Questions — Edexcel P3 (133 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel P3 2022 January Q7
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. 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
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
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
  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
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
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
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
  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
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
  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
  1. Find, in simplest form,
$$\int ( 2 \cos x - \sin x ) ^ { 2 } d x$$
Edexcel P3 2023 January Q9
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
  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.)