Questions — CAIE (7646 questions)

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CAIE P3 2022 June Q1
4 marks Moderate -0.3
1 Solve the equation \(\ln \left( \mathrm { e } ^ { 2 x } + 3 \right) = 2 x + \ln 3\), giving your answer correct to 3 decimal places.
CAIE P3 2022 June Q2
5 marks Moderate -0.3
2 Solve the equation \(3 \cos 2 \theta = 3 \cos \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2022 June Q3
5 marks Moderate -0.3
3 The polynomial \(a x ^ { 3 } + x ^ { 2 } + b x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by ( \(2 x - 1\) ) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).
CAIE P3 2022 June Q4
6 marks Standard +0.8
4 The equation of a curve is \(y = \cos ^ { 3 } x \sqrt { \sin x }\). It is given that the curve has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this stationary point, giving your answer correct to 3 significant figures.
CAIE P3 2022 June Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln x = 3 x - x ^ { 2 }\) has one real root.
  2. Verify by calculation that the root lies between 2 and 2.8.
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 June Q6
8 marks Moderate -0.3
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { y - x } ,$$ and \(y = 0\) when \(x = 0\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. Find the value of \(y\) when \(x = 1\), giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.
CAIE P3 2022 June Q7
9 marks Standard +0.3
7 The equation of a curve is \(x ^ { 3 } + 3 x ^ { 2 } y - y ^ { 3 } = 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + 2 x y } { y ^ { 2 } - x ^ { 2 } }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P3 2022 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( 3 x - 1 ) \left( x ^ { 2 } + 3 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int _ { 1 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in a simplified exact form.
CAIE P3 2022 June Q9
10 marks Standard +0.3
9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = - \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 5 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } + \mathbf { k } )$$ respectively, where \(a\) and \(b\) are constants.
  1. Given that \(l\) and \(m\) intersect, show that \(2 b - a = 4\).
  2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
  3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).
CAIE P3 2022 June Q10
11 marks Standard +0.8
10 The complex number \(- 1 + \sqrt { 7 } \mathrm { i }\) is denoted by \(u\). It is given that \(u\) is a root of the equation $$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$ where \(k\) is a real constant.
  1. Find the value of \(k\).
  2. Find the other two roots of the equation.
  3. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - u | = 2\).
  4. Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians.
    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 June Q1
4 marks Challenging +1.2
1 Find, in terms of \(a\), the set of values of \(x\) satisfying the inequality $$2 | 3 x + a | < | 2 x + 3 a |$$ where \(a\) is a positive constant.
CAIE P3 2022 June Q2
5 marks Moderate -0.3
2 Solve the equation \(\cos \left( \theta - 60 ^ { \circ } \right) = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2022 June Q3
5 marks Moderate -0.3
3
  1. Show that the equation \(\log _ { 3 } ( 2 x + 1 ) = 1 + 2 \log _ { 3 } ( x - 1 )\) can be written as a quadratic equation in \(x\).
  2. Hence solve the equation \(\log _ { 3 } ( 4 y + 1 ) = 1 + 2 \log _ { 3 } ( 2 y - 1 )\), giving your answer correct to 2 decimal places.
CAIE P3 2022 June Q4
7 marks Standard +0.3
4 The curve \(y = \mathrm { e } ^ { - 4 x } \tan x\) has two stationary points in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show it can be written in the form \(\sec ^ { 2 } x ( a + b \sin 2 x ) \mathrm { e } ^ { - 4 x }\), where \(a\) and \(b\) are constants.
  2. Hence find the exact \(x\)-coordinates of the two stationary points.
CAIE P3 2022 June Q5
8 marks Standard +0.3
5 The complex number \(3 - \mathrm { i }\) is denoted by \(u\).
  1. Show, on an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. State the type of quadrilateral formed by the points \(O , A , B\) and \(C\).
  2. Express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u ^ { * } } { u }\), or otherwise, prove that \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
CAIE P3 2022 June Q6
8 marks Standard +0.3
6 The parametric equations of a curve are \(x = \frac { 1 } { \cos t } , y = \ln \tan t\), where \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos t } { \sin ^ { 2 } t }\).
  2. Find the equation of the tangent to the curve at the point where \(y = 0\).
CAIE P3 2022 June Q7
10 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + 8 x - 3 } { ( x - 2 ) \left( 2 x ^ { 2 } + 3 \right) }\).
  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 2022 June Q8
9 marks Standard +0.3
8 At time \(t\) days after the start of observations, the number of insects in a population is \(N\). The variation in the number of insects is modelled by a differential equation of the form \(\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t\), where \(k\) is a constant and \(N\) is a continuous variable. It is given that when \(t = 0 , N = 100\).
  1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
  2. Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).
  3. Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.
CAIE P3 2022 June Q9
9 marks Standard +0.3
9 With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\).
  1. Find in degrees the acute angle between the directions of \(O A\) and \(l\).
  2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
  3. Hence find the position vector of the reflection of \(A\) in \(l\).
CAIE P3 2022 June Q10
10 marks Standard +0.8
10 The constant \(a\) is such that \(\int _ { 1 } ^ { a } x ^ { 2 } \ln x \mathrm {~d} x = 4\).
  1. Show that \(a = \left( \frac { 35 } { 3 \ln a - 1 } \right) ^ { \frac { 1 } { 3 } }\).
  2. Verify by calculation that \(a\) lies between 2.4 and 2.8.
  3. 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.
    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 June Q1
3 marks Standard +0.3
1 Solve the equation $$3 \mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { - 2 x } = 5$$ Give the answer correct to 3 decimal places.
CAIE P3 2023 June Q2
4 marks Moderate -0.8
2
  1. Sketch the graph of \(y = | 2 x + 3 |\).
  2. Solve the inequality \(3 x + 8 > | 2 x + 3 |\).
CAIE P3 2023 June Q3
4 marks Standard +0.8
3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 3 + x ) \sqrt { 1 + 4 x }\).
CAIE P3 2023 June Q4
6 marks Moderate -0.3
4
  1. Show that the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) can be expressed in the form $$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
  2. Hence solve the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2023 June Q5
8 marks Standard +0.3
5 The equation of a curve is \(x ^ { 2 } y - a y ^ { 2 } = 4 a ^ { 3 }\), where \(a\) is a non-zero constant.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x y } { 2 a y - x ^ { 2 } }\).
  2. Hence find the coordinates of the points where the tangent to the curve is parallel to the \(y\)-axis. [4]