Questions P3 (1243 questions)

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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.
CAIE P3 2012 June Q9
10 marks Standard +0.3
9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.
  1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
  2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
  3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).
CAIE P3 2021 November Q9
11 marks Standard +0.3
9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.
  1. Show that \(l\) and \(m\) are perpendicular.
  2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
  3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q1
3 marks Moderate -0.5
1 Find the set of values of \(x\) for which \(3 \left( 2 ^ { 3 x + 1 } \right) < 8\). Give your answer in a simplified exact form.
CAIE P3 2020 Specimen Q2
4 marks Moderate -0.8
2
  1. Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 Specimen Q3
4 marks Easy -1.3
3
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Solve the inequality \(3 x - 1 > | 2 x - 3 |\).
CAIE P3 2020 Specimen Q4
9 marks Standard +0.3
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t - 3 } , \quad y = 4 \ln t$$ where \(t > 0\). When \(t = a\) the gradient of the curve is 2 .
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
  2. Verify by calculation that this equation has a root between 1 and 2 .
  3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2020 Specimen Q5
7 marks Standard +0.8
5
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
  2. Show that \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).
CAIE P3 2020 Specimen Q6
8 marks Moderate -0.5
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the argument of \(\frac { u } { v }\).
    In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
CAIE P3 2020 Specimen Q7
9 marks Standard +0.3
7
  1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).