Questions — CAIE P3 (1110 questions)

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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.
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]
CAIE P3 2024 June Q3
5 marks Standard +0.3
The square roots of \(24 - 7i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7i\) in exact Cartesian form. [5]
CAIE P3 2024 June Q4
4 marks Moderate -0.5
\includegraphics{figure_4} The variables \(x\) and \(y\) satisfy the equation \(ky = e^{cx}\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((2.80, 0.372)\) and \((5.10, 2.21)\), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures. [4]
CAIE P3 2024 June Q5
5 marks Moderate -0.8
Express \(\frac{6x^2 - 2x + 2}{(x - 1)(2x + 1)}\) in partial fractions. [5]
CAIE P3 2024 June Q6
7 marks Standard +0.3
  1. On an Argand diagram shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(|z - 4 - 3i| \leqslant 2\) and \(\arg(z - 2 - i) \geqslant \frac{1}{4}\pi\). [5]
  2. Calculate the greatest value of \(\arg z\) for points in this region. [2]
CAIE P3 2024 June Q7
3 marks Easy -1.2
Let \(f(x) = 8x^3 + 54x^2 - 17x - 21\).
  1. Show that \(x + 7\) is a factor of \(f(x)\). [1]
  2. Find the quotient when \(f(x)\) is divided by \(x + 7\). [2]
CAIE P3 2024 June Q7
3 marks Standard +0.3
  1. Hence solve the equation $$8 \cos^3 \theta + 54 \cos^2 \theta - 17 \cos \theta - 21 = 0,$$ for \(0° \leqslant \theta \leqslant 360°\). [3]
CAIE P3 2024 June Q8
8 marks Challenging +1.2
  1. Express \(3 \cos 2x - \sqrt{3} \sin 2x\) in the form \(R \cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the exact values of \(R\) and \(\alpha\). [3]
  2. Hence find the exact value of \(\int_0^{\frac{1}{2}\pi} \frac{3}{(3 \cos 2x - \sqrt{3} \sin 2x)^2} \, dx\), simplifying your answer. [5]
CAIE P3 2024 June Q9
11 marks Challenging +1.2
\includegraphics{figure_9} A container in the shape of a cuboid has a square base of side \(x\) and a height of \((10 - x)\). It is given that \(x\) varies with time, \(t\), where \(t > 0\). The container decreases in volume at a rate which is inversely proportional to \(t\). When \(t = \frac{1}{10}\), \(x = \frac{1}{2}\) and the rate of decrease of \(x\) is \(\frac{20}{37}\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac{dx}{dt} = \frac{-1}{2t(20x - 3x^2)}$$ [5]
  2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(x\). [6]