Questions — Edexcel P3 (133 questions)

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Edexcel P3 2021 June Q3
  1. (i) Find
$$\int \frac { 12 } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) (a) Write \(\frac { 4 x + 3 } { x + 2 }\) in the form $$A + \frac { B } { x + 2 } \text { where } A \text { and } B \text { are constants to be found }$$ (b) Hence find, using algebraic integration, the exact value of $$\int _ { - 8 } ^ { - 5 } \frac { 4 x + 3 } { x + 2 } d x$$ giving your answer in simplest form.
Edexcel P3 2021 June Q4
4. The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 x + 6 } { x - 5 } & x \in \mathbb { R } , x \neq 5
\mathrm {~g} ( x ) = 5 - 2 x ^ { 2 } & x \in \mathbb { R } , x \leqslant 0 \end{array}$$
  1. Solve the equation $$\operatorname { fg } ( x ) = 3$$
  2. Find \(\mathrm { f } ^ { - 1 }\)
  3. Sketch and label, on the same axes, the curve with equation \(y = \mathrm { g } ( x )\) and the curve with equation \(y = \mathrm { g } ^ { - 1 } ( x )\). Show on your sketch the coordinates of the points where each curve meets or cuts the coordinate axes.
Edexcel P3 2021 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76205772-5395-4ab2-96f9-ad9803b8388c-16_582_737_248_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The growth of duckweed on a pond is being studied. The surface area of the pond covered by duckweed, \(A \mathrm {~m} ^ { 2 }\), at a time \(t\) days after the start of the study is modelled by the equation $$A = p q ^ { t } \quad \text { where } p \text { and } q \text { are positive constants }$$ Figure 1 shows the linear relationship between \(\log _ { 10 } A\) and \(t\).
The points \(( 0,0.32 )\) and \(( 8,0.56 )\) lie on the line as shown.
  1. Find, to 3 decimal places, the value of \(p\) and the value of \(q\). Using the model with the values of \(p\) and \(q\) found in part (a),
  2. find the rate of increase of the surface area of the pond covered by duckweed, in \(\mathrm { m } ^ { 2 }\) / day, exactly 6 days after the start of the study.
    Give your answer to 2 decimal places.
    \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-19_2649_1840_117_114}
Edexcel P3 2021 June Q6
6. Given that \(k\) is a positive constant,
  1. on separate diagrams, sketch the graph with equation
    1. \(y = k - 2 | x |\)
    2. \(y = \left| 2 x - \frac { k } { 3 } \right|\) Show on each sketch the coordinates, in terms of \(k\), of each point where the graph meets or cuts the axes.
  2. Hence find, in terms of \(k\), the values of \(x\) for which $$\left| 2 x - \frac { k } { 3 } \right| = k - 2 | x |$$ giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-23_2647_1840_118_111}
Edexcel P3 2021 June Q7
7. Given that $$x = 6 \sin ^ { 2 } 2 y \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { A \sqrt { \left( B x - x ^ { 2 } \right) } }$$ where \(A\) and \(B\) are integers to be found.
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Edexcel P3 2021 June Q8
8. A scientist is studying a population of fish in a lake. The number of fish, \(N\), in the population, \(t\) years after the start of the study, is modelled by the equation $$N = \frac { 600 \mathrm { e } ^ { 0.3 t } } { 2 + \mathrm { e } ^ { 0.3 t } } \quad t \geqslant 0$$ Use the equation of the model to answer parts (a), (b), (c), (d) and (e).
  1. Find the number of fish in the lake at the start of the study.
  2. Find the upper limit to the number of fish in the lake.
  3. Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.
  4. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { A \mathrm { e } ^ { 0.3 t } } { \left( 2 + \mathrm { e } ^ { 0.3 t } \right) ^ { 2 } }$$ where \(A\) is a constant to be found. Given that when \(t = T , \frac { \mathrm {~d} N } { \mathrm {~d} t } = 8\)
  5. find the value of \(T\) to one decimal place.
    (Solutions relying entirely on calculator technology are not acceptable.)
    \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-27_2644_1840_118_111}
Edexcel P3 2021 June Q9
  1. (a) Express \(12 \sin x - 5 \cos x\) in the form \(R \sin ( x - \alpha )\), 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.
The function g is defined by $$g ( \theta ) = 10 + 12 \sin \left( 2 \theta - \frac { \pi } { 6 } \right) - 5 \cos \left( 2 \theta - \frac { \pi } { 6 } \right) \quad \theta > 0$$ Find
(b) (i) the minimum value of \(\mathrm { g } ( \theta )\)
(ii) the smallest value of \(\theta\) at which the minimum value occurs. The function h is defined by $$\mathrm { h } ( \beta ) = 10 - ( 12 \sin \beta - 5 \cos \beta ) ^ { 2 }$$ (c) Find the range of h .
\includegraphics[max width=\textwidth, alt={}]{76205772-5395-4ab2-96f9-ad9803b8388c-32_2644_1837_118_114}
Edexcel P3 2022 June Q1
  1. The curve \(C\) has equation
$$y = ( 3 x - 2 ) ^ { 6 }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Given that the point \(P \left( \frac { 1 } { 3 } , 1 \right)\) lies on \(C\),
  2. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
Edexcel P3 2022 June Q2
2. The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 5 - x } { 3 x + 2 } & x \in \mathbb { R } , x \neq - \frac { 2 } { 3 }
\mathrm {~g} ( x ) = 2 x - 7 & x \in \mathbb { R } \end{array}$$
  1. Find the value of \(\mathrm { fg } ( 5 )\)
  2. Find \(\mathrm { f } ^ { - 1 }\)
  3. Solve the equation $$\mathrm { f } \left( \frac { 1 } { a } \right) = \mathrm { g } ( a + 3 )$$
Edexcel P3 2022 June Q3
3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Given that \(k\) is a positive constant,
  1. find $$\int \frac { 9 x } { 3 x ^ { 2 } + k } d x$$ Given also that $$\int _ { 2 } ^ { 5 } \frac { 9 x } { 3 x ^ { 2 } + k } \mathrm {~d} x = \ln 8$$
  2. find the value of \(k\)
Edexcel P3 2022 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44035bf8-f54c-472a-b969-b4fa4fa3d203-10_677_839_251_516} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The number of subscribers to an online video streaming service, \(N\), is modelled by the equation $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years since monitoring began.
The line in Figure 1 shows the linear relationship between \(t\) and \(\log _ { 10 } N\)
The line passes through the points \(( 0,3.08 )\) and \(( 5,3.85 )\) Using this information,
  1. find an equation for this line.
  2. Find the value of \(a\) and the value of \(b\), giving your answers to 3 significant figures. When \(t = T\) the number of subscribers is 500000 According to the model,
  3. find the value of \(T\)
Edexcel P3 2022 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44035bf8-f54c-472a-b969-b4fa4fa3d203-14_668_812_258_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | k x - 9 | - 2 \quad x \in \mathbb { R }$$ and \(k\) is a positive constant. The graph intersects the \(y\)-axis at the point \(A\) and has a minimum point at \(B\) as shown.
    1. Find the \(y\) coordinate of \(A\)
    2. Find, in terms of \(k\), the \(x\) coordinate of \(B\)
  2. Find, in terms of \(k\), the range of values of \(x\) that satisfy the inequality $$| k x - 9 | - 2 < 0$$ Given that the line \(y = 3 - 2 x\) intersects the graph \(y = \mathrm { f } ( x )\) at two distinct points,
  3. find the range of possible values of \(k\)
Edexcel P3 2022 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44035bf8-f54c-472a-b969-b4fa4fa3d203-18_579_643_255_653} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \section*{In this question you must show all stages of your working.} Solutions relying entirely on calculator technology are not acceptable. The function f is defined by $$f ( x ) = 5 \left( x ^ { 2 } - 2 \right) ( 4 x + 9 ) ^ { \frac { 1 } { 2 } } \quad x \geqslant - \frac { 9 } { 4 }$$
  1. Show that $$f ^ { \prime } ( x ) = \frac { k \left( 5 x ^ { 2 } + 9 x - 2 \right) } { ( 4 x + 9 ) ^ { \frac { 1 } { 2 } } }$$ where \(k\) is an integer to be found.
  2. Hence, find the values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\) Figure 3 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). The curve has a local maximum at the point \(P\)
  3. Find the exact coordinates of \(P\) The function g is defined by $$g ( x ) = 2 f ( x ) + 4 \quad - \frac { 9 } { 4 } \leqslant x \leqslant 0$$
  4. Find the range of g
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 $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$