Questions P3 (1203 questions)

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Edexcel P3 2022 June Q7
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
  1. Show that the equation $$2 \sin \theta \left( 3 \cot ^ { 2 } 2 \theta - 7 \right) = 13 \sec \theta$$ can be written as $$3 \operatorname { cosec } ^ { 2 } 2 \theta - 13 \operatorname { cosec } 2 \theta - 10 = 0$$
  2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\), the equation $$2 \sin \theta \left( 3 \cot ^ { 2 } 2 \theta - 7 \right) = 13 \sec \theta$$ giving your answers to 3 significant figures.
Edexcel P3 2022 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44035bf8-f54c-472a-b969-b4fa4fa3d203-26_579_467_219_749} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 is a graph showing the velocity of a sprinter during a 100 m race.
The sprinter's velocity during the race, \(v \mathrm {~ms} ^ { - 1 }\), is modelled by the equation $$v = 12 - \mathrm { e } ^ { t - 10 } - 12 \mathrm { e } ^ { - 0.75 t } \quad t \geqslant 0$$ where \(t\) seconds is the time after the sprinter begins to run. According to the model,
  1. find, using calculus, the sprinter's maximum velocity during the race. Given that the sprinter runs 100 m in \(T\) seconds, such that $$\int _ { 0 } ^ { T } v \mathrm {~d} t = 100$$
  2. show that \(T\) is a solution of the equation $$T = \frac { 1 } { 12 } \left( 116 - 16 \mathrm { e } ^ { - 0.75 T } + \mathrm { e } ^ { T - 10 } - \mathrm { e } ^ { - 10 } \right)$$ The iteration formula $$T _ { n + 1 } = \frac { 1 } { 12 } \left( 116 - 16 \mathrm { e } ^ { - 0.75 T _ { n } } + \mathrm { e } ^ { T _ { n } - 10 } - \mathrm { e } ^ { - 10 } \right)$$ is used to find an approximate value for \(T\) Using this iteration formula with \(T _ { 1 } = 10\)
  3. find, to 4 decimal places,
    1. the value of \(T _ { 2 }\)
    2. the time taken by the sprinter to run the race, according to the model.
Edexcel P3 2022 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44035bf8-f54c-472a-b969-b4fa4fa3d203-30_773_775_255_587} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows the curve with equation $$y = \frac { 1 + 2 \cos x } { 1 + \sin x } \quad - \frac { \pi } { 2 } < x < \frac { 3 \pi } { 2 }$$ The point \(M\), shown in Figure 5, is the minimum point on the curve.
  1. Show that the \(x\) coordinate of \(M\) is a solution of the equation $$2 \sin x + \cos x = - 2$$
  2. Hence find, to 3 significant figures, the \(x\) coordinate of \(M\).
Edexcel P3 2023 June Q1
1. $$g ( x ) = x ^ { 6 } + 2 x - 1000$$
  1. Show that \(\mathrm { g } ( x ) = 0\) has a root \(\alpha\) in the interval [3,4] Using the iteration formula $$x _ { n + 1 } = \sqrt [ 6 ] { 1000 - 2 x _ { n } } \quad \text { with } x _ { 1 } = 3$$
    1. find, to 4 decimal places, the value of \(x _ { 2 }\)
    2. find, by repeated iteration, the value of \(\alpha\). Give your answer to 4 decimal places.
Edexcel P3 2023 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bef290fb-fbac-4c9c-981e-5e323ac7182e-04_814_839_242_614} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the linear relationship between \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
The line passes through the points \(( 0,4 )\) and \(( 2,0 )\) as shown.
    1. Find an equation linking \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
    2. Hence find the exact value of \(T\) when \(x = 216\)
  2. Find an equation, not involving logs, linking \(T\) with \(x\)
Edexcel P3 2023 June Q3
  1. (i) Find \(\frac { \mathrm { d } } { \mathrm { d } x } \ln \left( \sin ^ { 2 } 3 x \right)\) writing your answer in simplest form.
    (ii) (a) Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( 3 x ^ { 2 } - 4 \right) ^ { 6 }\)
    (b) Hence show that
$$\int _ { 0 } ^ { \sqrt { 2 } } x \left( 3 x ^ { 2 } - 4 \right) ^ { 5 } \mathrm {~d} x = R$$ where \(R\) is an integer to be found.
(Solutions relying on calculator technology are not acceptable.)
Edexcel P3 2023 June Q4
  1. The function f is defined by
$$\mathrm { f } ( x ) = 2 x ^ { 2 } - 5 \quad x \geqslant 0 \quad x \in \mathbb { R }$$
  1. State the range of f On the following page there is a diagram, labelled Diagram 1, which shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
  2. On Diagram 1, sketch the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) meets the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(P\) Using algebra and showing your working,
  3. find the exact \(x\) coordinate of \(P\)
    \includegraphics[max width=\textwidth, alt={}]{bef290fb-fbac-4c9c-981e-5e323ac7182e-09_607_610_248_731}
    \section*{Diagram 1}
Edexcel P3 2023 June Q5
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Solve, for \(0 < x < \pi\)
    $$( x - 2 ) ( \sqrt { 3 } \sec x + 2 ) = 0$$
  2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$10 \sin \theta = 3 \cos 2 \theta$$
Edexcel P3 2023 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bef290fb-fbac-4c9c-981e-5e323ac7182e-14_752_794_251_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 | x - 2 | - 10$$ The vertex of the graph is at point \(P\), shown in Figure 2.
  1. Find the coordinates of \(P\)
  2. Find \(\mathrm { ff } ( 0 )\)
  3. Solve the inequality $$3 | x - 2 | - 10 < 5 x + 10$$
  4. Solve the equation $$\mathrm { f } ( | x | ) = 0$$
Edexcel P3 2023 June Q7
  1. A scientist is studying two different populations of bacteria.
The number of bacteria \(N\) in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that
  • there were 2500 bacteria in this population at the start of the study
  • there were 10000 bacteria 8 hours later
    1. find the exact value of \(A\) and the value of \(k\) to 4 significant figures.
The number of bacteria \(N\) in the second population is modelled by the equation $$N = 60000 \mathrm { e } ^ { - 0.6 t } \quad t \geqslant 0$$ where \(t\) is the time in hours from the start of the study.
  • Find the rate of decrease of bacteria in this population exactly 5 hours from the start of the study. Give your answer to 3 significant figures. When \(t = T\), the number of bacteria in the two different populations was the same.
  • Find the value of \(T\), giving your answer to 3 significant figures.
    (Solutions relying entirely on calculator technology are not acceptable.)
    •
    Edexcel P3 2023 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bef290fb-fbac-4c9c-981e-5e323ac7182e-22_687_698_255_685} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 1 ) ^ { 3 } e ^ { - 4 x }$$
    1. Show that $$\mathrm { f } ^ { \prime } ( x ) = A ( 2 x + 1 ) ^ { 2 } ( 1 - 4 x ) \mathrm { e } ^ { - 4 x }$$ where \(A\) is a constant to be found.
    2. Hence find the exact coordinates of the two stationary points on \(C\). The function g is defined by $$g ( x ) = 8 f ( x - 2 )$$
    3. Find the coordinates of the maximum stationary point on the curve with equation \(y = g ( x )\).
    Edexcel P3 2023 June Q9
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
    2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
    3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$
    Edexcel P3 2023 June Q10
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{bef290fb-fbac-4c9c-981e-5e323ac7182e-30_719_876_246_598} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of the curve with equation $$x = \frac { 2 y ^ { 2 } + 6 } { 3 y - 3 }$$
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) giving your answer as a fully simplified fraction. The tangents at points \(P\) and \(Q\) on the curve are parallel to the \(y\)-axis, as shown in Figure 4.
    2. Use the answer to part (a) to find the equations of these two tangents.
    Edexcel P3 2024 June Q1
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-02_520_474_246_797} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 | x - 5 | + 10$$ The point \(P\), shown in Figure 1, is the vertex of the graph.
    1. State the coordinates of \(P\)
    2. Use algebra to solve $$2 | x - 5 | + 10 > 6 x$$ (Solutions relying on calculator technology are not acceptable.)
    3. Find the point to which \(P\) is mapped, when the graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = 3 \mathrm { f } ( x - 2 )\)
    Edexcel P3 2024 June Q2
    2. $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x + 8 } { x - 2 }$$
    1. Write \(g ( x )\) in the form $$A x + B + \frac { C } { x - 2 }$$ where \(A , B\) and \(C\) are integers to be found.
    2. Hence use algebraic integration to show that $$\int _ { 4 } ^ { 8 } \mathrm {~g} ( x ) \mathrm { d } x = \alpha + \beta \ln 3$$ where \(\alpha\) and \(\beta\) are integers to be found.
    Edexcel P3 2024 June Q3
    1. (i) The variables \(x\) and \(y\) are connected by the equation
    $$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$ Sketch the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\)
    Show on your sketch the coordinates of the points of intersection of the graph with the axes.
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-08_888_885_744_552} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the linear relationship between \(\log _ { 3 } N\) and \(t\).
    Show that \(N = a b ^ { t }\) where \(a\) and \(b\) are constants to be found.
    Edexcel P3 2024 June Q4
    4. $$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$
    1. Write \(\mathrm { f } ( x )\) in the form $$a \sin 2 x + b \cos 2 x + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
    2. Use the answer to part (a) to write \(\mathrm { f } ( x )\) in the form $$R \sin ( 2 x + \alpha ) + c$$ where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
      Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 significant figures.
    3. Hence, or otherwise,
      1. state the maximum value of \(\mathrm { f } ( x )\)
      2. find the second smallest positive value of \(x\) at which a maximum value of \(\mathrm { f } ( x )\) occurs. Give your answer to 3 significant figures.
    Edexcel P3 2024 June Q5
    1. The functions \(f\) and \(g\) are defined by
    $$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0
    & \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$
    1. Find \(\mathrm { f } ^ { - 1 } ( 22 )\)
    2. Use differentiation to prove that g is an increasing function.
    3. Find \(\mathrm { g } ^ { - 1 }\)
    4. Find the range of fg
    Edexcel P3 2024 June Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-18_856_990_246_539} \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 $$y = \sqrt { 4 x - 7 }$$ The line \(l\), shown in Figure 3, is the normal to the curve at the point \(P ( 8,5 )\)
    1. Use calculus to show that an equation of \(l\) is $$5 x + 2 y - 50 = 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and \(l\).
    2. Use algebraic integration to find the exact area of \(R\).
    Edexcel P3 2024 June Q7
    1. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
    1. Given that $$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$ show that $$\tan x = - 2 - \sqrt { 3 }$$
    2. Hence or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$
    Edexcel P3 2024 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_499_551_246_758} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 is a graph showing the path of a golf ball after the ball has been hit until it first hits the ground. The vertical height, \(h\) metres, of the ball above the ground has been plotted against the horizontal distance travelled, \(x\) metres, measured from where the ball was hit. The ball travels a horizontal distance of \(d\) metres before it first hits the ground.
    The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
    The path of the ball is modelled by the equation $$h = 1.5 x - 0.5 x \mathrm { e } ^ { 0.02 x } \quad 0 \leqslant x \leqslant d$$ \section*{Use the model to answer parts (a), (b) and (c).}
    1. Find the value of \(d\), giving your answer to 2 decimal places.
      (Solutions relying entirely on calculator technology are not acceptable.)
    2. Show that the maximum value of \(h\) occurs when $$x = 50 \ln \left( \frac { 150 } { x + 50 } \right)$$ Using the iteration formula $$x _ { n + 1 } = 50 \ln \left( \frac { 150 } { x _ { n } + 50 } \right) \quad \text { with } x _ { 1 } = 30$$
      1. find the value of \(x _ { 2 }\) to 2 decimal places,
      2. find, by repeated iteration, the horizontal distance travelled by the golf ball before it reaches its maximum height. Give your answer to 2 decimal places.
        \includegraphics[max width=\textwidth, alt={}, center]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_2270_56_309_1981}
    Edexcel P3 2024 June Q9
    9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-30_714_1079_251_495} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} The curve shown in Figure 5 has equation $$x = 4 \sin ^ { 2 } y - 1 \quad 0 \leqslant y \leqslant \frac { \pi } { 2 }$$ The point \(P \left( k , \frac { \pi } { 3 } \right)\) lies on the curve.
    1. Verify that \(k = 2\)
      1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\)
      2. Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 \sqrt { x + 1 } \sqrt { 3 - x } }\) The normal to the curve at \(P\) cuts the \(x\)-axis at the point \(N\).
    3. Find the exact area of triangle \(O P N\), where \(O\) is the origin. Give your answer in the form \(a \pi + b \pi ^ { 2 }\) where \(a\) and \(b\) are constants.
    Edexcel P3 2020 October Q1
    1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation
    $$2 \cos 2 x = 7 \cos x$$ giving your solutions to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    Edexcel P3 2020 October Q2
    1. A scientist monitored the growth of bacteria on a dish over a 30 -day period.
    The area, \(N \mathrm {~mm} ^ { 2 }\), of the dish covered by bacteria, \(t\) days after monitoring began, is modelled by the equation $$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$
    1. Show that this equation may be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest integer and give the value of \(b\) to 3 significant figures.
    2. Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in \(\mathrm { mm } ^ { 2 }\), to 2 significant figures.
    Edexcel P3 2020 October Q3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{96948fd3-5438-4e95-b41b-2f649ca8dfac-06_828_828_210_557} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { \sqrt { 4 x - 1 } } \quad x > \frac { 1 } { 4 }$$
    1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
    2. Hence find the range of f.
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