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CAIE P2 2017 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-05_551_533_260_806} The variables \(x\) and \(y\) satisfy the equation \(y = \frac { K } { a ^ { 2 x } }\), where \(K\) and \(a\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.6,1.81 )\) and \(( 1.4,1.39 )\), as shown in the diagram. Find the values of \(K\) and \(a\) correct to 2 significant figures.
CAIE P2 2017 June Q8
11 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve are \(( 1,4 )\) and \(( 3,3 )\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
  2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
  3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).
CAIE P2 2018 June Q1
5 marks Moderate -0.3
1 Solve the equation \(3 \mathrm { e } ^ { 2 x } - 82 \mathrm { e } ^ { x } + 27 = 0\), giving your answers in the form \(k \ln 3\).
CAIE P2 2018 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-04_554_493_260_826} The variables \(x\) and \(y\) satisfy the equation \(y = A \times B ^ { \ln x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points (2.2, 4.908) and (5.9, 11.008), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 2 significant figures.
CAIE P2 2018 June Q3
5 marks Moderate -0.5
3 Without using a calculator, find the exact value of \(\int _ { 0 } ^ { 2 } 4 \mathrm { e } ^ { - x } \left( \mathrm { e } ^ { 3 x } + 1 \right) \mathrm { d } x\).
CAIE P2 2018 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-06_355_839_260_653} The diagram shows the curve with equation \(y = \frac { 5 \ln x } { 2 x + 1 }\). The curve crosses the \(x\)-axis at the point \(P\) and has a maximum point \(M\).
  1. Find the gradient of the curve at the point \(P\).
  2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \frac { x + 0.5 } { \ln x }\).
  3. Use an iterative formula based on the equation in part (ii) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Show the result of each iteration to 6 significant figures.
CAIE P2 2018 June Q5
7 marks Standard +0.3
5 The parametric equations of a curve are $$x = 2 \cos 2 \theta + 3 \sin \theta , \quad y = 3 \cos \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Find the gradient of the curve at the point for which \(\theta = 1\) radian.
  2. Find the value of \(\sin \theta\) at the point on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P2 2018 June Q6
9 marks Standard +0.3
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + 14 x + a + 1$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
  1. Use the factor theorem to find the value of \(a\) and hence factorise \(\mathrm { f } ( x )\) completely.
  2. Hence, without using a calculator, solve the equation \(\mathrm { f } ( 2 x ) = 3 \mathrm { f } ( x )\).
CAIE P2 2018 June Q7
11 marks
7
  1. Express \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Using your answer from part (i), solve the equation $$5 \cot \theta - 4 \operatorname { cosec } \theta = 2$$ for \(0 < \theta < 2 \pi\).
  3. Find \(\int \frac { 1 } { \left( 5 \cos \frac { 1 } { 2 } x - 2 \sin \frac { 1 } { 2 } x \right) ^ { 2 } } \mathrm {~d} x\).
    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 P2 2018 June Q1
4 marks Standard +0.3
1 Solve the inequality \(| 3 x - 2 | < | x + 5 |\).
CAIE P2 2018 June Q2
6 marks Moderate -0.3
2 A curve has equation \(y = 3 \ln ( 2 x + 9 ) - 2 \ln x\).
  1. Find the \(x\)-coordinate of the stationary point.
  2. Determine whether the stationary point is a maximum or minimum point.
CAIE P2 2018 June Q3
6 marks Standard +0.3
3
  1. Find the quotient when $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$ is divided by \(x ^ { 2 } + 6\) and show that the remainder is 1 .
  2. Show that the equation $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$ has no real roots.
CAIE P2 2018 June Q4
7 marks Standard +0.3
4
  1. Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = 4 \ln 2\).
  2. Hence solve the equation $$2 \ln \left( 2 ^ { u + 1 } \right) - \ln \left( 2 ^ { u } + 3 \right) = 4 \ln 2$$ giving the value of \(u\) correct to 4 significant figures.
CAIE P2 2018 June Q5
6 marks Standard +0.3
5 A curve has equation $$y ^ { 3 } \sin 2 x + 4 y = 8$$ Find the equation of the tangent to the curve at the point where it crosses the \(y\)-axis.
CAIE P2 2018 June Q6
11 marks Challenging +1.2
6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.
  1. Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
  2. Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2018 June Q7
10 marks Standard +0.8
7
  1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
  2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
  3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
    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 2002 June Q1
3 marks Standard +0.3
1 Prove the identity $$\cot \theta - \tan \theta \equiv 2 \cot 2 \theta$$
CAIE P3 2002 June Q2
4 marks Moderate -0.8
2 Expand \(( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2002 June Q3
4 marks Standard +0.3
3 The polynomial \(x ^ { 4 } + 4 x ^ { 2 } + x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x ^ { 2 } + x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
Find the value of \(a\) and the other quadratic factor of \(p ( x )\).
CAIE P3 2002 June Q4
5 marks Moderate -0.3
4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 } \left( x _ { n } + \frac { 1 } { x _ { n } ^ { 2 } } \right)$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\).
  1. Use this formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.
  2. State an equation satisfied by \(\alpha\), and hence find the exact value of \(\alpha\).
CAIE P3 2002 June Q5
7 marks Standard +0.3
5 The equation of a curve is \(y = 2 \cos x + \sin 2 x\). Find the \(x\)-coordinates of the stationary points on the curve for which \(0 < x < \pi\), and determine the nature of each of these stationary points.
CAIE P3 2002 June Q6
10 marks Standard +0.3
6 Let \(\mathrm { f } ( x ) = \frac { 4 x } { ( 3 x + 1 ) ( x + 1 ) ^ { 2 } }\).
  1. Express \(f ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 1 - \ln 2\).
CAIE P3 2002 June Q7
10 marks Standard +0.3
7 In a certain chemical process a substance is being formed, and \(t\) minutes after the start of the process there are \(m\) grams of the substance present. In the process the rate of increase of \(m\) is proportional to \(( 50 - m ) ^ { 2 }\). When \(t = 0 , m = 0\) and \(\frac { \mathrm { d } m } { \mathrm {~d} t } = 5\).
  1. Show that \(m\) satisfies the differential equation $$\frac { \mathrm { d } m } { \mathrm {~d} t } = 0.002 ( 50 - m ) ^ { 2 }$$
  2. Solve the differential equation, and show that the solution can be expressed in the form $$m = 50 - \frac { 500 } { t + 10 }$$
  3. Calculate the mass of the substance when \(t = 10\), and find the time taken for the mass to increase from 0 to 45 grams.
  4. State what happens to the mass of the substance as \(t\) becomes very large.
CAIE P3 2002 June Q8
10 marks Standard +0.3
8 The straight line \(l\) passes through the points \(A\) and \(B\) whose position vectors are \(\mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) respectively. The plane \(p\) has equation \(x + 3 y - 2 z = 3\).
  1. Given that \(l\) intersects \(p\), find the position vector of the point of intersection.
  2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = 1\).
CAIE P3 2002 June Q9
11 marks Standard +0.3
9 The complex number \(1 + i \sqrt { } 3\) is denoted by \(u\).
  1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Hence, or otherwise, find the modulus and argument of \(u ^ { 2 }\) and \(u ^ { 3 }\).
  2. Show that \(u\) is a root of the equation \(z ^ { 2 } - 2 z + 4 = 0\), and state the other root of this equation.
  3. Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities $$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$