Questions P3 (1203 questions)

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CAIE P3 2024 November Q11
11 Let \(\mathrm { f } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } - 3 \mathrm { e } ^ { x } + 2 }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-16_2718_35_107_2012}
  2. Use the substitution \(u = e ^ { x }\) and partial fractions to find the exact value of \(\int _ { \ln 3 } ^ { \ln 5 } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-18_2718_42_107_2007}
CAIE P3 2024 November Q1
1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
  1. On the Argand diagram below, sketch the locus of the points representing \(z\).
  2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}
CAIE P3 2024 November Q2
2 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 4\).
  1. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { \frac { 4 } { 5 - 2 x _ { n } } }$$ converges, then it converges to a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. The equation has a root close to 1.2 . Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2024 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).
  1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
  2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour.
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}
CAIE P3 2024 November Q4
4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2024 November Q5
5
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
  2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).
CAIE P3 2024 November Q6
6 The lines \(l\) and \(m\) have vector equations $$l : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { k } ) \quad \text { and } \quad m : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) .$$ Lines \(l\) and \(m\) intersect at the point \(P\).
  1. State the coordinates of \(P\).
  2. Find the exact value of the cosine of the acute angle between \(l\) and \(m\).
  3. The point \(A\) on line \(I\) has coordinates ( \(0,1,1\) ). The point \(B\) on line \(m\) has coordinates ( \(0,2 , - 8\) ). Find the exact area of triangle \(A P B\).
CAIE P3 2024 November Q7
7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
  2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.
CAIE P3 2024 November Q8
4 marks
8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
  3. State the set of values of \(x\) for which the expansion in part (b) is valid.
CAIE P3 2024 November Q9
9
  1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
  2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).
CAIE P3 2024 November Q10
10 A water tank is in the shape of a cuboid with base area \(40000 \mathrm {~cm} ^ { 2 }\). At time \(t\) minutes the depth of water in the tank is \(h \mathrm {~cm}\). Water is pumped into the tank at a rate of \(50000 \mathrm {~cm} ^ { 3 }\) per minute. Water is leaking out of the tank through a hole in the bottom at a rate of \(600 \mathrm {~cm} ^ { 3 }\) per minute.
  1. Show that \(200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 250 - 3 h\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-17_2723_33_99_22}
  2. It is given that when \(t = 0 , h = 50\). Find the time taken for the depth of water in the tank to reach 80 cm . Give your answer correct to 2 significant figures.
CAIE P3 2024 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_565_634_260_717} The diagram shows the curve \(y = 2 \sin x \sqrt { 2 + \cos x }\), for \(0 \leqslant x \leqslant 2 \pi\), and its minimum point \(M\), where \(x = a\).
  1. Find the value of \(a\) correct to 2 decimal places.
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_2716_38_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-19_2726_33_97_22}
  2. Use the substitution \(u = 2 + \cos x\) to find the exact area of the shaded region \(R\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
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}
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