Questions P3 (1243 questions)

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CAIE P3 2020 Specimen Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-14_485_716_262_676} In the diagram, \(O A B C\) is a pyramid in which \(O A = 2\) units, \(O B = 4\) units and \(O C = 2\) units. The edge \(O C\) is vertical, the base \(O A B\) is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O B\) and \(O C\) respectively. The midpoints of \(A B\) and \(B C\) are \(M\) and \(N\) respectively.
  1. Express the vectors \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Calculate the angle between the directions of \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\).
  3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-16_307_593_269_735} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \sin x\), find the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2020 Specimen Q10
11 marks Standard +0.5
10 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\).
The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x ) .$$
  2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE P3 2020 June Q7
9 marks Challenging +1.2
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)\).
CAIE P3 2021 June Q10
10 marks Standard +0.3
  1. Given that the sum of the areas of the shaded sectors is \(90 \%\) of the area of the trapezium, show that \(x\) satisfies the equation \(x = 0.9 ( 2 - \cos x ) \sin x\).
  2. Verify by calculation that \(x\) lies between 0.5 and 0.7 .
  3. Show that if a sequence of values in the interval \(0 < x < \frac { 1 } { 2 } \pi\) given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( 2 - \frac { x _ { n } } { 0.9 \sin x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (a).
  4. Use this iterative formula to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q10
10 marks Standard +0.3
  1. Find the exact coordinates of \(M\).
  2. Using the substitution \(u = 3 - 2 x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt { 13 }\), where \(a\) is a rational number. [5]
CAIE P3 2021 November Q9
10 marks Standard +0.3
  1. Find the \(x\)-coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
  2. Using the substitution \(u = \sqrt { x }\), show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 3 } \ln 5\).
CAIE P3 2024 November Q6
8 marks Standard +0.3
  1. Given that the \(x\)-coordinate of \(M\) lies in the interval \(\frac { 1 } { 2 } \pi < x < \frac { 3 } { 4 } \pi\), find the exact coordinates of \(M\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_2718_35_107_2012} \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-11_2725_35_99_20}
  2. Find the exact area of the region \(R\).
CAIE P3 2012 June Q7
8 marks Standard +0.3
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing the complex numbers \(u , 1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\) respectively.
  3. By considering the arguments of \(1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\), show that $$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$
CAIE P3 2015 June Q8
10 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [5]
CAIE P3 2016 June Q9
9 marks Challenging +1.2
  1. Sketch this diagram and state fully the geometrical relationship between \(O B\) and \(A C\).
  2. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex number \(\frac { u } { v }\).
  3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
CAIE P3 2017 June Q5
8 marks Moderate -0.3
  1. Show that \(x\) satisfies the equation \(x = \frac { 1 } { 3 } ( \pi + \sin x )\).
  2. Verify by calculation that \(x\) lies between 1 and 1.5.
  3. Use an iterative formula based on the equation in part (i) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2017 March Q8
10 marks Standard +0.3
  1. Showing all your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( z ) = 0\).
  2. Find the other three roots of the equation \(\mathrm { p } ( z ) = 0\).
CAIE P3 2013 November Q7
10 marks Standard +0.3
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
CAIE P3 2016 November Q9
10 marks Standard +0.3
  1. Solve the equation \(( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 1 - \mathrm { i } | \leqslant 2\) and \(- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
CAIE P3 2016 November Q7
9 marks Standard +0.3
  1. Find the modulus and argument of \(z\).
  2. Express each of the following in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact:
    1. \(z + 2 z ^ { * }\);
    2. \(\frac { z ^ { * } } { \mathrm { i } z }\).
    3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z ^ { * }\) and \(\mathrm { i } z\) respectively. Prove that angle \(A O B\) is equal to \(\frac { 1 } { 6 } \pi\).
CAIE P3 2017 November Q8
10 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2019 November Q7
9 marks Standard +0.8
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the equation of the plane containing \(l\) and \(m\).
CAIE P3 2019 November Q6
7 marks Standard +0.3
  1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
  2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
  3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
Edexcel P3 2022 October Q8
9 marks Standard +0.3
  1. Express \(8 \sin x - 15 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures. $$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
  2. Find
    1. the minimum value of \(\mathrm { f } ( x )\)
    2. the smallest value of \(x\) at which this minimum value occurs.
  3. State the \(y\) coordinate of the minimum points on the curve with equation $$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
  4. State the smallest value of \(x\) at which a maximum point occurs for the curve with equation $$y = - \mathrm { f } ( 2 x ) \quad x > 0$$ \section*{8. In this question you must show all stages of your working.
    In this question you must show all stages of your working.}
CAIE P3 2019 June Q5
7 marks Standard +0.3
5 Throughout this question the use of a calculator is not permitted. It is given that the complex number \(- 1 + ( \sqrt { } 3 ) \mathrm { i }\) is a root of the equation $$k x ^ { 3 } + 5 x ^ { 2 } + 10 x + 4 = 0$$ where \(k\) is a real constant.
  1. Write down another root of the equation.
  2. Find the value of \(k\) and the third root of the equation.
CAIE P3 2013 November Q8
10 marks Standard +0.3
8 Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
CAIE P3 2014 November Q5
8 marks Standard +0.3
5 Throughout this question the use of a calculator is not permitted. The complex numbers \(w\) and \(z\) satisfy the relation $$w = \frac { z + \mathrm { i } } { \mathrm { i } z + 2 }$$
  1. Given that \(z = 1 + \mathrm { i }\), find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Given instead that \(w = z\) and the real part of \(z\) is negative, find \(z\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2024 June Q1
4 marks Moderate -0.3
Solve the equation \(8^{3-6x} = 4 \times 5^{-2x}\). Give your answer correct to 3 decimal places. [4]
CAIE P3 2024 June Q2
5 marks Standard +0.3
Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]