Questions — Edexcel (10514 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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 Paper 1 2024 June Q14
9 marks Moderate -0.3
  1. A balloon is being inflated.
In a simple model,
  • the balloon is modelled as a sphere
  • the rate of increase of the radius of the balloon is inversely proportional to the square root of the radius of the balloon
At time \(t\) seconds, the radius of the balloon is \(r \mathrm {~cm}\).
  1. Write down a differential equation to model this situation. At the instant when \(t = 10\)
    • the radius is 16 cm
    • the radius is increasing at a rate of \(0.9 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\)
    • Solve the differential equation to show that
    $$r ^ { \frac { 3 } { 2 } } = 5.4 t + 10$$
  2. Hence find the radius of the balloon when \(t = 20\) Give your answer to the nearest millimetre.
  3. Suggest a limitation of the model.
Edexcel Paper 1 2024 June Q15
6 marks Standard +0.3
  1. Show that \(k ^ { 2 } - 4 k + 5\) is positive for all real values of \(k\).
  2. A student was asked to prove by contradiction that "There are no positive integers \(x\) and \(y\) such that \(( 3 x + 2 y ) ( 2 x - 5 y ) = 28\) " The start of the student's proof is shown below. Assume that positive integers \(x\) and \(y\) exist such that $$\left. \begin{array} { c } ( 3 x + 2 y ) ( 2 x - 5 y ) = 28 \\ \text { If } 3 x + 2 y = 14 \text { and } 2 x - 5 y = 2 \\ 3 x + 2 y = 14 \\ 2 x - 5 y = 2 \end{array} \right\} \Rightarrow x = \frac { 74 } { 19 } , y = \frac { 22 } { 19 } \text { Not integers }$$ Show the calculations and statements needed to complete the proof.
Edexcel Paper 1 2020 October Q1
5 marks Moderate -0.8
  1. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + 8 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
  2. Explain how you could use \(x = \frac { 1 } { 32 }\) in the expansion to find an approximation for \(\sqrt { 5 }\) There is no need to carry out the calculation.
Edexcel Paper 1 2020 October Q2
3 marks Easy -1.2
  1. By taking logarithms of both sides, solve the equation
$$4 ^ { 3 p - 1 } = 5 ^ { 210 }$$ giving the value of \(p\) to one decimal place.
Edexcel Paper 1 2020 October Q3
4 marks Moderate -0.5
  1. Relative to a fixed origin \(O\)
  • point \(A\) has position vector \(2 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }\)
  • point \(B\) has position vector \(3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\)
  • point \(C\) has position vector \(2 \mathbf { i } - 16 \mathbf { j } + 4 \mathbf { k }\)
    1. Find \(\overrightarrow { A B }\)
    2. Show that quadrilateral \(O A B C\) is a trapezium, giving reasons for your answer.
Edexcel Paper 1 2020 October Q4
5 marks Moderate -0.3
  1. The function f is defined by
$$f ( x ) = \frac { 3 x - 7 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$
  1. Find \(f ^ { - 1 } ( 7 )\)
  2. Show that \(\operatorname { ff } ( x ) = \frac { a x + b } { x - 3 }\) where \(a\) and \(b\) are integers to be found.
    VI4V SIHI NI ILIUM ION OCVIAV SIHI NI III IM I O N OOVJAV SIHI NI III M M ION OC
Edexcel Paper 1 2020 October Q5
6 marks Moderate -0.8
  1. A car has six forward gears.
The fastest speed of the car
  • in \(1 ^ { \text {st } }\) gear is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
  • in \(6 ^ { \text {th } }\) gear is \(115 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,
  1. find the fastest speed of the car in \(3 { } ^ { \text {rd } }\) gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
  2. find the fastest speed of the car in \(5 ^ { \text {th } }\) gear.
Edexcel Paper 1 2020 October Q6
7 marks Standard +0.3
  1. Express \(\sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside a room on a given day is modelled by the equation $$\theta = 5 + \sin \left( \frac { \pi t } { 12 } - 3 \right) + 2 \cos \left( \frac { \pi t } { 12 } - 3 \right) \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
    Using the equation of the model and your answer to part (a),
  2. deduce the maximum temperature of the room during this day,
  3. find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.
Edexcel Paper 1 2020 October Q7
5 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-16_868_805_242_632} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\).
The curve \(C\) meets \(l\) at the points \(( - 2,13 )\) and \(( 0,25 )\) as shown.
The shaded region \(R\) is bounded by \(C\) and \(l\) as shown in Figure 1.
Given that
  • \(\mathrm { f } ( x )\) is a quadratic function in \(x\)
  • ( \(- 2,13\) ) is the minimum turning point of \(y = \mathrm { f } ( x )\) use inequalities to define \(R\).
Edexcel Paper 1 2020 October Q8
2 marks Easy -1.2
  1. A new smartphone was released by a company.
The company monitored the total number of phones sold, \(n\), at time \(t\) days after the phone was released. The company observed that, during this time,
the rate of increase of \(n\) was proportional to \(n\) Use this information to write down a suitable equation for \(n\) in terms of \(t\).
(You do not need to evaluate any unknown constants in your equation.)
Edexcel Paper 1 2020 October Q9
9 marks Standard +0.3
9.
\includegraphics[max width=\textwidth, alt={}]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-22_602_752_246_657}
\section*{Figure 2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 4 \left( x ^ { 2 } - 2 \right) \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R }$$
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 8 \left( 2 + x - x ^ { 2 } \right) \mathrm { e } ^ { - 2 x }\)
  2. Hence find, in simplest form, the exact coordinates of the stationary points of \(C\). The function g and the function h are defined by $$\begin{array} { l l } \mathrm { g } ( x ) = 2 \mathrm { f } ( x ) & x \in \mathbb { R } \\ \mathrm {~h} ( x ) = 2 \mathrm { f } ( x ) - 3 & x \geqslant 0 \end{array}$$
  3. Find (i) the range of \(g\) (ii) the range of h
Edexcel Paper 1 2020 October Q10
10 marks Standard +0.8
  1. Use the substitution \(x = u ^ { 2 } + 1\) to show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \int _ { p } ^ { q } \frac { 6 \mathrm {~d} u } { u ( 3 + 2 u ) }$$ where \(p\) and \(q\) are positive constants to be found.
  2. Hence, using algebraic integration, show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \ln a$$ where \(a\) is a rational constant to be found.
Edexcel Paper 1 2020 October Q11
8 marks Standard +0.8
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-30_738_837_242_614} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 100\) Circle \(C _ { 2 }\) has equation \(( x - 15 ) ^ { 2 } + y ^ { 2 } = 40\) The circles meet at points \(A\) and \(B\) as shown in Figure 3.
  1. Show that angle \(A O B = 0.635\) radians to 3 significant figures, where \(O\) is the origin. The region shown shaded in Figure 3 is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
  2. Find the perimeter of the shaded region, giving your answer to one decimal place.
Edexcel Paper 1 2020 October Q12
8 marks Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\operatorname { cosec } \theta - \sin \theta \equiv \cos \theta \cot \theta \quad \theta \neq ( 180 n ) ^ { \circ } \quad n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve for \(0 < x < 180 ^ { \circ }\) $$\operatorname { cosec } x - \sin x = \cos x \cot \left( 3 x - 50 ^ { \circ } \right)$$
Edexcel Paper 1 2020 October Q13
7 marks Challenging +1.8
  1. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = \frac { k \left( a _ { n } + 2 \right) } { a _ { n } } \quad n \in \mathbb { N }$$ where \(k\) is a constant.
Given that
  • the sequence is a periodic sequence of order 3
  • \(a _ { 1 } = 2\)
    1. show that
$$k ^ { 2 } + k - 2 = 0$$
  • For this sequence explain why \(k \neq 1\)
  • Find the value of $$\sum _ { r = 1 } ^ { 80 } a _ { r }$$
    •
    Edexcel Paper 1 2020 October Q14
    10 marks Standard +0.3
    1. A large spherical balloon is deflating.
    At time \(t\) seconds the balloon has radius \(r \mathrm {~cm}\) and volume \(V \mathrm {~cm} ^ { 3 }\) The volume of the balloon is modelled as decreasing at a constant rate.
    1. Using this model, show that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = - \frac { k } { r ^ { 2 } }$$ where \(k\) is a positive constant. Given that
      • the initial radius of the balloon is 40 cm
      • after 5 seconds the radius of the balloon is 20 cm
      • the volume of the balloon continues to decrease at a constant rate until the balloon is empty
      • solve the differential equation to find a complete equation linking \(r\) and \(t\).
      • Find the limitation on the values of \(t\) for which the equation in part (b) is valid.
    Edexcel Paper 1 2020 October Q15
    7 marks Challenging +1.2
    1. The curve \(C\) has equation
    $$x ^ { 2 } \tan y = 9 \quad 0 < y < \frac { \pi } { 2 }$$
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 18 x } { x ^ { 4 } + 81 }$$
    2. Prove that \(C\) has a point of inflection at \(x = \sqrt [ 4 ] { 27 }\)
    Edexcel Paper 1 2020 October Q16
    4 marks Standard +0.8
    1. Prove by contradiction that there are no positive integers \(p\) and \(q\) such that
    $$4 p ^ { 2 } - q ^ { 2 } = 25$$
    Edexcel Paper 1 2021 October Q1
    3 marks Easy -1.2
    1. $$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$ Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
    You must make your method clear.
    Edexcel Paper 1 2021 October Q2
    4 marks Easy -1.8
    1. Given that
    $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$
    1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
      • meets the \(y\)-axis at the point \(P\)
      • has a minimum turning point at the point \(Q\)
      • Write down
        1. the coordinates of \(P\)
        2. the coordinates of \(Q\)
    Edexcel Paper 1 2021 October Q3
    6 marks Standard +0.3
    1. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
    $$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where \(k\) is an integer.
    Given that \(u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0\)
    1. show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
    2. Find the value of \(k\), giving a reason for your answer.
    3. Find the value of \(u _ { 3 }\)
    Edexcel Paper 1 2021 October Q4
    9 marks Standard +0.3
    1. The curve with equation \(y = \mathrm { f } ( x )\) where
    $$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at \(x = \alpha\)
    1. Show that \(\alpha\) is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for \(\alpha\).
      Starting with \(x _ { 1 } = 0.3\)
    2. calculate, giving each answer to 4 decimal places,
      1. the value of \(x _ { 2 }\)
      2. the value of \(x _ { 4 }\) Using a suitable interval and a suitable function that should be stated,
    3. show that \(\alpha\) is 0.341 to 3 decimal places.
    Edexcel Paper 1 2021 October Q5
    6 marks Moderate -0.8
    1. In this question you should show all stages of your working.
    \section*{Solutions relying entirely on calculator technology are not acceptable.} A company made a profit of \(\pounds 20000\) in its first year of trading, Year 1
    A model for future trading predicts that the yearly profit will increase by \(8 \%\) each year, so that the yearly profits will form a geometric sequence. According to the model,
    1. show that the profit for Year 3 will be \(\pounds 23328\)
    2. find the first year when the yearly profit will exceed £65000
    3. find the total profit for the first 20 years of trading, giving your answer to the nearest £1000
    Edexcel Paper 1 2021 October Q6
    5 marks Moderate -0.3
    6. Figure 1 Figure 1 shows a sketch of triangle \(A B C\).
    Given that
    • \(\overrightarrow { A B } = - 3 \mathbf { i } - 4 \mathbf { j } - 5 \mathbf { k }\)
    • \(\overrightarrow { B C } = \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
      1. find \(\overrightarrow { A C }\)
      2. show that \(\cos A B C = \frac { 9 } { 10 }\)
    Edexcel Paper 1 2021 October Q7
    9 marks Standard +0.3
    1. The circle \(C\) has equation
    $$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$
    1. Find
      1. the coordinates of the centre of \(C\),
      2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
        Given that \(l\) is a tangent to \(C\),
    2. find the possible values of \(k\), giving your answers as simplified surds.