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Edexcel Paper 1 2023 June Q8
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
    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
    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
    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
    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
    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.
    Edexcel Paper 1 2023 June Q14
    1. Prove, using algebra, that
    $$( n + 1 ) ^ { 3 } - n ^ { 3 }$$ is odd for all \(n \in \mathbb { N }\)
    Edexcel Paper 1 2023 June Q15
    1. A curve has equation \(y = \mathrm { f } ( x )\), where
    $$\mathrm { f } ( x ) = \frac { 7 x \mathrm { e } ^ { x } } { \sqrt { \mathrm { e } ^ { 3 x } - 2 } } \quad x > \ln \sqrt [ 3 ] { 2 }$$
    1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 7 \mathrm { e } ^ { x } \left( \mathrm { e } ^ { 3 x } ( 2 - x ) + A x + B \right) } { 2 \left( \mathrm { e } ^ { 3 x } - 2 \right) ^ { \frac { 3 } { 2 } } }$$ where \(A\) and \(B\) are constants to be found.
    2. Hence show that the \(x\) coordinates of the turning points of the curve are solutions of the equation $$x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }$$ The equation \(x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) has two positive roots \(\alpha\) and \(\beta\) where \(\beta > \alpha\)
      A student uses the iteration formula $$x _ { n + 1 } = \frac { 2 \mathrm { e } ^ { 3 x _ { n } } - 4 } { \mathrm { e } ^ { 3 x _ { n } } + 4 }$$ in an attempt to find approximations for \(\alpha\) and \(\beta\)
      Diagram 1 shows a plot of part of the curve with equation \(y = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) and part of the line with equation \(y = x\) Using Diagram 1 on page 42
    3. draw a staircase diagram to show that the iteration formula starting with \(x _ { 1 } = 1\) can be used to find an approximation for \(\beta\) Use the iteration formula with \(x _ { 1 } = 1\), to find, to 3 decimal places,
      1. the value of \(x _ { 2 }\)
      2. the value of \(\beta\) Using a suitable interval and a suitable function that should be stated
    5. show that \(\alpha = 0.432\) to 3 decimal places. Only use the copy of Diagram 1 if you need to redraw your answer to part (c). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_736_812_372_143} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_738_815_370_1114} \captionsetup{labelformat=empty} \caption{copy of Diagram 1}
      \end{figure}
    Edexcel Paper 1 2024 June Q1
    1. $$g ( x ) = 3 x ^ { 3 } - 20 x ^ { 2 } + ( k + 17 ) x + k$$ where \(k\) is a constant.
    Given that \(( x - 3 )\) is a factor of \(\mathrm { g } ( x )\), find the value of \(k\).
    Edexcel Paper 1 2024 June Q2
    1. (a) Find, in ascending powers of \(x\), the first four terms of the binomial expansion of
    $$( 1 - 9 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
    (b) Give a reason why \(x = - \frac { 2 } { 9 }\) should not be used in the expansion to find an approximation to \(\sqrt { 3 }\)
    Edexcel Paper 1 2024 June Q3
    3. $$\mathrm { f } ( x ) = x + \tan \left( \frac { 1 } { 2 } x \right) \quad \pi < x < \frac { 3 \pi } { 2 }$$ Given that the equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\)
    1. show that \(\alpha\) lies in the interval [3.6, 3.7]
    2. Find \(\mathrm { f } ^ { \prime } ( x )\)
    3. Using 3.7 as a first approximation for \(\alpha\), apply the Newton-Raphson method once to obtain a second approximation for \(\alpha\). Give your answer to 3 decimal places.
    Edexcel Paper 1 2024 June Q4
    1. Given that \(y = x ^ { 2 }\), use differentiation from first principles to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x\)
    Edexcel Paper 1 2024 June Q5
    1. The function f is defined by
    $$f ( x ) = \frac { 2 x - 3 } { x ^ { 2 } + 4 } \quad x \in \mathbb { R }$$
    1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { a x ^ { 2 } + b x + c } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be found.
    2. Hence, using algebra, find the values of \(x\) for which f is decreasing. You must show each step in your working.
    Edexcel Paper 1 2024 June Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e116a86f-63e0-4e80-b49c-d9f3c819ce15-12_680_677_246_696} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the graph with equation $$y = 3 | x - 2 | + 5$$ The vertex of the graph is at the point \(P\), shown in Figure 1.
    1. Find the coordinates of \(P\).
    2. Solve the equation $$16 - 4 x = 3 | x - 2 | + 5$$ A line \(l\) has equation \(y = k x + 4\) where \(k\) is a constant.
      Given that \(l\) intersects \(y = 3 | x - 2 | + 5\) at 2 distinct points,
    3. find the range of values of \(k\).
    Edexcel Paper 1 2024 June Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e116a86f-63e0-4e80-b49c-d9f3c819ce15-14_495_711_243_641} \captionsetup{labelformat=empty} \caption{Diagram not drawn to scale.}
    \end{figure} Figure 2 Figure 2 shows a cylindrical tank of height 1.5 m .
    Initially the tank is full of water.
    The water starts to leak from a small hole, at a point \(L\), in the side of the tank.
    While the tank is leaking, the depth, \(H\) metres, of the water in the tank is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - 0.12 \mathrm { e } ^ { - 0.2 t }$$ where \(t\) hours is the time after the leak starts.
    Using the model,
    1. show that $$H = A \mathrm { e } ^ { - 0.2 t } + B$$ where \(A\) and \(B\) are constants to be found,
    2. find the time taken for the depth of the water to decrease to 1.2 m . Give your answer in hours and minutes, to the nearest minute. In the long term, the water level in the tank falls to the same height as the hole.
    3. Find, according to the model, the height of the hole from the bottom of the tank.
    Edexcel Paper 1 2024 June Q8
    1. The functions f and g are defined by
    $$\begin{array} { l l } f ( x ) = 4 - 3 x ^ { 2 } & x \in \mathbb { R }
    g ( x ) = \frac { 5 } { 2 x - 9 } & x \in \mathbb { R } , x \neq \frac { 9 } { 2 } \end{array}$$
    1. Find fg(2)
    2. Find \(\mathrm { g } ^ { - 1 }\)
      1. Find \(\mathrm { gf } ( x )\), giving your answer as a simplified fraction.
      2. Deduce the range of \(\operatorname { gf } ( x )\). The function h is defined by $$h ( x ) = 2 x ^ { 2 } - 6 x + k \quad x \in \mathbb { R }$$ where \(k\) is a constant.
    4. Find the range of values of \(k\) for which the equation $$\mathrm { f } ( x ) = \mathrm { h } ( x )$$ has no real solutions.
    Edexcel Paper 1 2024 June Q9
    1. The first 3 terms of a geometric sequence are
    $$3 ^ { 4 k - 5 } \quad 9 ^ { 7 - 2 k } \quad 3 ^ { 2 ( k - 1 ) }$$ where \(k\) is a constant.
    1. Using algebra and making your reasoning clear, prove that \(k = \frac { 5 } { 2 }\)
    2. Hence find the sum to infinity of the geometric sequence.
    Edexcel Paper 1 2024 June Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e116a86f-63e0-4e80-b49c-d9f3c819ce15-24_872_1285_246_392} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    Figure 3 shows a sketch of part of the curve with equation $$y = 8 x - x ^ { \frac { 5 } { 2 } } \quad x \geqslant 0$$ The curve crosses the \(x\)-axis at the point \(A\).
    1. Verify that the \(x\) coordinate of \(A\) is 4 The line \(l _ { 1 }\) is the tangent to the curve at \(A\).
    2. Use calculus to show that an equation of line \(l _ { 1 }\) is $$12 x + y = 48$$ The line \(l _ { 2 }\) has equation \(y = 8 x\)
      The region \(R\), shown shaded in Figure 3, is bounded by the curve, the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
    3. Use algebraic integration to find the exact area of \(R\).
    Edexcel Paper 1 2024 June Q11
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e116a86f-63e0-4e80-b49c-d9f3c819ce15-28_451_899_239_584} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows the design of a badge.
    The shape \(A B C O A\) is a semicircle with centre \(O\) and diameter 10 cm .
    \(O B\) is the arc of a circle with centre \(A\) and radius 5 cm .
    The region \(R\), shown shaded in Figure 4, is bounded by the arc \(O B\), the arc \(B C\) and the line \(O C\). Find the exact area of \(R\).
    Give your answer in the form \(( a \sqrt { 3 } + b \pi ) \mathrm { cm } ^ { 2 }\), where \(a\) and \(b\) are rational numbers.
    Edexcel Paper 1 2024 June Q12
    1. (a) Express \(140 \cos \theta - 480 \sin \theta\) in the form \(K \cos ( \theta + \alpha )\)
      where \(K > 0\) and \(0 < \alpha < 90 ^ { \circ }\)
      State the value of \(K\) and give the value of \(\alpha\), in degrees, to 2 decimal places.
    A scientist studies the number of rabbits and the number of foxes in a wood for one year. The number of rabbits, \(R\), is modelled by the equation $$R = A + 140 \cos ( 30 t ) ^ { \circ } - 480 \sin ( 30 t ) ^ { \circ }$$ where \(t\) months is the time after the start of the year and \(A\) is a constant.
    Given that, during the year, the maximum number of rabbits in the wood is 1500
    (b) (i) find a complete equation for this model.
    (ii) Hence write down the minimum number of rabbits in the wood during the year according to the model. The actual number of rabbits in the wood is at its minimum value in the middle of April.
    (c) Use this information to comment on the model for the number of rabbits. The number of foxes, \(F\), in the wood during the same year is modelled by the equation $$F = 100 + 70 \sin ( 30 t + 70 ) ^ { \circ }$$ The number of foxes is at its minimum value after \(T\) months.
    (d) Find, according to the models, the number of rabbits in the wood at time \(T\) months.
    Edexcel Paper 1 2024 June Q13
    1. (a) Given that \(a\) is a positive constant, use the substitution \(x = a \sin ^ { 2 } \theta\) to show that
    $$\int _ { 0 } ^ { a } x ^ { \frac { 1 } { 2 } } \sqrt { a - x } \mathrm {~d} x = \frac { 1 } { 2 } a ^ { 2 } \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$ (b) Hence use algebraic integration to show that $$\int _ { 0 } ^ { a } x ^ { \frac { 1 } { 2 } } \sqrt { a - x } d x = k \pi a ^ { 2 }$$ where \(k\) is a constant to be found.
    Edexcel Paper 1 2024 June Q14
    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
    1. (i) Show that \(k ^ { 2 } - 4 k + 5\) is positive for all real values of \(k\).
      (ii) 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
    1. (a) 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.
    (b) 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
    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.