Questions — Edexcel (10514 questions)

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Edexcel Paper 1 2019 June Q9
5 marks Standard +0.3
  1. Given that \(a > b > 0\) and that \(a\) and \(b\) satisfy the equation
$$\log a - \log b = \log ( a - b )$$
  1. show that $$a = \frac { b ^ { 2 } } { b - 1 }$$ (3)
  2. Write down the full restriction on the value of \(b\), explaining the reason for this restriction.
Edexcel Paper 1 2019 June Q10
6 marks Standard +0.3
  1. Prove that for all \(n \in \mathbb { N } , n ^ { 2 } + 2\) is not divisible by 4
  2. "Given \(x \in \mathbb { R }\), the value of \(| 3 x - 28 |\) is greater than or equal to the value of ( \(x - 9\) )." State, giving a reason, if the above statement is always true, sometimes true or never true.
    (2)
Edexcel Paper 1 2019 June Q11
7 marks Moderate -0.3
  1. A competitor is running a 20 kilometre race.
She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be \(5 \%\) greater than the time that she took to complete the previous kilometre. Using the model,
  1. show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
  2. show that her estimated time, in minutes, to run the \(r\) th kilometre, for \(5 \leqslant r \leqslant 20\), is $$6 \times 1.05 ^ { r - 4 }$$
  3. estimate the total time, in minutes and seconds, that she will take to complete the race.
Edexcel Paper 1 2019 June Q12
10 marks Standard +0.3
12. $$\mathrm { f } ( x ) = 10 \mathrm { e } ^ { - 0.25 x } \sin x , \quad x \geqslant 0$$
  1. Show that the \(x\) coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\) satisfy the equation \(\tan x = 4\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-34_687_1029_495_518} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
  2. Sketch the graph of \(H\) against \(t\) where $$\mathrm { H } ( t ) = \left| 10 \mathrm { e } ^ { - 0.25 t } \sin t \right| \quad t \geqslant 0$$ showing the long-term behaviour of this curve. The function \(\mathrm { H } ( t )\) is used to model the height, in metres, of a ball above the ground \(t\) seconds after it has been kicked. Using this model, find
  3. the maximum height of the ball above the ground between the first and second bounce.
  4. Explain why this model should not be used to predict the time of each bounce.
Edexcel Paper 1 2019 June Q13
11 marks Standard +0.3
  1. The curve \(C\) with equation
$$y = \frac { p - 3 x } { ( 2 x - q ) ( x + 3 ) } \quad x \in \mathbb { R } , x \neq - 3 , x \neq 2$$ where \(p\) and \(q\) are constants, passes through the point \(\left( 3 , \frac { 1 } { 2 } \right)\) and has two vertical asymptotes
with equations \(x = 2\) and \(x = - 3\) with equations \(x = 2\) and \(x = - 3\)
    1. Explain why you can deduce that \(q = 4\)
    2. Show that \(p = 15\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-38_616_889_842_587} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 shows a sketch of part of the curve \(C\). The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
  2. Show that the exact value of the area of \(R\) is \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are rational constants to be found.
Edexcel Paper 1 2019 June Q14
7 marks Standard +0.3
  1. The curve \(C\), in the standard Cartesian plane, is defined by the equation
$$x = 4 \sin 2 y \quad \frac { - \pi } { 4 } < y < \frac { \pi } { 4 }$$ The curve \(C\) passes through the origin \(O\)
  1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the origin.
    1. Use the small angle approximation for \(\sin 2 y\) to find an equation linking \(x\) and \(y\) for points close to the origin.
    2. Explain the relationship between the answers to (a) and (b)(i).
  3. Show that, for all points \(( x , y )\) lying on \(C\), $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { a \sqrt { b - x ^ { 2 } } }$$ where \(a\) and \(b\) are constants to be found.
Edexcel Paper 1 2022 June Q1
4 marks Moderate -0.8
  1. The point \(P ( - 2 , - 5 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\)
Find the point to which \(P\) is mapped, when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation
  1. \(y = f ( x ) + 2\)
  2. \(y = | f ( x ) |\)
  3. \(y = 3 f ( x - 2 ) + 2\)
Edexcel Paper 1 2022 June Q2
3 marks Moderate -0.8
  1. \(\mathrm { f } ( x ) = ( x - 4 ) \left( x ^ { 2 } - 3 x + k \right) - 42\) where \(k\) is a constant Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(k\).
Edexcel Paper 1 2022 June Q3
5 marks Moderate -0.8
  1. A circle has equation
$$x ^ { 2 } + y ^ { 2 } - 10 x + 16 y = 80$$
  1. Find
    1. the coordinates of the centre of the circle,
    2. the radius of the circle. Given that \(P\) is the point on the circle that is furthest away from the origin \(O\),
  2. find the exact length \(O P\)
Edexcel Paper 1 2022 June Q4
3 marks Moderate -0.8
  1. Express \(\lim _ { \delta x \rightarrow 0 } \sum _ { x = 2.1 } ^ { 6.3 } \frac { 2 } { x } \delta x\) as an integral.
  2. Hence show that $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 2.1 } ^ { 6.3 } \frac { 2 } { x } \delta x = \ln k$$ where \(k\) is a constant to be found.
Edexcel Paper 1 2022 June Q5
6 marks Moderate -0.8
  1. The height, \(h\) metres, of a tree, \(t\) years after being planted, is modelled by the equation
$$h ^ { 2 } = a t + b \quad 0 \leqslant t < 25$$ where \(a\) and \(b\) are constants.
Given that
  • the height of the tree was 2.60 m , exactly 2 years after being planted
  • the height of the tree was 5.10 m , exactly 10 years after being planted
    1. find a complete equation for the model, giving the values of \(a\) and \(b\) to 3 significant figures.
Given that the height of the tree was 7 m , exactly 20 years after being planted
  • evaluate the model, giving reasons for your answer.
    •
    Edexcel Paper 1 2022 June Q6
    6 marks Moderate -0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-12_528_812_251_628} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a cubic expression in \(X\). The curve
    • passes through the origin
    • has a maximum turning point at \(( 2,8 )\)
    • has a minimum turning point at \(( 6,0 )\)
      1. Write down the set of values of \(x\) for which
    $$\mathrm { f } ^ { \prime } ( x ) < 0$$ The line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at only one point.
  • Find the set of values of \(k\), giving your answer in set notation.
  • Find the equation of \(C\). You may leave your answer in factorised form.
    •
    Edexcel Paper 1 2022 June Q7
    5 marks Standard +0.8
    1. Given that \(p\) and \(q\) are integers such that use algebra to prove by contradiction that at least one of \(p\) or \(q\) is even.
    2. Given that \(x\) and \(y\) are integers such that
    Edexcel Paper 1 2022 June Q8
    8 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-16_522_673_248_696} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A car stops at two sets of traffic lights.
    Figure 2 shows a graph of the speed of the car, \(v \mathrm {~ms} ^ { - 1 }\), as it travels between the two sets of traffic lights. The car takes \(T\) seconds to travel between the two sets of traffic lights.
    The speed of the car is modelled by the equation $$v = ( 10 - 0.4 t ) \ln ( t + 1 ) \quad 0 \leqslant t \leqslant T$$ where \(t\) seconds is the time after the car leaves the first set of traffic lights.
    According to the model,
    1. find the value of \(T\)
    2. show that the maximum speed of the car occurs when $$t = \frac { 26 } { 1 + \ln ( t + 1 ) } - 1$$ Using the iteration formula $$t _ { n + 1 } = \frac { 26 } { 1 + \ln \left( t _ { n } + 1 \right) } - 1$$ with \(t _ { 1 } = 7\)
      1. find the value of \(t _ { 3 }\) to 3 decimal places,
      2. find, by repeated iteration, the time taken for the car to reach maximum speed.
    Edexcel Paper 1 2022 June Q9
    6 marks Standard +0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-20_406_515_246_776} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of a parallelogram \(P Q R S\).
    Given that
    • \(\overrightarrow { P Q } = 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\)
    • \(\overrightarrow { Q R } = 5 \mathbf { i } - 2 \mathbf { k }\)
      1. show that parallelogram \(P Q R S\) is a rhombus.
      2. Find the exact area of the rhombus \(P Q R S\).
    Edexcel Paper 1 2022 June Q10
    8 marks Moderate -0.3
    1. A scientist is studying the number of bees and the number of wasps on an island.
    The number of bees, measured in thousands, \(N _ { b }\), is modelled by the equation $$N _ { b } = 45 + 220 \mathrm { e } ^ { 0.05 t }$$ where \(t\) is the number of years from the start of the study.
    According to the model,
    1. find the number of bees at the start of the study,
    2. show that, exactly 10 years after the start of the study, the number of bees was increasing at a rate of approximately 18 thousand per year. The number of wasps, measured in thousands, \(N _ { w }\), is modelled by the equation $$N _ { w } = 10 + 800 \mathrm { e } ^ { - 0.05 t }$$ where \(t\) is the number of years from the start of the study.
      When \(t = T\), according to the models, there are an equal number of bees and wasps.
    3. Find the value of \(T\) to 2 decimal places.
    Edexcel Paper 1 2022 June Q11
    7 marks Standard +0.3
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-28_647_855_244_605} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 2 x ^ { 3 } + 10 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 42 x - 15 x ^ { 2 } - 7 \quad x > 0$$
    1. Verify that the curves intersect at \(x = \frac { 1 } { 2 }\) The curves intersect again at the point \(P\)
    2. Using algebra and showing all stages of working, find the exact \(x\) coordinate of \(P\)
    Edexcel Paper 1 2022 June Q12
    5 marks Standard +0.3
    1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    Show that $$\int _ { 1 } ^ { \mathrm { e } ^ { 2 } } x ^ { 3 } \ln x \mathrm {~d} x = a \mathrm { e } ^ { 8 } + b$$ where \(a\) and \(b\) are rational constants to be found.
    Edexcel Paper 1 2022 June Q13
    7 marks Easy -1.2
    1. In an arithmetic series, the first term is \(a\) and the common difference is \(d\). Show that $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$
    2. James saves money over a number of weeks to buy a printer that costs \(\pounds 64\) He saves \(\pounds 10\) in week \(1 , \pounds 9.20\) in week \(2 , \pounds 8.40\) in week 3 and so on, so that the weekly amounts he saves form an arithmetic sequence. Given that James takes \(n\) weeks to save exactly \(\pounds 64\)
    1. show that $$n ^ { 2 } - 26 n + 160 = 0$$
    2. Solve the equation $$n ^ { 2 } - 26 n + 160 = 0$$
    3. Hence state the number of weeks James takes to save enough money to buy the printer, giving a brief reason for your answer.
    Edexcel Paper 1 2022 June Q14
    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. Given that $$2 \sin \left( x - 60 ^ { \circ } \right) = \cos \left( x - 30 ^ { \circ } \right)$$ show that $$\tan x = 3 \sqrt { 3 }$$
    2. Hence or otherwise solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$2 \sin 2 \theta = \cos \left( 2 \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
    Edexcel Paper 1 2022 June Q15
    10 marks Standard +0.3
    15. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-42_444_739_244_662} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} A company makes toys for children.
    Figure 5 shows the design for a solid toy that looks like a piece of cheese.
    The toy is modelled so that
    • face \(A B C\) is a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(A\)
    • angle \(B A C = 0.8\) radians
    • faces \(A B C\) and \(D E F\) are congruent
    • edges \(A D , C F\) and \(B E\) are perpendicular to faces \(A B C\) and \(D E F\)
    • edges \(A D , C F\) and \(B E\) have length \(h \mathrm {~cm}\)
    Given that the volume of the toy is \(240 \mathrm {~cm} ^ { 3 }\)
    1. show that the surface area of the toy, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.8 r ^ { 2 } + \frac { 1680 } { r }$$ making your method clear. Using algebraic differentiation,
    2. find the value of \(r\) for which \(S\) has a stationary point.
    3. Prove, by further differentiation, that this value of \(r\) gives the minimum surface area of the toy.
    Edexcel Paper 1 2022 June Q16
    9 marks Challenging +1.2
    16. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-46_770_999_242_534} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \sin ^ { 2 } t \quad y = 2 \sin 2 t + 3 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 6, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 4\)
    1. Show that the area of \(R\) is given by $$\int _ { 0 } ^ { a } \left( 8 - 8 \cos 4 t + 48 \sin ^ { 2 } t \cos t \right) \mathrm { d } t$$ where \(a\) is a constant to be found.
    2. Hence, using algebraic integration, find the exact area of \(R\).
    Edexcel Paper 1 2023 June Q1
    4 marks Moderate -0.8
    1. Find
    $$\int \frac { x ^ { \frac { 1 } { 2 } } ( 2 x - 5 ) } { 3 } \mathrm {~d} x$$ writing each term in simplest form.
    Edexcel Paper 1 2023 June Q2
    6 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. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
    Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),
    1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
    2. Hence
      1. find the value of \(a\)
      2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$
    Edexcel Paper 1 2023 June Q3
    3 marks Easy -1.2
    1. Relative to a fixed origin \(O\)
    • the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
    • the point \(B\) has position vector \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) where \(a\) is a positive integer.
      1. Show that \(| \overrightarrow { O A } | = \sqrt { 38 }\)
      2. Find the smallest value of \(a\) for which
    $$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$