Questions — Edexcel AS Paper 1 (150 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel AS Paper 1 2022 June Q6
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 3 - \frac { 2 x } { 9 } \right) ^ { 8 }$$ giving each term in simplest form. $$f ( x ) = \left( \frac { x - 1 } { 2 x } \right) \left( 3 - \frac { 2 x } { 9 } \right) ^ { 8 }$$ (b) Find the coefficient of \(x ^ { 2 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.
Edexcel AS Paper 1 2022 June Q7
  1. (a) Factorise completely \(9 x - x ^ { 3 }\)
The curve \(C\) has equation $$y = 9 x - x ^ { 3 }$$ (b) Sketch \(C\) showing the coordinates of the points at which the curve cuts the \(x\)-axis. The line \(l\) has equation \(y = k\) where \(k\) is a constant.
Given that \(C\) and \(l\) intersect at 3 distinct points,
(c) find the range of values for \(k\), writing your answer in set notation. Solutions relying on calculator technology are not acceptable.
Edexcel AS Paper 1 2022 June Q8
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} The air pressure, \(P \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), inside a car tyre, \(t\) minutes from the instant when the tyre developed a puncture is given by the equation $$P = k + 1.4 \mathrm { e } ^ { - 0.5 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ where \(k\) is a constant.
Given that the initial air pressure inside the tyre was \(2.2 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)
  1. state the value of \(k\). From the instant when the tyre developed the puncture,
  2. find the time taken for the air pressure to fall to \(1 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\) Give your answer in minutes to one decimal place.
  3. Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture.
    Give your answer in \(\mathrm { kg } / \mathrm { cm } ^ { 2 }\) per minute to 3 significant figures.
Edexcel AS Paper 1 2022 June Q9
  1. (a) Given that \(p = \log _ { 3 } x\), where \(x > 0\), find in simplest form in terms of \(p\),
    1. \(\log _ { 3 } \left( \frac { x } { 9 } \right)\)
    2. \(\log _ { 3 } ( \sqrt { x } )\)
      (b) Hence, or otherwise, solve
    $$2 \log _ { 3 } \left( \frac { x } { 9 } \right) + 3 \log _ { 3 } ( \sqrt { x } ) = - 11$$ giving your answer as a simplified fraction. Solutions relying on calculator technology are not acceptable.
Edexcel AS Paper 1 2022 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d31369fa-9532-4a09-b67d-a3a3cbf7d586-30_639_878_246_596} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 3 } x ^ { 2 } - 2 \sqrt { x } + 3 \quad x \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 4
The line \(l\) is the tangent to \(C\) at \(P\).
  1. Show that \(l\) has equation $$13 x - 6 y - 26 = 0$$ The region \(R\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the curve \(C\), the line \(l\) and the \(x\)-axis.
  2. Find the exact area of \(R\).
Edexcel AS Paper 1 2022 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d31369fa-9532-4a09-b67d-a3a3cbf7d586-34_833_1033_248_516} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 32 = 0$$ and the line \(l\) with equation $$2 y + x + 6 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\),
    2. the radius of \(C\).
  2. Find the shortest distance between \(C\) and \(l\).
Edexcel AS Paper 1 2022 June Q12
  1. A company makes drinks containers out of metal.
The containers are modelled as closed cylinders with base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\) and the capacity of each container is \(355 \mathrm {~cm} ^ { 3 }\) The metal used
  • for the circular base and the curved side costs 0.04 pence/ \(\mathrm { cm } ^ { 2 }\)
  • for the circular top costs 0.09 pence/ \(\mathrm { cm } ^ { 2 }\)
Both metals used are of negligible thickness.
  1. Show that the total cost, \(C\) pence, of the metal for one container is given by $$C = 0.13 \pi r ^ { 2 } + \frac { 28.4 } { r }$$
  2. Use calculus to find the value of \(r\) for which \(C\) is a minimum, giving your answer to 3 significant figures.
  3. Using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} r ^ { 2 } }\) prove that the cost is minimised for the value of \(r\) found in part (b).
  4. Hence find the minimum value of \(C\), giving your answer to the nearest integer.
Edexcel AS Paper 1 2022 June Q13
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\frac { 1 } { \cos \theta } + \tan \theta \equiv \frac { \cos \theta } { 1 - \sin \theta } \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } \quad n \in \mathbb { Z }$$ Given that \(\cos 2 x \neq 0\)
  2. solve for \(0 < x < 90 ^ { \circ }\) $$\frac { 1 } { \cos 2 x } + \tan 2 x = 3 \cos 2 x$$ giving your answers to one decimal place.
Edexcel AS Paper 1 2022 June Q14
  1. (i) A student states
    "if \(x ^ { 2 }\) is greater than 9 then \(x\) must be greater than 3 "
Determine whether or not this statement is true, giving a reason for your answer.
(ii) Prove that for all positive integers \(n\), $$n ^ { 3 } + 3 n ^ { 2 } + 2 n$$ is divisible by 6
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.