)
    3. use algebra to find the \(x\) coordinates of the other points of intersection between \(C _ { 1 }\) and \(C _ { 2 }\)
      (Solutions relying on calculator technology are not acceptable.)
    Edexcel AS Paper 1 2024 June Q6
    1. The binomial expansion of
    $$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.
    1. Show that \(a = - \frac { 5 } { 8 }\)
    2. Hence find the value of \(k\) Using the expansion and making your method clear,
    3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.
    Edexcel AS Paper 1 2024 June Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-18_614_878_296_555} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A chimney emits smoke particles.
    On a particular day, the concentration of smoke particles in the air emitted by this chimney, \(P\) parts per million, is measured at various distances, \(x \mathrm {~km}\), from the chimney. Figure 2 shows a sketch of the linear relationship between \(\log _ { 10 } P\) and \(x\) that is used to model this situation. The line passes through the point ( \(0,3.3\) ) and the point ( \(6,2.1\) )
    1. Find a complete equation for the model in the form $$P = a b ^ { x }$$ where \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) each to 4 significant figures.
    2. With reference to the model, interpret the value of \(a b\)
    Edexcel AS Paper 1 2024 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23689deb-7eed-4022-848f-1278231a4056-20_915_924_303_580} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of the curve \(C\) with equation $$y = x ^ { 3 } - 14 x + 23$$ The line \(l\) is the tangent to \(C\) at the point \(A\), also shown in Figure 3.
    Given that \(l\) has equation \(y = - 2 x + 7\)
    1. show, using calculus, that the \(x\) coordinate of \(A\) is 2 The line \(l\) cuts \(C\) again at the point \(B\).
    2. Verify that the \(x\) coordinate of \(B\) is - 4 The finite region, \(R\), shown shaded in Figure 3, is bounded by \(C\) and \(l\).
      Using algebraic integration,
    3. show that the area of \(R\) is 108