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Edexcel P3 2023 October Q5
7 marks Standard +0.3
  1. The curve \(C\) has equation
$$y = \frac { \ln \left( x ^ { 2 } + k \right) } { x ^ { 2 } + k } \quad x \in \mathbb { R }$$ where \(k\) is a positive constant.
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x \left( B - \ln \left( x ^ { 2 } + k \right) \right) } { \left( x ^ { 2 } + k \right) ^ { 2 } }$$ where \(A\) and \(B\) are constants to be found. Given that \(C\) has exactly three turning points,
  2. find the \(x\) coordinate of each of these points. Give your answer in terms of \(k\) where appropriate.
  3. find the upper limit to the value for \(k\).
Edexcel P3 2023 October Q6
6 marks Moderate -0.3
  1. An area of sea floor is being monitored.
The area of the sea floor, \(S \mathrm {~km} ^ { 2 }\), covered by coral reefs is modelled by the equation $$S = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of years after monitoring began.
Given that $$\log _ { 10 } S = 4.5 - 0.006 t$$
  1. find, according to the model, the area of sea floor covered by coral reefs when \(t = 2\)
  2. find a complete equation for the model in the form $$S = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 3 significant figures.
  3. With reference to the model, interpret the value of the constant \(q\)
Edexcel P3 2023 October Q7
10 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08291ac1-bdd4-4241-8959-7c89318fa5eb-18_554_1129_248_468} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = e ^ { - x ^ { 2 } } \left( 2 x ^ { 2 } - 3 \right) ^ { 2 }$$
  1. Find the range of f
  2. Show that $$\mathrm { f } ^ { \prime } ( x ) = 2 x \left( 2 x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x ^ { 2 } } \left( A - B x ^ { 2 } \right)$$ where \(A\) and \(B\) are constants to be found. Given that the line \(y = k\), where \(k\) is a constant, \(k > 0\), intersects the curve at exactly two distinct points,
  3. find the exact range of values of \(k\)
Edexcel P3 2023 October Q8
8 marks Standard +0.8
  1. (a) Prove that
$$2 \operatorname { cosec } ^ { 2 } 2 \theta ( 1 - \cos 2 \theta ) \equiv 1 + \tan ^ { 2 } \theta$$ (b) Hence solve for \(0 < x < 360 ^ { \circ }\), where \(x \neq ( 90 n ) ^ { \circ } , n \in \mathbb { N }\), the equation $$2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = 4 + 3 \sec x$$ giving your answers to one decimal place.
(Solutions relying entirely on calculator technology are not acceptable.)
Edexcel P3 2023 October Q9
9 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08291ac1-bdd4-4241-8959-7c89318fa5eb-26_613_729_386_667} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation $$y = | 2 - 4 \ln ( x + 1 ) | \quad x > k$$ where \(k\) is a constant.
Given that the curve
  • has an asymptote at \(x = k\)
  • cuts the \(y\)-axis at point \(A\)
  • meets the \(x\)-axis at point \(B\)
    as shown in Figure 2,
    1. state the value of \(k\)
      1. find the \(y\) coordinate of \(A\)
      2. find the exact \(x\) coordinate of \(B\)
    3. Using algebra and showing your working, find the set of values of \(x\) such that
$$| 2 - 4 \ln ( x + 1 ) | > 3$$
Edexcel P3 2023 October Q10
9 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
A curve \(C\) has equation $$x = \sin ^ { 2 } 4 y \quad 0 \leqslant y \leqslant \frac { \pi } { 8 } \quad 0 \leqslant x \leqslant 1$$ The point \(P\) with \(x\) coordinate \(\frac { 1 } { 4 }\) lies on \(C\)
  1. Find the exact \(y\) coordinate of \(P\)
  2. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\)
  3. Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { q + r ( x + s ) ^ { 2 } } }$$ where \(q , r\) and \(s\) are constants to be found. Using the answer to part (c),
    1. state the \(x\) coordinate of the point where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is a minimum,
    2. state the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at this point.
Edexcel P3 2018 Specimen Q1
4 marks Moderate -0.8
  1. Express
$$\frac { 6 x + 4 } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.
Edexcel P3 2018 Specimen Q2
8 marks Standard +0.3
2. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) } \quad x \neq - 3$$ The equation \(x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0\) has a single root which is between 1 and 2
  2. Use the iteration formula $$x _ { n + 1 } = \sqrt { \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) } \quad n \geqslant 0$$ with \(x _ { 0 } = 1\) to find, to 2 decimal places, the value of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) The root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
  3. By choosing a suitable interval, prove that \(\alpha = 1.272\) to 3 decimal places.
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Edexcel P3 2018 Specimen Q3
5 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-08_542_540_269_696} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 \quad x \geqslant 0$$
  1. Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots,
  2. state the set of possible values for \(k\).
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JAHAM ION OOVI4V SIHIL NI JIIIM ION OC
Edexcel P3 2018 Specimen Q4
7 marks Moderate -0.3
4. (i) Find $$\int _ { 5 } ^ { 13 } \frac { 1 } { ( 2 x - 1 ) } \mathrm { d } x$$ writing your answer in its simplest form.
(ii) Use integration to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin 2 x + \sec \frac { 1 } { 3 } x \tan \frac { 1 } { 3 } x \mathrm {~d} x$$
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Edexcel P3 2018 Specimen Q5
6 marks Moderate -0.3
5. Given that $$y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } \quad x \neq 1$$ show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( x - 1 ) ^ { 3 } }\), where \(k\) is a constant to be found.
(6)
Edexcel P3 2018 Specimen Q6
14 marks Moderate -0.3
  1. The functions f and g are defined by
$$\mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 \quad x \in \mathbb { R }$$ $$\mathrm { g } : x \mapsto \ln x \quad x > 0$$
  1. State the range of f .
  2. Find \(\mathrm { fg } ( x )\), giving \(y\) our answer in its simplest form.
  3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
  4. Find \(\mathrm { f } ^ { - 1 }\) stating its domain.
  5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.
Edexcel P3 2018 Specimen Q7
7 marks Standard +0.3
  1. The point \(P\) lies on the curve with equation
$$x = ( 4 y - \sin 2 y ) ^ { 2 }$$ Given that \(P\) has \(( x , y )\) coordinates \(\left( p , \frac { \pi } { 2 } \right)\), where \(p\) is a constant,
  1. find the exact value of \(p\) The tangent to the curve at \(P\) cuts the \(y\)-axis at the point \(A\).
  2. Use calculus to find the coordinates of \(A\).
Edexcel P3 2018 Specimen Q8
7 marks Moderate -0.8
8. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted.
\(N\) and \(T\) are expected to satisfy a relationship of the form $$N = a T ^ { b } \quad \text { where } a \text { and } b \text { are constants }$$
  1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-24_1223_1043_895_461} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
  2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
  3. With reference to the model, interpret the value of the constant \(a\).
Edexcel P3 2018 Specimen Q9
9 marks Standard +0.8
9. (a) Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\) $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.
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Edexcel P3 2018 Specimen Q10
8 marks Moderate -0.3
10. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.
  1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
  2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
  3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.
Edexcel C34 2014 January Q1
6 marks Moderate -0.3
1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) > 0\) You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2014 January Q2
6 marks Standard +0.3
2. Solve, for \(0 \leqslant x \leqslant 270 ^ { \circ }\), the equation $$\frac { \tan 2 x + \tan 50 ^ { \circ } } { 1 - \tan 2 x \tan 50 ^ { \circ } } = 2$$ Give your answers in degrees to 2 decimal places.
(6)
\includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-05_104_95_2613_1786}
Edexcel C34 2014 January Q3
10 marks Standard +0.3
3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$
  1. find the values of the constants \(A , B , C\) and \(D\).
  2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
Edexcel C34 2014 January Q4
11 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-10_606_613_285_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-10_602_608_287_1062} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$ Figure 2 shows a sketch of part of the graph \(y = \mathrm { g } ( x )\), where $$\operatorname { g } ( x ) = \frac { x + 9 } { 2 x + 3 } , \quad x \geqslant 0$$
  1. Find the value of \(\mathrm { fg } ( 1 )\)
  2. State the range of g
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain. Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
  4. state the range of possible values of \(k\).
Edexcel C34 2014 January Q5
9 marks Standard +0.3
  1. (a) Prove, by using logarithms, that
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( 2 ^ { x } \right) = 2 ^ { x } \ln 2$$ The curve \(C\) has the equation $$2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y + 12 = 4 \times 2 ^ { x }$$ The point \(P\), with coordinates \(( 2,0 )\), lies on \(C\).
(b) Find an equation of the tangent to \(C\) at \(P\).
Edexcel C34 2014 January Q6
9 marks Standard +0.8
6. Given that the binomial expansion, in ascending powers of \(x\), of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is \(\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots\)
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find the coefficient of \(x ^ { 6 }\)
Edexcel C34 2014 January Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-20_689_712_248_680} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ( 1 + x ) \ln x , \quad x > 0$$ The curve has a minimum turning point at \(A\).
  1. Find f'(x)
  2. Hence show that the \(x\) coordinate of \(A\) is the solution of the equation $$x = \mathrm { e } ^ { - \frac { 1 + x } { 1 + 2 x } }$$
  3. Use the iteration formula $$x _ { n + 1 } = \mathrm { e } ^ { - \frac { 1 + x _ { n } } { 1 + 2 x _ { n } } } , \quad x _ { 0 } = 0.46$$ to find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
  4. Use your answer to part (c) to estimate the coordinates of \(A\) to 2 decimal places.
Edexcel C34 2014 January Q8
10 marks Standard +0.3
8. (a) Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)
  1. \(2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3\)
  2. \(\tan \theta + \cot \theta = 5\) Give your answers to 3 significant figures.
Edexcel C34 2014 January Q9
15 marks Standard +0.3
9. (a) Use the substitution \(u = 4 - \sqrt { } x\) to find $$\int \frac { \mathrm { d } x } { 4 - \sqrt { } x }$$ A team of scientists is studying a species of slow growing tree.
The rate of change in height of a tree in this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 4 - \sqrt { } h } { 20 }$$ where \(h\) is the height in metres and \(t\) is the time measured in years after the tree is planted.
(b) Find the range in values of \(h\) for which the height of a tree in this species is increasing.
(c) Given that one of these trees is 1 metre high when it is planted, calculate the time it would take to reach a height of 10 metres. Write your answer to 3 significant figures.
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