Questions P3 (1203 questions)

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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. \(\int \frac { \mathrm { e } ^ { 2 x } } { \left( \mathrm { e } ^ { 2 x } - 1 \right) ^ { 3 } } \mathrm {~d} x \quad x \neq 0\)
    VIIV SIHI NI JIIIM IONOOVIUV SIHI NI III M M I ON OOVI4V SIHI NI IIIYM ION OC
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.
    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. \(\alpha\)