Questions — CAIE (7279 questions)

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CAIE P2 2002 June Q5
  1. Find the exact coordinates of \(P\).
  2. Show that the \(x\)-coordinates of \(Q\) and \(R\) satisfy the equation $$x = \frac { 1 } { 4 } e ^ { x } .$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 4 } e ^ { x _ { n } }$$ with initial value \(x _ { 1 } = 0\), to find the \(x\)-coordinate of \(Q\) correct to 2 decimal places, showing the value of each approximation that you calculate.
CAIE P2 2010 June Q6
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has only one root.
  2. Verify by calculation that this root lies between \(x = 1.3\) and \(x = 1.4\).
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
  4. Use the iterative formula \(x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2014 June Q5
  1. Prove that \(\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }\).
  2. Hence
    (a) find the exact value of \(\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi\),
    (b) evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta\).
CAIE P2 2003 November Q4
  1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(\alpha\).
  2. Hence show that one solution of the equation $$\cos \theta + ( \sqrt { } 3 ) \sin \theta = \sqrt { } 2$$ is \(\theta = \frac { 7 } { 12 } \pi\), and find the other solution in the interval \(0 < \theta < 2 \pi\).
  3. By sketching a suitable pair of graphs, for \(x < 0\), show that exactly one root of the equation \(x ^ { 2 } = 2 ^ { x }\) is negative.
  4. Verify by calculation that this root lies between - 1.0 and - 0.5 .
  5. Use the iterative formula $$x _ { n + 1 } = - \sqrt { } \left( 2 ^ { x _ { n } } \right)$$ to determine this root correct to 2 significant figures, showing the result of each iteration.
CAIE P2 2007 November Q7
  1. Prove the identity $$( \cos x + 3 \sin x ) ^ { 2 } \equiv 5 - 4 \cos 2 x + 3 \sin 2 x$$
  2. Using the identity, or otherwise, find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \cos x + 3 \sin x ) ^ { 2 } d x$$
CAIE P2 2017 November Q5
  1. Show that the \(x\)-coordinate of \(Q\) satisfies the equation \(x = \frac { 9 } { 8 } - \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
  2. Use an iterative formula based on the equation in part (i) to find the \(x\)-coordinate of \(Q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P3 2020 June Q7
  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
  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
  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
  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
  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
  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 2013 June Q6
  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 \(0 ^ { \circ } < x < 360 ^ { \circ }\).
  3. Express \(\frac { 1 } { x ^ { 2 } ( 2 x + 1 ) }\) in the form \(\frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 2 x + 1 }\).
  4. The variables \(x\) and \(y\) satisfy the differential equation $$y = x ^ { 2 } ( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$ and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms.
    (a) The complex number \(w\) is such that \(\operatorname { Re } w > 0\) and \(w + 3 w ^ { * } = \mathrm { i } w ^ { 2 }\), where \(w ^ { * }\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    (b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 2 i | \leqslant 2\) and \(0 \leqslant \arg ( z + 2 ) \leqslant \frac { 1 } { 4 } \pi\). Calculate the greatest value of \(| z |\) for points in this region, giving your answer correct to 2 decimal places.
CAIE P3 2015 June Q8
  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
  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
  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
  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
  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
  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
  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:
    (a) \(z + 2 z ^ { * }\);
    (b) \(\frac { z ^ { * } } { \mathrm { i } z }\).
  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
  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
  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
  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 FP1 2017 November Q8
  1. Find the value of \(I _ { 2 }\).
  2. Show that, for \(n > 2\), $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
  3. The curve \(C\) has equation \(y = \sec ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2019 November Q5
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).