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CAIE P3 2014 June Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_391_826_1946_657} The diagram shows the curve \(y = x \cos \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y + 4 \sin \frac { 1 } { 2 } x = 0\).
  2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis.
CAIE P3 2014 June Q9
10 marks Standard +0.3
9 The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \(( 1 - 0.01 N )\). When \(t = 0 , N = 20\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.32\).
  1. Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.02 N ( 1 - 0.01 N )$$
  2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\).
  3. Find the time at which the population will be double its value at \(t = 0\).
CAIE P3 2014 June Q10
12 marks Standard +0.3
10 Referred to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 3 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }$$
  1. Find the exact value of the cosine of angle \(B A C\).
  2. Hence find the exact value of the area of triangle \(A B C\).
  3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2014 June Q1
3 marks Moderate -0.5
1 Solve the equation \(\log _ { 10 } ( x + 9 ) = 2 + \log _ { 10 } x\).
CAIE P3 2014 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 2014 June Q3
6 marks Standard +0.3
3
  1. Show that the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ can be written in the form $$2 \tan ^ { 2 } x + ( \sqrt { } 3 ) \tan x - 1 = 0$$
  2. Hence solve the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2014 June Q4
7 marks Standard +0.3
4 The equation \(x = \frac { 10 } { \mathrm { e } ^ { 2 x } - 1 }\) has one positive real root, denoted by \(\alpha\).
  1. Show that \(\alpha\) lies between \(x = 1\) and \(x = 2\).
  2. Show that if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 1 + \frac { 10 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 June Q5
7 marks Standard +0.3
5 The variables \(x\) and \(\theta\) satisfy the differential equation $$2 \cos ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \sqrt { } ( 2 x + 1 )$$ and \(x = 0\) when \(\theta = \frac { 1 } { 4 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta\).
CAIE P3 2014 June Q6
7 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).
CAIE P3 2014 June Q7
9 marks Standard +0.3
7
  1. The complex number \(\frac { 3 - 5 \mathrm { i } } { 1 + 4 \mathrm { i } }\) is denoted by \(u\). Showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 - \mathrm { i } | \leqslant 1\) and \(| z - \mathrm { i } | \leqslant | z - 2 |\).
    2. Calculate the maximum value of \(\arg z\) for points lying in the shaded region.
CAIE P3 2014 June Q8
9 marks Standard +0.8
8 Let \(f ( x ) = \frac { 6 + 6 x } { ( 2 - x ) \left( 2 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 2 + x ^ { 2 } }\).
  2. Show that \(\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 3 \ln 3\).
CAIE P3 2014 June Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_639_387_1749_879} The diagram shows the curve \(y = \mathrm { e } ^ { 2 \sin x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \sin x\), find the exact value of the area of the shaded region bounded by the curve and the axes.
  2. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
CAIE P3 2014 June Q10
12 marks Standard +0.3
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + \lambda ( 3 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\) and the plane \(p\) has equation \(2 x + 3 y - 5 z = 18\).
  1. Find the position vector of the point of intersection of \(l\) and \(p\).
  2. Find the acute angle between \(l\) and \(p\).
  3. A second plane \(q\) is perpendicular to the plane \(p\) and contains the line \(l\). Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2015 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(2 ^ { 5 x } = 3 ^ { 2 x + 1 }\), giving the answer correct to 3 significant figures.
CAIE P3 2015 June Q2
4 marks Moderate -0.5
2 Use the trapezium rule with three intervals to find an approximation to $$\int _ { 0 } ^ { 3 } \left| 3 ^ { x } - 10 \right| \mathrm { d } x$$
CAIE P3 2015 June Q3
6 marks Standard +0.3
3 Show that, for small values of \(x ^ { 2 }\), $$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$ where the value of the constant \(k\) is to be determined.
CAIE P3 2015 June Q4
7 marks Standard +0.3
4 The equation of a curve is $$y = 3 \cos 2 x + 7 \sin x + 2$$ Find the \(x\)-coordinates of the stationary points in the interval \(0 \leqslant x \leqslant \pi\). Give each answer correct to 3 significant figures.
CAIE P3 2015 June Q5
8 marks Standard +0.3
5
  1. Find \(\int \left( 4 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \sin \left( x + \frac { 1 } { 6 } \pi \right) } { \sin x } \mathrm {~d} x\).
CAIE P3 2015 June Q6
9 marks Standard +0.3
6 The straight line \(l _ { 1 }\) passes through the points \(( 0,1,5 )\) and \(( 2 , - 2,1 )\). The straight line \(l _ { 2 }\) has equation \(\mathbf { r } = 7 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\).
  1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
  2. Find the acute angle between the direction of the line \(l _ { 2 }\) and the direction of the \(x\)-axis.
CAIE P3 2015 June Q7
9 marks Standard +0.3
7 Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x \left( 3 y ^ { 2 } + 10 y + 3 \right)$$ obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2015 June Q8
9 marks Standard +0.3
8 The complex number \(w\) is defined by \(w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }\).
  1. Without using a calculator, show that \(w = 2 + 4 \mathrm { i }\).
  2. It is given that \(p\) is a real number such that \(\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi\). Find the set of possible values of \(p\).
  3. The complex conjugate of \(w\) is denoted by \(w ^ { * }\). The complex numbers \(w\) and \(w ^ { * }\) are represented in an Argand diagram by the points \(S\) and \(T\) respectively. Find, in the form \(| z - a | = k\), the equation of the circle passing through \(S , T\) and the origin.
CAIE P3 2015 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { 2 - x }\) and its maximum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) is 2 .
  2. Find the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm { e } ^ { 2 - x } \mathrm {~d} x\).
CAIE P3 2015 June Q10
10 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_515_508_1105_815} The diagram shows part of the curve with parametric equations $$x = 2 \ln ( t + 2 ) , \quad y = t ^ { 3 } + 2 t + 3$$
  1. Find the gradient of the curve at the origin.
  2. At the point \(P\) on the curve, the value of the parameter is \(p\). It is given that the gradient of the curve at \(P\) is \(\frac { 1 } { 2 }\).
    (a) Show that \(p = \frac { 1 } { 3 p ^ { 2 } + 2 } - 2\).
    (b) By first using an iterative formula based on the equation in part (a), determine the coordinates of the point \(P\). Give the result of each iteration to 5 decimal places and each coordinate of \(P\) correct to 2 decimal places.
CAIE P3 2015 June Q1
3 marks Moderate -0.8
1 Use the trapezium rule with three intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \ln ( 1 + \sin x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
CAIE P3 2015 June Q2
4 marks Moderate -0.8
2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.