Questions P3 (1243 questions)

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CAIE P3 2024 November Q3
5 marks Moderate -0.5
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
5 marks Moderate -0.5
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
6 marks Standard +0.3
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
7 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Moderate -0.3
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 marks Standard +0.8
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
6 marks Standard +0.3
  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
8 marks Moderate -0.3
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
5 marks Moderate -0.8
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
11 marks Standard +0.3
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
8 marks Standard +0.8
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
9 marks Standard +0.3
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
11 marks Standard +0.3
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
10 marks Moderate -0.3
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
7 marks Standard +0.3
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.
    Leave blankQ9
    END
Edexcel P3 2021 January Q1
3 marks Moderate -0.8
  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
6 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Moderate -0.3
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
11 marks Standard +0.3
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
8 marks Standard +0.8
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
9 marks Standard +0.8
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.