Questions — CAIE P3 (1110 questions)

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CAIE P3 2024 June Q5
6 marks Standard +0.3
5
  1. It is given that the equation \(\mathrm { e } ^ { 2 x } = 5 + \cos 3 x\) has only one root.
    Show by calculation that this root lies in the interval \(0.7 < x < 0.8\).
  2. Show that if a sequence of values in the interval \(0.7 < x < 0.8\) given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 5 + \cos 3 x _ { n } \right)$$ converges then it converges to the root of the equation in part (a).
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2024 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-08_351_1031_264_516} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { - \mathrm { ax } }\), where \(a\) is a positive constant, and its maximum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 2 } { a } } x e ^ { - a x } d x\).
CAIE P3 2024 June Q7
9 marks Standard +0.3
7
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta\).
  2. Hence find the exact value of \(\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta\).
CAIE P3 2024 June Q8
12 marks Standard +0.3
8 The points \(A , B\) and \(C\) have position vectors \(\overrightarrow { \mathrm { OA } } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } , \overrightarrow { \mathrm { OB } } = 5 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { \mathrm { OC } } = 8 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }\), where \(O\) is the origin. The line \(l _ { 1 }\) passes through \(B\) and \(C\).
  1. Find a vector equation for \(l _ { 1 }\).
    The line \(l _ { 2 }\) has equation \(\mathbf { r } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\).
  2. Find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  3. The point \(D\) on \(l _ { 2 }\) is such that \(\mathrm { AB } = \mathrm { BD }\). Find the position vector of \(D\). \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-13_58_1545_388_349}
CAIE P3 2024 June Q9
10 marks Standard +0.3
9 The complex numbers \(z\) and \(\omega\) are defined by \(z = 1 - i\) and \(\omega = - 3 + 3 \sqrt { 3 } i\).
  1. Express \(z \omega\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real and in exact surd form.
  2. Express \(z\) and \(\omega\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\) in each case.
  3. On an Argand diagram, the points representing \(\omega\) and \(z \omega\) are \(A\) and \(B\) respectively. Prove that \(O A B\) is an isosceles right-angled triangle, where \(O\) is the origin.
  4. Using your answers to part (b), prove that \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).
CAIE P3 2024 June Q10
10 marks Challenging +1.2
10
  1. By writing \(y = \sec ^ { 3 } \theta\) as \(\frac { 1 } { \cos ^ { 3 } \theta }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = 3 \sin \theta \sec ^ { 4 } \theta\).
  2. The variables \(x\) and \(\theta\) satisfy the differential equation $$\left( x ^ { 2 } + 9 \right) \sin \theta \frac { d \theta } { d x } = ( x + 3 ) \cos ^ { 4 } \theta$$ It is given that \(x = 3\) when \(\theta = \frac { 1 } { 3 } \pi\).
    Solve the differential equation to find the value of \(\cos \theta\) when \(x = 0\). Give your answer correct to 3 significant figures.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2020 March Q1
4 marks Moderate -0.8
1
  1. Sketch the graph of \(y = | x - 2 |\).
  2. Solve the inequality \(| x - 2 | < 3 x - 4\).
CAIE P3 2020 March Q2
4 marks Moderate -0.3
2 Solve the equation \(\ln 3 + \ln ( 2 x + 5 ) = 2 \ln ( x + 2 )\). Give your answer in a simplified exact form.
CAIE P3 2020 March Q3
7 marks Standard +0.3
3
  1. By sketching a suitable pair of graphs, show that the equation \(\sec x = 2 - \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.8 and 1 .
  3. Use the iterative formula \(x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { 4 - x _ { n } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 March Q4
7 marks Moderate -0.3
4 Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\). Give your answer in a simplified exact form.
CAIE P3 2020 March Q5
7 marks Standard +0.3
5
  1. Show that \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 2 \cot 2 x\).
  2. Hence solve the equation \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 4\), for \(0 < x < \pi\).
CAIE P3 2020 March Q6
8 marks Standard +0.3
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$ It is given that \(y = 0\) when \(x = 1\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. State what happens to the value of \(y\) as \(x\) tends to infinity.
CAIE P3 2020 March Q7
9 marks Standard +0.8
7 The equation of a curve is \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 5\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + y ^ { 2 } } { y ^ { 2 } - 2 x y }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P3 2020 March Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{8f81a526-783c-4321-b540-c9deccfee17b-12_639_713_262_715} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 3\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The point \(M\) on \(A B\) is such that \(M B = 2 A M\). The midpoint of \(F G\) is \(N\).
  1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Find a vector equation for the line through \(M\) and \(N\).
  3. Find the position vector of \(P\), the foot of the perpendicular from \(D\) to the line through \(M\) and \(N\). [4]
CAIE P3 2020 March Q9
10 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).
  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 2020 March Q10
10 marks Standard +0.3
10
  1. The complex numbers \(v\) and \(w\) satisfy the equations $$v + \mathrm { i } w = 5 \quad \text { and } \quad ( 1 + 2 \mathrm { i } ) v - w = 3 \mathrm { i } .$$ Solve the equations for \(v\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z - 2 - 3 \mathrm { i } | = 1\).
    2. Calculate the least value of \(\arg z\) for points on this locus.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2022 March Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x + 3 | > 3 | x + 2 |\).
CAIE P3 2022 March Q2
4 marks Moderate -0.5
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 - 3 \mathrm { i } | \leqslant 2\) and \(\arg z \leqslant \frac { 3 } { 4 } \pi\).
CAIE P3 2022 March Q3
5 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-05_666_800_260_667} The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y ^ { 2 } = C\), where \(n\) and \(C\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0.31,1.21 )\) and \(( 1.06,0.91 )\), as shown in the diagram. Find the value of \(n\) and find the value of \(C\) correct to 2 decimal places.
CAIE P3 2022 March Q4
5 marks Moderate -0.3
4 The parametric equations of a curve are $$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
CAIE P3 2022 March Q5
6 marks Standard +0.8
5 The angles \(\alpha\) and \(\beta\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) and are such that $$\tan ( \alpha + \beta ) = 2 \quad \text { and } \quad \tan \alpha = 3 \tan \beta .$$ Find the possible values of \(\alpha\) and \(\beta\).
CAIE P3 2022 March Q6
6 marks Challenging +1.2
6 Find the complex numbers \(w\) which satisfy the equation \(w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1\) and are such that \(\operatorname { Re } w \leqslant 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2022 March Q7
7 marks Standard +0.3
7
  1. By sketching a suitable pair of graphs, show that the equation \(4 - x ^ { 2 } = \sec \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \pi\).
  2. Verify by calculation that this root lies between 1 and 2 .
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 4 - \sec \frac { 1 } { 2 } x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 March Q8
8 marks Standard +0.3
8
  1. Find the quotient and remainder when \(8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7\) is divided by \(4 x ^ { 2 } + 1\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\).
CAIE P3 2022 March Q9
9 marks Standard +0.3
9 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\).
Solve the differential equation and find the exact value of \(y\) when \(x = 3\), giving your answer in a simplified form.