Questions — CAIE P3 (1070 questions)

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CAIE P3 2021 March Q2
2 The polynomial \(a x ^ { 3 } + 5 x ^ { 2 } - 4 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 2 . Find the values of \(a\) and \(b\).
CAIE P3 2021 March Q3
3 By first expressing the equation \(\tan \left( x + 45 ^ { \circ } \right) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2021 March Q4
4 The variables \(x\) and \(y\) satisfy the differential equation $$( 1 - \cos x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y \sin x$$ It is given that \(y = 4\) when \(x = \pi\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2 \pi\).
CAIE P3 2021 March Q5
5
  1. Express \(\sqrt { 7 } \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(\sqrt { 7 } \sin 2 \theta + 2 \cos 2 \theta = 1\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2021 March Q6
6 Let \(\mathrm { f } ( x ) = \frac { 5 a } { ( 2 x - a ) ( 3 a - x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { a } ^ { 2 a } \mathrm { f } ( x ) \mathrm { d } x = \ln 6\).
    \(7 \quad\) Two lines have equations \(\mathbf { r } = \left( \begin{array} { l } 1
    3
    2 \end{array} \right) + s \left( \begin{array} { r } 2
    - 1
    3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 2
    1
    4 \end{array} \right) + t \left( \begin{array} { r } 1
    - 1
    4 \end{array} \right)\).
  3. Show that the lines are skew.
  4. Find the acute angle between the directions of the two lines.
CAIE P3 2021 March Q8
8 The complex numbers \(u\) and \(v\) are defined by \(u = - 4 + 2 \mathrm { i }\) and \(v = 3 + \mathrm { i }\).
  1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Hence express \(\frac { u } { v }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
    In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(2 u + v\) respectively.
  3. State fully the geometrical relationship between \(O A\) and \(B C\).
  4. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
CAIE P3 2021 March Q9
9 Let \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { 2 x } + 1 } { \mathrm { e } ^ { 2 x } - 1 }\), for \(x > 0\).
  1. The equation \(x = \mathrm { f } ( x )\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5.
  2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  3. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the exact value of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = - 8\).
CAIE P3 2021 March Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{149a8d28-8d2a-4b01-bed0-f16f1e201f32-18_372_675_264_735} The diagram shows the curve \(y = \sin 2 x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis.
  2. Find the exact \(x\)-coordinate of \(M\).
    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
1 Solve the inequality \(| 2 x + 3 | > 3 | x + 2 |\).
CAIE P3 2022 March Q2
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
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
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
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 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
  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
  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 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.
CAIE P3 2022 March Q10
10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) respectively. The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\).
  1. Find a vector equation for the line through \(A\) and \(B\).
  2. Find the acute angle between the directions of \(A B\) and \(l\), giving your answer in degrees.
  3. Show that the line through \(A\) and \(B\) does not intersect the line \(l\).
CAIE P3 2022 March Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-16_556_698_274_712} The diagram shows the curve \(y = \sin x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 significant figures.
  2. Using the substitution \(u = \cos x\), find the area of the shaded region enclosed by the curve and the \(x\)-axis in the first quadrant, giving your answer in a simplified exact form.
    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 2023 March Q1
1 It is given that \(x = \ln ( 2 y - 3 ) - \ln ( y + 4 )\).
Express \(y\) in terms of \(x\).
CAIE P3 2023 March Q2
2
  1. On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(- \frac { 1 } { 3 } \pi \leqslant \arg ( z - 1 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 3 } \pi\) and \(\operatorname { Re } z \leqslant 3\).
  2. Calculate the least value of \(\arg z\) for points in the region from (a). Give your answer in radians correct to 3 decimal places.
CAIE P3 2023 March Q3
3 The polynomial \(2 x ^ { 4 } + a x ^ { 3 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - x + 1\) the remainder is \(3 x + 2\). Find the values of \(a\) and \(b\).
CAIE P3 2023 March Q4
4 Solve the equation $$\frac { 5 z } { 1 + 2 \mathrm { i } } - z z ^ { * } + 30 + 10 \mathrm { i } = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2023 March Q5
5 The parametric equations of a curve are $$x = t \mathrm { e } ^ { 2 t } , \quad y = t ^ { 2 } + t + 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 t }\).
  2. Hence show that the normal to the curve, where \(t = - 1\), passes through the point \(\left( 0,3 - \frac { 1 } { \mathrm { e } ^ { 4 } } \right)\).
CAIE P3 2023 March Q6
6
  1. Express \(5 \sin \theta + 12 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. Hence solve the equation \(5 \sin 2 x + 12 \cos 2 x = 6\) for \(0 \leqslant x \leqslant \pi\).