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 P4 2023 January Q2
6 marks Standard +0.3
  1. A set of points \(P ( x , y )\) is defined by the parametric equations
$$x = \frac { t - 1 } { 2 t + 1 } \quad y = \frac { 6 } { 2 t + 1 } \quad t \neq - \frac { 1 } { 2 }$$
  1. Show that all points \(P ( x , y )\) lie on a straight line.
  2. Hence or otherwise, find the \(x\) coordinate of the point of intersection of this line and the line with equation \(y = x + 12\)
Edexcel P4 2023 January Q3
5 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-08_419_665_255_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} Figure 1 shows a sketch of the curve with equation $$y = \sqrt { \frac { 3 x } { 3 x ^ { 2 } + 5 } } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = \sqrt { 5 }\) and \(x = 5\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Use integration to find the exact volume of the solid generated. Give your answer in the form \(a \ln b\), where \(a\) is an irrational number and \(b\) is a prime number.
Edexcel P4 2023 January Q4
9 marks Standard +0.8
  1. (a) Using the substitution \(u = \sqrt { 2 x + 1 }\), show that
$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } \mathrm { e } ^ { \sqrt { 2 x + 1 } } \mathrm {~d} x$$ may be expressed in the form $$\int _ { a } ^ { b } k u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u$$ where \(a\), \(b\) and \(k\) are constants to be found.
(b) Hence find, by algebraic integration, the exact value of $$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } e ^ { \sqrt { 2 x + 1 } } d x$$ giving your answer in simplest form.
Edexcel P4 2023 January Q5
7 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-14_940_881_251_593} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation $$y ^ { 2 } = 2 x ^ { 2 } + 15 x + 10 y$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve is not defined for values of \(x\) in the interval ( \(p , q\) ), as shown in Figure 2.
  2. Using your answer to part (a) or otherwise, find the value of \(p\) and the value of \(q\).
    (Solutions relying entirely on calculator technology are not acceptable.)
Edexcel P4 2023 January Q6
8 marks Standard +0.3
  1. Relative to a fixed origin \(O\).
  • the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
  • the point \(B\) has position vector \(8 \mathbf { i } + 3 \mathbf { j } - 7 \mathbf { k }\)
The line \(l\) passes through \(A\) and \(B\).
    1. Find \(\overrightarrow { A B }\)
    2. Find a vector equation for the line \(l\) The point \(C\) has position vector \(3 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k }\) The point \(P\) lies on \(l\) Given that \(\overrightarrow { C P }\) is perpendicular to \(l\)
  2. find the position vector of the point \(P\)
Edexcel P4 2023 January Q7
12 marks Standard +0.3
  1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a spherical balloon with radius \(r \mathrm {~cm}\) is given by the formula
$$V = \frac { 4 } { 3 } \pi r ^ { 3 }$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) giving your answer in simplest form. At time \(t\) seconds, the volume of the balloon is increasing according to the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 900 } { ( 2 t + 3 ) ^ { 2 } } \quad t \geqslant 0$$ Given that \(V = 0\) when \(t = 0\)
    1. solve this differential equation to show that $$V = \frac { 300 t } { 2 t + 3 }$$
    2. Hence find the upper limit to the volume of the balloon.
  3. Find the radius of the balloon at \(t = 3\), giving your answer in cm to 3 significant figures.
  4. Find the rate of increase of the radius of the balloon at \(t = 3\), giving your answer to 2 significant figures. Show your working and state the units of your answer.
Edexcel P4 2023 January Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-26_582_773_255_648} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} A curve \(C\) has parametric equations $$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) with parameter \(t = \frac { \pi } { 4 }\) lies on \(C\).
The line \(l\) is the normal to \(C\) at \(P\), as shown in Figure 3.
  1. Show, using calculus, that an equation for \(l\) is $$8 y + 2 x = 17$$ The region \(S\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
  2. Find, using calculus, the exact area of \(S\).
Edexcel P4 2023 January Q9
8 marks Challenging +1.2
  1. A student was asked to prove, for \(p \in \mathbb { N }\), that
    "if \(p ^ { 3 }\) is a multiple of 3 , then \(p\) must be a multiple of 3 "
The start of the student's proof by contradiction is shown in the box below. Assumption:
There exists a number \(p , p \in \mathbb { N }\), such that \(p ^ { 3 }\) is a multiple of 3 , and \(p\) is NOT a multiple of 3 Let \(p = 3 k + 1 , k \in \mathbb { N }\). $$\text { Consider } \begin{aligned} p ^ { 3 } = ( 3 k + 1 ) ^ { 3 } & = 27 k ^ { 3 } + 27 k ^ { 2 } + 9 k + 1 \\ & = 3 \left( 9 k ^ { 3 } + 9 k ^ { 2 } + 3 k \right) + 1 \quad \text { which is not a multiple of } 3 \end{aligned}$$
  1. Show the calculations and statements that are required to complete the proof.
  2. Hence prove, by contradiction, that \(\sqrt [ 3 ] { 3 }\) is an irrational number.
Edexcel P4 2024 January Q1
4 marks Moderate -0.8
  1. Find, in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), the binomial expansion of
$$( 1 - 4 x ) ^ { - 3 } \quad | x | < \frac { 1 } { 4 }$$ fully simplifying each term.
Edexcel P4 2024 January Q2
10 marks Standard +0.3
  1. Given that
$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form \(p \ln q + r\) where \(p\), \(q\) and \(r\) are rational numbers.
Edexcel P4 2024 January Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-08_815_849_248_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has equation $$y ^ { 2 } x + 3 y = 4 x ^ { 2 } + k \quad y > 0$$ where \(k\) is a constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) The point \(P ( p , 2 )\), where \(p\) is a constant, lies on \(C\).
    Given that \(P\) is the minimum turning point on \(C\),
  2. find
    1. the value of \(p\)
    2. the value of \(k\)
Edexcel P4 2024 January Q4
5 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-12_595_588_248_740} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A cone, shown in Figure 2, has
  • fixed height 5 cm
  • base radius \(r \mathrm {~cm}\)
  • slant height \(l \mathrm {~cm}\)
    1. Find an expression for \(l\) in terms of \(r\)
Given that the base radius is increasing at a constant rate of 3 cm per minute,
  • find the rate at which the total surface area of the cone is changing when the radius of the cone is 1.5 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per minute to one decimal place.
    [0pt] [The total surface area, \(S\), of a cone is given by the formula \(S = \pi r ^ { 2 } + \pi r l\) ]
    •
    Edexcel P4 2024 January Q5
    9 marks Standard +0.3
    1. (a) Find \(\int x ^ { 2 } \cos 2 x d x\) (b) Hence solve the differential equation
    $$\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( \frac { t \cos t } { y } \right) ^ { 2 }$$ giving your answer in the form \(y ^ { n } = \mathrm { f } ( t )\) where \(n\) is an integer.
    Edexcel P4 2024 January Q6
    14 marks Standard +0.3
    1. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
    $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 3 \mathbf { i } + p \mathbf { j } + 7 \mathbf { k } ) + \lambda ( 2 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 8 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) + \mu ( 4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect,
    1. find the value of \(p\),
    2. find the position vector of the point of intersection.
    3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in degrees to one decimal place. The point \(A\) lies on \(l _ { 1 }\) with parameter \(\lambda = 2\) The point \(B\) lies on \(l _ { 2 }\) with \(\overrightarrow { A B }\) perpendicular to \(l _ { 2 }\)
    4. Find the coordinates of \(B\)
    Edexcel P4 2024 January Q7
    8 marks
    1. (a) Using the substitution \(u = 4 x + 2 \sin 2 x\), or otherwise, show that
    $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 4 x + 2 \sin 2 x } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 8 } \left( \mathrm { e } ^ { 2 \pi } - 1 \right)$$ Figure 3 The curve shown in Figure 3, has equation $$y = 6 \mathrm { e } ^ { 2 x + \sin 2 x } \cos x$$ The region \(R\), shown shaded in Figure 3, is bounded by the positive \(x\)-axis, the positive \(y\)-axis and the curve. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
    (b) Use the answer to part (a) to find the volume of the solid formed, giving the answer in simplest form.
    Edexcel P4 2024 January Q8
    4 marks Standard +0.8
    1. Use proof by contradiction to prove that the curve with equation
    $$y = 2 x + x ^ { 3 } + \cos x$$ has no stationary points.
    Edexcel P4 2024 January Q9
    12 marks Standard +0.8
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-28_597_1020_251_525} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \sec t \quad y = \sqrt { 3 } \tan \left( t + \frac { \pi } { 3 } \right) \quad \frac { \pi } { 6 } < t < \frac { \pi } { 2 }$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\)
    2. Find an equation for the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    3. Show that all points on \(C\) satisfy the equation $$y = \frac { A x ^ { 2 } + B \sqrt { 3 x ^ { 2 } - 3 } } { 4 - 3 x ^ { 2 } }$$ where \(A\) and \(B\) are constants to be found.
    Edexcel C4 Q5
    Standard +0.3
    5. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a0bd937d-b92e-41d0-abfa-ec83ccda058a-007_586_1079_260_427}
    \end{figure} Figure 1 shows the graph of the curve with equation $$y = x \mathrm { e } ^ { 2 x } , \quad x \geqslant 0$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
    1. Use integration to find the exact value for the area of \(R\).
    2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and 0.8 .
      \(x\)00.20.40.60.81
      \(y = x \mathrm { e } ^ { 2 x }\)00.298361.992077.38906
    3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures.
    Edexcel C4 2006 January Q1
    7 marks Standard +0.3
    1. A curve \(C\) is described by the equation
    $$3 x ^ { 2 } + 4 y ^ { 2 } - 2 x + 6 x y - 5 = 0$$ Find an equation of the tangent to \(C\) at the point \(( 1 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    Edexcel C4 2006 January Q2
    7 marks Moderate -0.3
    2. (a) Given that \(y = \sec x\), complete the table with the values of \(y\) corresponding to \(x = \frac { \pi } { 16 } , \frac { \pi } { 8 }\) and \(\frac { \pi } { 4 }\).
    \(x\)0\(\frac { \pi } { 16 }\)\(\frac { \pi } { 8 }\)\(\frac { 3 \pi } { 16 }\)\(\frac { \pi } { 4 }\)
    \(y\)11.20269
    (b) Use the trapezium rule, with all the values for \(y\) in the completed table, to obtain an estimate for \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x\). Show all the steps of your working, and give your answer to 4 decimal places. The exact value of \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec x \mathrm {~d} x\) is \(\ln ( 1 + \sqrt { } 2 )\).
    (c) Calculate the \% error in using the estimate you obtained in part (b).
    Edexcel C4 2006 January Q3
    8 marks Standard +0.3
    3. Using the substitution \(u ^ { 2 } = 2 x - 1\), or otherwise, find the exact value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { ( 2 x - 1 ) } } \mathrm { d } x$$ (8)
    (8)
    Edexcel C4 2006 January Q4
    8 marks Standard +0.8
    4. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9bf05d7e-7bb9-40f6-b626-69a8a6eda5a5-05_556_723_299_632}
    \end{figure} Figure 1 shows the finite shaded region, \(R\), which is bounded by the curve \(y = x \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis. The region \(R\) is rotated through 360 degrees about the \(x\)-axis.
    Use integration by parts to find an exact value for the volume of the solid generated.
    (8)
    Edexcel C4 2006 January Q5
    11 marks Standard +0.3
    5. $$f ( x ) = \frac { 3 x ^ { 2 } + 16 } { ( 1 - 3 x ) ( 2 + x ) ^ { 2 } } = \frac { A } { ( 1 - 3 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 3 } .$$
    1. Find the values of \(A\) and \(C\) and show that \(B = 0\).
    2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Simplify each term.
    Edexcel C4 2006 January Q6
    10 marks Standard +0.3
    6. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = 8 \mathbf { i } + 12 \mathbf { j } + 14 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) ,$$ where \(\lambda\) is a parameter. The point \(A\) has coordinates (4, 8, a), where \(a\) is a constant. The point \(B\) has coordinates ( \(b , 13,13\) ), where \(b\) is a constant. Points \(A\) and \(B\) lie on the line \(l _ { 1 }\).
    1. Find the values of \(a\) and \(b\). Given that the point \(O\) is the origin, and that the point \(P\) lies on \(l _ { 1 }\) such that \(O P\) is perpendicular to \(l _ { 1 }\),
    2. find the coordinates of \(P\).
    3. Hence find the distance \(O P\), giving your answer as a simplified surd.
    Edexcel C4 2006 January Q7
    12 marks Standard +0.3
    7. The volume of a spherical balloon of radius \(r \mathrm {~cm}\) is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).
    1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\). The volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1000 } { ( 2 t + 1 ) ^ { 2 } } , \quad t \geqslant 0$$
    2. Using the chain rule, or otherwise, find an expression in terms of \(r\) and \(t\) for \(\frac { \mathrm { d } r } { \mathrm {~d} t }\).
    3. Given that \(V = 0\) when \(t = 0\), solve the differential equation \(\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1000 } { ( 2 t + 1 ) ^ { 2 } }\), to obtain \(V\) in terms of \(t\).
    4. Hence, at time \(t = 5\),
      1. find the radius of the balloon, giving your answer to 3 significant figures,
      2. show that the rate of increase of the radius of the balloon is approximately \(2.90 \times 10 ^ { - 2 } \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).