Questions — Edexcel (9685 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 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
Sort by: Default | Easiest first | Hardest first
Edexcel AS Paper 1 2020 June Q14
9 marks Standard +0.8
  1. A curve has equation \(y = \mathrm { g } ( x )\).
Given that
  • \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
  • the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
  • the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
    1. find \(\mathrm { g } ( x )\),
    2. prove that the stationary point at \(( 2,9 )\) is a maximum.
Edexcel AS Paper 1 2022 June Q1
4 marks Easy -1.2
  1. Find
$$\int \left( 8 x ^ { 3 } - \frac { 3 } { 2 \sqrt { x } } + 5 \right) \mathrm { d } x$$ giving your answer in simplest form.
Edexcel AS Paper 1 2022 June Q2
7 marks Moderate -0.8
2. $$f ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + 2 x + 15$$
  1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the constants \(a\), \(b\) and \(c\) such that $$f ( x ) = ( x + 3 ) \left( a x ^ { 2 } + b x + c \right)$$
  3. Hence show that \(\mathrm { f } ( x ) = 0\) has only one real root.
  4. Write down the real root of the equation \(\mathrm { f } ( x - 5 ) = 0\)
Edexcel AS Paper 1 2022 June Q3
6 marks Moderate -0.8
  1. The triangle \(P Q R\) is such that \(\overrightarrow { P Q } = 3 \mathbf { i } + 5 \mathbf { j }\) and \(\overrightarrow { P R } = 13 \mathbf { i } - 15 \mathbf { j }\)
    1. Find \(\overrightarrow { Q R }\)
    2. Hence find \(| \overrightarrow { Q R } |\) giving your answer as a simplified surd.
    The point \(S\) lies on the line segment \(Q R\) so that \(Q S : S R = 3 : 2\)
  2. Find \(\overrightarrow { P S }\)
Edexcel AS Paper 1 2022 June Q4
6 marks Standard +0.3
4. Figure 1 Figure 1 shows a sketch of triangle \(A B C\) with \(A B = ( x + 2 ) \mathrm { cm } , B C = ( 3 x + 10 ) \mathrm { cm }\), \(A C = 7 x \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = \theta ^ { \circ }\)
    1. Show that \(17 x ^ { 2 } - 35 x - 48 = 0\)
    2. Hence find the value of \(x\).
  2. Hence find the value of \(\theta\) giving your answer to one decimal place.
Edexcel AS Paper 1 2022 June Q5
10 marks Moderate -0.8
  1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation
$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$
  1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
  2. With reference to the model, interpret
    1. the value of the constant \(p\),
    2. the value of the constant \(q\).
  3. Find, according to the model,
    1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
    2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
  4. State one reason why this may not be a realistic model in the long term.
Edexcel AS Paper 1 2022 June Q6
6 marks Standard +0.3
  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
7 marks Standard +0.8
  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
6 marks Moderate -0.3
  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
6 marks Moderate -0.3
  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 marks Standard +0.3
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
8 marks Moderate -0.3
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
12 marks Standard +0.3
  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
8 marks Standard +0.3
  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
4 marks Standard +0.3
  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
6 marks Moderate -0.8
  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
4 marks Moderate -0.8
  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
4 marks Moderate -0.3
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
    5 marks Easy -1.2
    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
    5 marks Moderate -0.8
    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
    5 marks Standard +0.3
    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
    8 marks Easy -1.2
    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
    5 marks Moderate -0.8
    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
    5 marks Moderate -0.3
    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
    8 marks Moderate -0.8
    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 \(\frac { p } { q }\)