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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
    $$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$
    Edexcel Paper 1 2023 June Q4
    5 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.
    The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\) Given that
    • \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 1 } { 2 } \cos x\)
    • the curve has a stationary point with \(x\) coordinate \(\alpha\)
    • \(\alpha\) is small
      1. use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.
    The point \(P ( 0,3 )\) lies on \(C\)
  • Find the equation of the tangent to the curve at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
    •
    Edexcel Paper 1 2023 June Q5
    6 marks Moderate -0.8
    1. A continuous curve has equation \(y = \mathrm { f } ( x )\).
    The table shows corresponding values of \(x\) and \(y\) for this curve, where \(a\) and \(b\) are constants.
    \(x\)33.23.43.63.84
    \(y\)\(a\)16.8\(b\)20.218.713.5
    The trapezium rule is used, with all the \(y\) values in the table, to find an approximate area under the curve between \(x = 3\) and \(x = 4\) Given that this area is 17.59
    1. show that \(a + 2 b = 51\) Given also that the sum of all the \(y\) values in the table is 97.2
    2. find the value of \(a\) and the value of \(b\)
    Edexcel Paper 1 2023 June Q6
    6 marks Moderate -0.3
    6. $$a = \log _ { 2 } x \quad b = \log _ { 2 } ( x + 8 )$$ Express in terms of \(a\) and/or \(b\)
    1. \(\log _ { 2 } \sqrt { x }\)
    2. \(\log _ { 2 } \left( x ^ { 2 } + 8 x \right)\)
    3. \(\log _ { 2 } \left( 8 + \frac { 64 } { x } \right)\) Give your answer in simplest form.
    Edexcel Paper 1 2023 June Q7
    8 marks Moderate -0.3
    1. The function f is defined by
    $$f ( x ) = 3 + \sqrt { x - 2 } \quad x \in \mathbb { R } \quad x > 2$$
    1. State the range of f
    2. Find f-1 The function \(g\) is defined by $$g ( x ) = \frac { 15 } { x - 3 } \quad x \in \mathbb { R } \quad x \neq 3$$
    3. Find \(g f ( 6 )\)
    4. Find the exact value of the constant \(a\) for which $$\mathrm { f } \left( a ^ { 2 } + 2 \right) = \mathrm { g } ( a )$$
    Edexcel Paper 1 2023 June Q8
    10 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-18_505_1301_257_572} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the plan view of a stage.
    The plan view shows two congruent triangles \(A B O\) and \(G F O\) joined to a sector \(O C D E O\) of a circle, centre \(O\), where
    • angle \(C O E = 2.3\) radians
    • arc length \(C D E = 27.6 \mathrm {~m}\)
    • \(A O G\) is a straight line of length 15 m
      1. Show that \(O C = 12 \mathrm {~m}\).
      2. Show that the size of angle \(A O B\) is 0.421 radians correct to 3 decimal places.
    Given that the total length of the front of the stage, \(B C D E F\), is 35 m ,
  • find the total area of the stage, giving your answer to the nearest square metre.
    •
    Edexcel Paper 1 2023 June Q9
    7 marks Standard +0.3
    1. The first three terms of a geometric sequence are
    $$3 k + 4 \quad 12 - 3 k \quad k + 16$$ where \(k\) is a constant.
    1. Show that \(k\) satisfies the equation $$3 k ^ { 2 } - 62 k + 40 = 0$$ Given that the sequence converges,
      1. find the value of \(k\), giving a reason for your answer,
      2. find the value of \(S _ { \infty }\)
    Edexcel Paper 1 2023 June Q10
    9 marks Standard +0.8
    1. A circle \(C\) has equation
    $$x ^ { 2 } + y ^ { 2 } + 6 k x - 2 k y + 7 = 0$$ where \(k\) is a constant.
    1. Find in terms of \(k\),
      1. the coordinates of the centre of \(C\)
      2. the radius of \(C\) The line with equation \(y = 2 x - 1\) intersects \(C\) at 2 distinct points.
    2. Find the range of possible values of \(k\).
    Edexcel Paper 1 2023 June Q11
    7 marks Standard +0.3
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-28_590_739_219_671} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The value, \(V\) pounds, of a mobile phone, \(t\) months after it was bought, is modelled by $$V = a b ^ { t }$$ where \(a\) and \(b\) are constants.
    Figure 2 shows the linear relationship between \(\log _ { 10 } V\) and \(t\).
    The line passes through the points \(( 0,3 )\) and \(( 10,2.79 )\) as shown.
    Using these points,
    1. find the initial value of the phone,
    2. find a complete equation for \(V\) in terms of \(t\), giving the exact value of \(a\) and giving the value of \(b\) to 3 significant figures. Exactly 2 years after it was bought, the value of the phone was \(\pounds 320\)
    3. Use this information to evaluate the reliability of the model.
    Edexcel Paper 1 2023 June Q12
    5 marks Standard +0.8
    12. $$y = \sin x$$ where \(x\) is measured in radians.
    Use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cos x$$ You may
    • use without proof the formula for \(\sin ( A \pm B )\)
    • assume that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)
    Edexcel Paper 1 2023 June Q13
    7 marks Moderate -0.5
    1. On a roller coaster ride, passengers travel in carriages around a track.
    On the ride, carriages complete multiple circuits of the track such that
    • the maximum vertical height of a carriage above the ground is 60 m
    • a carriage starts a circuit at a vertical height of 2 m above the ground
    • the ground is horizontal
    The vertical height, \(H \mathrm {~m}\), of a carriage above the ground, \(t\) seconds after the carriage starts the first circuit, is modelled by the equation $$H = a - b ( t - 20 ) ^ { 2 }$$ where \(a\) and \(b\) are positive constants.
    1. Find a complete equation for the model.
    2. Use the model to determine the height of the carriage above the ground when \(t = 40\) In an alternative model, the vertical height, \(H \mathrm {~m}\), of a carriage above the ground, \(t\) seconds after the carriage starts the first circuit, is given by $$H = 29 \cos ( 9 t + \alpha ) ^ { \circ } + \beta \quad 0 \leqslant \alpha < 360 ^ { \circ }$$ where \(\alpha\) and \(\beta\) are constants.
    3. Find a complete equation for the alternative model. Given that the carriage moves continuously for 2 minutes,
    4. give a reason why the alternative model would be more appropriate.