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Edexcel P3 2021 January Q8
7 marks Moderate -0.3
  1. The percentage, \(P\), of the population of a small country who have access to the internet, is modelled by the equation
$$P = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years after the start of 2005
Using the data for the years between the start of 2005 and the start of 2010, a graph is plotted of \(\log _ { 10 } P\) against \(t\). The points are found to lie approximately on a straight line with gradient 0.09 and intercept 0.68 on the \(\log _ { 10 } P\) axis.
  1. Find, according to the model, the value of \(a\) and the value of \(b\), giving your answers to 2 decimal places.
  2. In the context of the model, give a practical interpretation of the constant \(a\).
  3. Use the model to estimate the percentage of the population who had access to the internet at the start of 2015
Edexcel P3 2021 January Q9
4 marks Standard +0.3
9. Find
  1. \(\int \frac { 3 x - 2 } { 3 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
  2. \(\int \frac { \mathrm { e } ^ { 2 x } } { \left( \mathrm { e } ^ { 2 x } - 1 \right) ^ { 3 } } \mathrm {~d} x \quad x \neq 0\)
    VIIV SIHI NI JIIIM IONOOVIUV SIHI NI III M M I ON OOVI4V SIHI NI IIIYM ION OC
Edexcel P3 2021 January Q10
10 marks Standard +0.3
10. The curve \(C\) has equation $$x = 3 \sec ^ { 2 } 2 y \quad x > 3 \quad 0 < y < \frac { \pi } { 4 }$$
  1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  2. Hence show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q x \sqrt { x - 3 } }$$ where \(p\) is irrational and \(q\) is an integer, stating the values of \(p\) and \(q\).
  3. Find the equation of the normal to \(C\) at the point where \(y = \frac { \pi } { 12 }\), giving your answer in the form \(y = m x + c\), giving \(m\) and \(c\) as exact irrational numbers.
    END
    VI4V SIHI NI JIIIM IONOOVIAV SIHI NI JIIIM ION OOVI4V SIHI NI IIIYM ION OC
Edexcel P3 2022 January Q1
4 marks Moderate -0.8
  1. Find, using calculus, the \(x\) coordinate of the stationary point on the curve with equation
$$y = ( 2 x + 5 ) e ^ { 3 x }$$
Edexcel P3 2022 January Q2
5 marks Moderate -0.3
2. (a) Show that the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ can be written in the form $$\sin 2 \theta = k$$ where \(k\) is a constant to be found.
(b) Hence find the smallest positive solution of the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ giving your answer, in degrees, to one decimal place.
Edexcel P3 2022 January Q3
6 marks Moderate -0.3
3. (i) Find, in simplest form, $$\int ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x$$ (ii) Show, by algebraic integration, that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$ where \(a\) is a rational constant to be found.
Edexcel P3 2022 January Q4
7 marks Standard +0.3
4. The growth of a weed on the surface of a pond is being studied. The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), is modelled by the equation $$A = \frac { 80 p \mathrm { e } ^ { 0.15 t } } { p \mathrm { e } ^ { 0.15 t } + 4 }$$ where \(p\) is a positive constant and \(t\) is the number of days after the start of the study.
Given that
  • \(30 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed at the start of the study
  • \(50 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed \(T\) days after the start of the study
    1. show that \(p = 2.4\)
    2. find the value of \(T\), giving your answer to one decimal place.
      (Solutions relying entirely on graphical or numerical methods are not acceptable.)
The weed grows until it covers the surface of the pond.
  • Find, according to the model, the maximum possible surface area of the pond.
    •
    Edexcel P3 2022 January Q5
    9 marks Standard +0.3
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3272b4c-d8dc-4f32-add9-153de90f4d0a-10_620_622_210_662} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = 6 \ln ( 2 x + 3 ) - \frac { 1 } { 2 } x ^ { 2 } + 4 \quad x > - \frac { 3 } { 2 }$$ The curve cuts the negative \(x\)-axis at the point \(P\), as shown in Figure 1.
    1. Show that the \(x\) coordinate of \(P\) lies in the interval \([ - 1.25 , - 1.2 ]\) The curve cuts the positive \(x\)-axis at the point \(Q\), also shown in Figure 1.
      Using the iterative formula $$x _ { n + 1 } = \sqrt { 12 \ln \left( 2 x _ { n } + 3 \right) + 8 } \text { with } x _ { 1 } = 6$$
      1. find, to 4 decimal places, the value of \(x _ { 2 }\)
      2. find, by continued iteration, the \(x\) coordinate of \(Q\). Give your answer to 4 decimal places. The curve has a maximum turning point at \(M\), as shown in Figure 1.
    3. Using calculus and showing each stage of your working, find the \(x\) coordinate of \(M\).
    Edexcel P3 2022 January Q6
    11 marks Standard +0.3
    6. The function f is defined by $$f ( x ) = \frac { 5 x - 3 } { x - 4 } \quad x > 4$$
    1. Show, by using calculus, that f is a decreasing function.
    2. Find \(\mathrm { f } ^ { - 1 }\)
      1. Show that \(\mathrm { ff } ( x ) = \frac { a x + b } { x + c }\) where \(a , b\) and \(c\) are constants to be found.
      2. Deduce the range of ff.
    Edexcel P3 2022 January Q7
    10 marks Standard +0.3
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3272b4c-d8dc-4f32-add9-153de90f4d0a-18_720_746_210_591} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { 1 } { 2 } | 2 x + 7 | - 10$$
    1. State the coordinates of the vertex, V, of the graph.
    2. Solve, using algebra, $$\frac { 1 } { 2 } | 2 x + 7 | - 10 \geqslant \frac { 1 } { 3 } x + 1$$
    3. Sketch the graph with equation $$y = | \mathrm { f } ( x ) |$$ stating the coordinates of the local maximum point and each local minimum point.
    Edexcel P3 2022 January Q8
    8 marks Moderate -0.3
    8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$
    1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
    2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
    3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)
    Edexcel P3 2022 January Q9
    8 marks Standard +0.3
    9. In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.
    1. Solve, for \(0 < x \leqslant \pi\), the equation $$2 \sec ^ { 2 } x - 3 \tan x = 2$$ giving the answers, as appropriate, to 3 significant figures.
    2. Prove that $$\frac { \sin 3 \theta } { \sin \theta } - \frac { \cos 3 \theta } { \cos \theta } \equiv 2$$
      VIAV SIHI NI III IM I ON OCVIIIV SIHI NI JIIIM I ON OOVAYV SIHI NI JIIYM ION OC
    Edexcel P3 2022 January Q10
    7 marks Challenging +1.2
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3272b4c-d8dc-4f32-add9-153de90f4d0a-30_661_743_210_603} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation $$x = y \mathrm { e } ^ { 2 y } \quad y \in \mathbb { R }$$
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x ( 1 + 2 y ) }$$ Given that the straight line with equation \(x = k\), where \(k\) is a constant, cuts \(C\) at exactly two points,
    2. find the range of possible values for \(k\).
    Edexcel P3 2023 January Q1
    6 marks Moderate -0.8
    1. The functions f and g are defined by
    $$\begin{array} { l l l } \mathrm { f } ( x ) = 9 - x ^ { 2 } & x \in \mathbb { R } & x \geqslant 0 \\ \mathrm {~g} ( x ) = \frac { 3 } { 2 x + 1 } & x \in \mathbb { R } & x \geqslant 0 \end{array}$$
    1. Write down the range of f
    2. Find the value of fg(1.5)
    3. Find \(\mathrm { g } ^ { - 1 }\)
    Edexcel P3 2023 January Q2
    6 marks Standard +0.3
    2. $$f ( x ) = \cos x + 2 \sin x$$
    1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), 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. $$g ( x ) = 3 - 7 f ( 2 x )$$
    2. Using the answer to part (a),
      1. write down the exact maximum value of \(\mathrm { g } ( x )\),
      2. find the smallest positive value of \(x\) for which this maximum value occurs, giving your answer to 2 decimal places.
    Edexcel P3 2023 January Q3
    5 marks Moderate -0.5
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5abaa077-1da4-4023-b442-194f6972095b-06_648_885_287_591} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The line \(l\) in Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(x\).
    The line passes through the points \(( 0,1.5 )\) and \(( - 4.8,0 )\) as shown.
    1. Write down an equation for \(l\).
    2. Hence, or otherwise, express \(y\) in the form \(k b ^ { x }\), giving the values of the constants \(k\) and \(b\) to 3 significant figures.
    Edexcel P3 2023 January Q4
    7 marks Standard +0.3
    4. $$f ( x ) = \frac { 2 x ^ { 4 } + 15 x ^ { 3 } + 35 x ^ { 2 } + 21 x - 4 } { ( x + 3 ) ^ { 2 } } \quad x \in \mathbb { R } \quad x > - 3$$
    1. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that $$\mathrm { f } ( x ) = A x ^ { 2 } + B x + C + \frac { D } { ( x + 3 ) ^ { 2 } }$$
    2. Hence find, $$\int \mathrm { f } ( x ) \mathrm { d } x$$
    Edexcel P3 2023 January Q5
    9 marks Challenging +1.2
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Prove that $$\cot ^ { 2 } x - \tan ^ { 2 } x \equiv 4 \cot 2 x \operatorname { cosec } 2 x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
    2. Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$4 \cot 2 \theta \operatorname { cosec } 2 \theta = 2 \tan ^ { 2 } \theta$$ giving your answers to 2 decimal places.
    Edexcel P3 2023 January Q6
    8 marks Moderate -0.3
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5abaa077-1da4-4023-b442-194f6972095b-16_652_835_292_616} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the graph with equation $$y = | 3 x - 5 a | - 2 a$$ where \(a\) is a positive constant.
    The graph
    • cuts the \(y\)-axis at the point \(P\)
    • cuts the \(x\)-axis at the points \(Q\) and \(R\)
    • has a minimum point at \(S\)
      1. Find, in simplest form in terms of \(a\), the coordinates of
        1. point \(P\)
        2. points \(Q\) and \(R\)
        3. point \(S\)
      2. Find, in simplest form in terms of \(a\), the values of \(x\) for which
    $$| 3 x - 5 a | - 2 a = | x - 2 a |$$
    Edexcel P3 2023 January Q7
    9 marks Standard +0.3
    1. The curve \(C\) has equation
    $$x = 3 \tan \left( y - \frac { \pi } { 6 } \right) \quad x \in \mathbb { R } \quad - \frac { \pi } { 3 } < y < \frac { 2 \pi } { 3 }$$
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { x ^ { 2 } + b }$$ where \(a\) and \(b\) are integers to be found. The point \(P\) with \(y\) coordinate \(\frac { \pi } { 3 }\) lies on \(C\).
      Given that the tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\).
    2. find, in simplest form, the exact \(x\) coordinate of \(Q\).
    Edexcel P3 2023 January Q8
    5 marks Moderate -0.8
    1. Find, in simplest form,
    $$\int ( 2 \cos x - \sin x ) ^ { 2 } d x$$
    Edexcel P3 2023 January Q9
    11 marks Standard +0.3
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5abaa077-1da4-4023-b442-194f6972095b-26_659_783_287_641} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \sqrt { 3 + 4 \mathrm { e } ^ { x ^ { 2 } } } \quad x \geqslant 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(P\) with \(x\) coordinate \(\alpha\) lies on \(C\).
      Given that the tangent to \(C\) at \(P\) passes through the origin, as shown in Figure 3,
    2. show that \(x = \alpha\) is a solution of the equation $$4 x ^ { 2 } e ^ { x ^ { 2 } } - 4 e ^ { x ^ { 2 } } - 3 = 0$$
    3. Hence show that \(\alpha\) lies between 1 and 2
    4. Show that the equation in part (b) can be written in the form $$x = \frac { 1 } { 2 } \sqrt { 4 + 3 \mathrm { e } ^ { - x ^ { 2 } } }$$ The iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { 4 + 3 \mathrm { e } ^ { - x _ { n } ^ { 2 } } }$$ with \(x _ { 1 } = 1\) is used to find an approximation for \(\alpha\).
    5. Use the iteration formula to find, to 4 decimal places, the value of
      1. \(X _ { 3 }\)
      2. \(\alpha\)
    Edexcel P3 2023 January Q10
    9 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.
    A population of fruit flies is being studied.
    The number of fruit flies, \(F\), in the population, \(t\) days after the start of the study, is modelled by the equation $$F = \frac { 350 \mathrm { e } ^ { k t } } { 9 + \mathrm { e } ^ { k t } }$$ where \(k\) is a constant.
    Use the equation of the model to answer parts (a), (b) and (c).
    1. Find the number of fruit flies in the population at the start of the study. Given that there are 200 fruit flies in the population 15 days after the start of the study,
    2. show that \(k = \frac { 1 } { 15 } \ln 12\) Given also that, when \(t = T\), the number of fruit flies in the population is increasing at a rate of 10 per day,
    3. find the possible values of \(T\), giving your answers to one decimal place.
    Edexcel P3 2024 January Q1
    4 marks Moderate -0.8
    1. The point \(P ( - 4 , - 3 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\)
    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. \(y = \mathrm { f } ( 2 x )\)
    2. \(y = 3 \mathrm { f } ( x - 1 )\)
    3. \(y = | f ( x ) |\)
    Edexcel P3 2024 January Q2
    6 marks Moderate -0.3
    1. A curve has equation \(y = \mathrm { f } ( x )\) where
    $$\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 2 } + 4 x - 7 \quad x \in \mathbb { R }$$
    1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [2,3]
    2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt [ 3 ] { \frac { 5 x ^ { 2 } - 4 x + 7 } { x } }$$ The iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { \frac { 5 x _ { n } ^ { 2 } - 4 x _ { n } + 7 } { x _ { n } } }$$ is used to find \(\alpha\)
    3. Starting with \(x _ { 1 } = 2\) and using the iterative formula,
      1. find, to 4 decimal places, the value of \(x _ { 2 }\)
      2. find, to 4 decimal places, the value of \(\alpha\)