Questions P3 (1203 questions)

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Edexcel P3 2023 January Q10
  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
  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
  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 \(\alpha\)
Edexcel P3 2024 January Q3
  1. The amount of money raised for a charity is being monitored.
The total amount raised in the \(t\) months after monitoring began, \(\pounds D\), is modelled by the equation $$\log _ { 10 } D = 1.04 + 0.38 t$$
  1. Write this equation in the form $$D = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give each value to 4 significant figures. When \(t = T\), the total amount of money raised is \(\pounds 45000\)
    According to the model,
  2. find the value of \(T\), giving your answer to 3 significant figures. The charity aims to raise a total of \(\pounds 350000\) within the first 12 months of monitoring.
    According to the model,
  3. determine whether or not the charity will achieve its aim.
Edexcel P3 2024 January Q4
  1. The function f is defined by
$$f ( x ) = \frac { 2 x ^ { 2 } - 32 } { 3 x ^ { 2 } + 7 x - 20 } + \frac { 8 } { 3 x - 5 } \quad x \in \mathbb { R } \quad x > 2$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 5 }\)
  2. Show, using calculus, that f is a decreasing function. You must make your reasoning clear. The function g is defined by $$g ( x ) = 3 + 2 \ln x \quad x \geqslant 1$$
  3. Find \(\mathrm { g } ^ { - 1 }\)
  4. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 5$$
Edexcel P3 2024 January Q5
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} The temperature, \(T ^ { \circ } \mathrm { C }\), of the air in a room \(t\) minutes after a heat source is switched off, is modelled by the equation $$T = 10 + A \mathrm { e } ^ { - B t }$$ where \(A\) and \(B\) are constants.
Given that the temperature of the air in the room at the instant the heat source was switched off was \(18 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\) Given also that, exactly 45 minutes after the heat source was switched off, the temperature of the air in the room was \(16 ^ { \circ } \mathrm { C }\),
  2. find the value of \(B\) to 3 significant figures. Using the values for \(A\) and \(B\),
  3. find, according to the model, the rate of change of the temperature of the air in the room exactly two minutes after the heat source was switched off.
    Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.
  4. Explain why, according to the model, the temperature of the air in the room cannot fall to \(5 ^ { \circ } \mathrm { C }\)
Edexcel P3 2024 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76989f19-2624-4e86-a8ee-4978dd1014c2-14_741_844_258_612} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 e ^ { 3 \sin x } \cos x \quad 0 \leqslant x \leqslant 2 \pi$$ The curve intersects the \(x\)-axis at point \(R\), as shown in Figure 1.
  1. State the coordinates of \(R\) The curve has two turning points, at point \(P\) and point \(Q\), also shown in Figure 1.
  2. Show that, at points \(P\) and \(Q\), $$a \sin ^ { 2 } x + b \sin x + c = 0$$ where \(a\), \(b\) and \(c\) are integers to be found.
  3. Hence find the \(x\) coordinate of point \(Q\), giving your answer to 3 decimal places.
Edexcel P3 2024 January Q7
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = \frac { 16 } { 9 ( 3 x - k ) } \quad x \neq \frac { k } { 3 }$$ where \(k\) is a positive constant not equal to 3
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving your answer in simplest form in terms of \(k\). The point \(P\) with \(x\) coordinate 1 lies on \(C\).
    Given that the gradient of the curve at \(P\) is - 12
  2. find the two possible values of \(k\) Given also that \(k < 3\)
  3. find the equation of the normal to \(C\) at \(P\), writing your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
  4. show, using algebraic integration that, $$\int _ { 1 } ^ { 3 } \frac { 16 } { 9 ( 3 x - k ) } d x = \lambda \ln 10$$ where \(\lambda\) is a constant to be found.
Edexcel P3 2024 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76989f19-2624-4e86-a8ee-4978dd1014c2-22_652_634_255_717} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The graph shown in Figure 2 has equation $$y = a - | 2 x - b |$$ where \(a\) and \(b\) are positive constants, \(a > b\)
  1. Find, giving your answer in terms of \(a\) and \(b\),
    1. the coordinates of the maximum point of the graph,
    2. the coordinates of the point of intersection of the graph with the \(y\)-axis,
    3. the coordinates of the points of intersection of the graph with the \(x\)-axis. On page 24 there is a copy of Figure 2 called Diagram 1.
  2. On Diagram 1, sketch the graph with equation $$y = | x | - 1$$ Given that the graphs \(y = | x | - 1\) and \(y = a - | 2 x - b |\) intersect at \(x = - 3\) and \(x = 5\)
  3. find the value of \(a\) and the value of \(b\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{76989f19-2624-4e86-a8ee-4978dd1014c2-24_675_652_1959_712} \captionsetup{labelformat=empty} \caption{Diagram 1}
    \end{figure}
Edexcel P3 2024 January 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 the equation $$\frac { 3 \sin \theta \cos \theta } { \cos \theta + \sin \theta } = ( 2 + \sec 2 \theta ) ( \cos \theta - \sin \theta )$$ can be written in the form $$3 \sin 2 \theta - 4 \cos 2 \theta = 2$$
  2. Hence solve for \(\pi < x < \frac { 3 \pi } { 2 }\) $$\frac { 3 \sin x \cos x } { \cos x + \sin x } = ( 2 + \sec 2 x ) ( \cos x - \sin x )$$ giving the answer to 3 significant figures.
Edexcel P3 2021 June Q1
  1. The curve \(C\) has equation
$$y = x ^ { 2 } \cos \left( \frac { 1 } { 2 } x \right) \quad 0 < x \leqslant \pi$$ The curve has a stationary point at the point \(P\).
  1. Show, using calculus, that the \(x\) coordinate of \(P\) is a solution of the equation $$x = 2 \arctan \left( \frac { 4 } { x } \right)$$ Using the iteration formula $$x _ { n + 1 } = 2 \arctan \left( \frac { 4 } { x _ { n } } \right) \quad x _ { 1 } = 2$$
  2. find the value of \(x _ { 2 }\) and the value of \(x _ { 6 }\), giving your answers to 3 decimal places.
Edexcel P3 2021 June Q2
2. (a) Show that $$\frac { 1 - \cos 2 x } { 2 \sin 2 x } \equiv k \tan x \quad x \neq ( 90 n ) ^ { \circ } \quad n \in \mathbb { Z }$$ where \(k\) is a constant to be found.
(b) Hence solve, for \(0 < \theta < 90 ^ { \circ }\) $$\frac { 9 ( 1 - \cos 2 \theta ) } { 2 \sin 2 \theta } = 2 \sec ^ { 2 } \theta$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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