Questions — Edexcel P3 (133 questions)

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Edexcel P3 2020 January Q1
  1. A population of a rare species of toad is being studied.
The number of toads, \(N\), in the population, \(t\) years after the start of the study, is modelled by the equation $$N = \frac { 900 \mathrm { e } ^ { 0.12 t } } { 2 \mathrm { e } ^ { 0.12 t } + 1 } \quad t \geqslant 0 , t \in \mathbb { R }$$ According to this model,
  1. calculate the number of toads in the population at the start of the study,
  2. find the value of \(t\) when there are 420 toads in the population, giving your answer to 2 decimal places.
  3. Explain why, according to this model, the number of toads in the population can never reach 500
Edexcel P3 2020 January Q2
2. The function \(f\) and the function \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 12 } { x + 1 } & x > 0 , x \in \mathbb { R }
\mathrm {~g} ( x ) = \frac { 5 } { 2 } \ln x & x > 0 , x \in \mathbb { R } \end{array}$$
  1. Find, in simplest form, the value of \(\mathrm { fg } \left( \mathrm { e } ^ { 2 } \right)\)
  2. Find f-1
  3. Hence, or otherwise, find all real solutions of the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )$$
Edexcel P3 2020 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-08_599_883_299_536} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(\log _ { 10 } x\)
The line passes through the points \(( 0,4 )\) and \(( 6,0 )\) as shown.
  1. Find an equation linking \(\log _ { 10 } y\) with \(\log _ { 10 } x\)
  2. Hence, or otherwise, express \(y\) in the form \(p x ^ { q }\), where \(p\) and \(q\) are constants to be found.
Edexcel P3 2020 January Q4
4. (i) $$f ( x ) = \frac { ( 2 x + 5 ) ^ { 2 } } { x - 3 } \quad x \neq 3$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) in the form \(\frac { P ( x ) } { Q ( x ) }\) where \(P ( x )\) and \(Q ( x )\) are fully factorised quadratic expressions.
  2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
    (ii) $$g ( x ) = x \sqrt { \sin 4 x } \quad 0 \leqslant x < \frac { \pi } { 4 }$$ The curve with equation \(y = g ( x )\) has a maximum at the point \(M\). Show that the \(x\) coordinate of \(M\) satisfies the equation $$\tan 4 x + k x = 0$$ where \(k\) is a constant to be found.
Edexcel P3 2020 January Q5
5. (a) Use the substitution \(t = \tan x\) to show that the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$5 t ^ { 4 } - 24 t ^ { 2 } - 5 = 0$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ Show each stage of your working and give your answers to one decimal place.
Edexcel P3 2020 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-18_736_1102_258_427} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 2 x - 5 | + 3 \quad x \geqslant 0$$ The vertex of the graph is at point \(P\) as shown.
  1. State the coordinates of \(P\).
  2. Solve the equation \(\mathrm { f } ( x ) = 3 x - 2\) Given that the equation $$f ( x ) = k x + 2$$ where \(k\) is a constant, has exactly two roots,
  3. find the range of values of \(k\).
Edexcel P3 2020 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-22_707_1047_264_463} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 2 \cos 3 x - 3 x + 4 \quad x > 0$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(P\), as shown in Figure 3.
Given that the \(x\) coordinate of \(P\) is \(\alpha\),
  1. show that \(\alpha\) lies between 0.8 and 0.9 The iteration formula $$x _ { n + 1 } = \frac { 1 } { 3 } \arccos \left( 1.5 x _ { n } - 2 \right)$$ can be used to find an approximate value for \(\alpha\).
  2. Using this iteration formula with \(x _ { 1 } = 0.8\) find, to 4 decimal places, the value of
    1. \(X _ { 2 }\)
    2. \(X _ { 5 }\) The point \(Q\) and the point \(R\) are local minimum points on the curve, as shown in Figure 3.
      Given that the \(x\) coordinates of \(Q\) and \(R\) are \(\beta\) and \(\lambda\) respectively, and that they are the two smallest values of \(x\) at which local minima occur,
  3. find, using calculus, the exact value of \(\beta\) and the exact value of \(\lambda\).
Edexcel P3 2020 January Q8
8. (i) Find, using algebraic integration, the exact value of $$\int _ { 3 } ^ { 42 } \frac { 2 } { 3 x - 1 } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) $$\mathrm { h } ( x ) = \frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 1 } { ( x - 1 ) ^ { 2 } } \quad x > 1$$ Given \(\mathrm { h } ( x ) = A x + B + \frac { C } { ( x - 1 ) ^ { 2 } }\) where \(A , B\) and \(C\) are constants to be found, find $$\int \mathrm { h } ( x ) \mathrm { d } x$$ \includegraphics[max width=\textwidth, alt={}, center]{1c700103-ecab-4a08-b411-3f445ed88885-26_2258_47_312_1985}
Edexcel P3 2020 January Q9
9. $$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$
  1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta + \alpha )\), 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 curve with equation \(y = \cos \theta\) is transformed onto the curve with equation \(y = \mathrm { f } ( \theta )\) by a sequence of two transformations. Given that the first transformation is a stretch and the second a translation,
    1. describe fully the transformation that is a stretch,
    2. describe fully the transformation that is a translation. Given $$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
  3. find the range of g.
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    Q9

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Edexcel P3 2021 January Q1
  1. Find
$$\int \frac { x ^ { 2 } - 5 } { 2 x ^ { 3 } } \mathrm {~d} x \quad x > 0$$ giving your answer in simplest form.
Edexcel P3 2021 January Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-04_903_1148_123_399} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in \mathbb { R }\) and \(\mathrm { f } ( x )\) is a polynomial. The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\) There is a maximum turning point at \(( 1,2 )\) and a minimum turning point at \(( 3,0 )\) On separate diagrams, sketch the curve with equation
  1. \(y = 3 f ( 2 x )\)
  2. \(y = \mathrm { f } ( - x ) - 1\) On each sketch, show clearly the coordinates of
    • the point where the curve crosses the \(y\)-axis
    • any maximum or minimum turning points
Edexcel P3 2021 January Q3
3. $$f ( x ) = 3 - \frac { x - 2 } { x + 1 } + \frac { 5 x + 26 } { 2 x ^ { 2 } - 3 x - 5 } \quad x > 4$$
  1. Show that $$\mathrm { f } ( x ) = \frac { a x + b } { c x + d } \quad x > 4$$ where \(a , b , c\) and \(d\) are integers to be found.
  2. Hence find \(\mathrm { f } ^ { - 1 } ( x )\)
  3. Find the domain of \(\mathrm { f } ^ { - 1 }\)
Edexcel P3 2021 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-10_646_762_264_593} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | 3 x + a | + a$$ and where \(a\) is a positive constant. The graph has a vertex at the point \(P\), as shown in Figure 2 .
  1. Find, in terms of \(a\), the coordinates of \(P\).
  2. Sketch the graph with equation \(y = g ( x )\), where $$g ( x ) = | x + 5 a |$$ On your sketch, show the coordinates, in terms of \(a\), of each point where the graph cuts or meets the coordinate axes. The graph with equation \(y = \mathrm { g } ( x )\) intersects the graph with equation \(y = \mathrm { f } ( x )\) at two points.
  3. Find, in terms of \(a\), the coordinates of the two points. \includegraphics[max width=\textwidth, alt={}, center]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel P3 2021 January Q5
5. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside an oven, \(t\) minutes after the oven is switched on, is given by $$\theta = A - 180 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature inside the oven is initially \(18 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\). The temperature inside the oven, 5 minutes after the oven is switched on, is \(90 ^ { \circ } \mathrm { C }\).
  2. Show that \(k = p \ln q\) where \(p\) and \(q\) are rational numbers to be found. Hence find
  3. the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,
  4. the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.
Edexcel P3 2021 January Q6
6. $$\mathrm { f } ( x ) = x \cos \left( \frac { x } { 3 } \right) \quad x > 0$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\)
  2. Show that the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\) can be written as $$x = k \arctan \left( \frac { k } { x } \right)$$ where \(k\) is an integer to be found.
  3. Starting with \(x _ { 1 } = 2.5\) use the iteration formula $$x _ { n + 1 } = k \arctan \left( \frac { k } { x _ { n } } \right)$$ with the value of \(k\) found in part (b), to calculate the values of \(x _ { 2 }\) and \(x _ { 6 }\) giving your answers to 3 decimal places.
  4. Using a suitable interval and a suitable function that should be stated, show that a root of \(\mathrm { f } ^ { \prime } ( x ) = 0\) is 2.581 correct to 3 decimal places.
    In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Edexcel P3 2021 January Q7
7. (a) Prove that $$\frac { \sin 2 x } { \cos x } + \frac { \cos 2 x } { \sin x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$7 + \frac { \sin 4 \theta } { \cos 2 \theta } + \frac { \cos 4 \theta } { \sin 2 \theta } = 3 \cot ^ { 2 } 2 \theta$$ giving your answers in radians to 3 significant figures where appropriate.
Edexcel P3 2021 January Q8
  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
9. Find
  1. \(\int \frac { 3 x - 2 } { 3 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
  2. \(\int \frac { \mathrm { e } ^ { 2 x } } { \left( \mathrm { e } ^ { 2 x } - 1 \right) ^ { 3 } } \mathrm {~d} x \quad x \neq 0\)
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Edexcel P3 2021 January Q10
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
  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
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
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
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
    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
    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.