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CAIE P3 2010 June Q5
6 marks Standard +0.3
5 Given that \(y = 0\) when \(x = 1\), solve the differential equation $$x y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 4 ,$$ obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
CAIE P3 2010 June Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-3_380_641_258_751} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The angle \(B O C\) is \(x\) radians. The area of the shaded segment is a quarter of the area of the semicircle.
  1. Show that \(x\) satisfies the equation $$x = \frac { 3 } { 4 } \pi - \sin x$$
  2. This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 June Q7
9 marks Challenging +1.2
7 The complex number \(2 + 2 \mathrm { i }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\).
  2. Sketch an Argand diagram showing the points representing the complex numbers 1, i and \(u\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 | \leqslant | z - \mathrm { i } |\) and \(| z - u | \leqslant 1\).
  3. Using your diagram, calculate the value of \(| z |\) for the point in this region for which \(\arg z\) is least.
CAIE P3 2010 June Q8
9 marks Standard +0.8
8
  1. Express \(\frac { 2 } { ( x + 1 ) ( x + 3 ) }\) in partial fractions.
  2. Using your answer to part (i), show that $$\left( \frac { 2 } { ( x + 1 ) ( x + 3 ) } \right) ^ { 2 } \equiv \frac { 1 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x + 1 } + \frac { 1 } { x + 3 } + \frac { 1 } { ( x + 3 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 4 } { ( x + 1 ) ^ { 2 } ( x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 7 } { 12 } - \ln \frac { 3 } { 2 }\).
CAIE P3 2010 June Q9
9 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-4_611_895_255_625} The diagram shows the curve \(y = \sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\).
  1. By first differentiating \(\frac { 1 - x } { 1 + x }\), obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that the gradient of the normal to the curve at the point \(( x , y )\) is \(( 1 + x ) \sqrt { } \left( 1 - x ^ { 2 } \right)\).
  2. The gradient of the normal to the curve has its maximum value at the point \(P\) shown in the diagram. Find, by differentiation, the \(x\)-coordinate of \(P\).
CAIE P3 2010 June Q10
12 marks Standard +0.3
10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Show that \(l\) and \(m\) intersect.
  2. Calculate the acute angle between the lines.
  3. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P3 2010 June Q1
4 marks Moderate -0.8
1 Solve the equation $$\frac { 2 ^ { x } + 1 } { 2 ^ { x } - 1 } = 5$$ giving your answer correct to 3 significant figures.
CAIE P3 2010 June Q2
5 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).
CAIE P3 2010 June Q3
7 marks Standard +0.3
3 It is given that \(\cos a = \frac { 3 } { 5 }\), where \(0 ^ { \circ } < a < 90 ^ { \circ }\). Showing your working and without using a calculator to evaluate \(a\),
  1. find the exact value of \(\sin \left( a - 30 ^ { \circ } \right)\),
  2. find the exact value of \(\tan 2 a\), and hence find the exact value of \(\tan 3 a\).
CAIE P3 2010 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625} The diagram shows the curve \(y = \frac { \sin x } { x }\) for \(0 < x \leqslant 2 \pi\), and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) satisfies the equation $$x = \tan x$$
  2. The iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$ can be used to determine the \(x\)-coordinate of \(M\). Use this formula to determine the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 June Q5
8 marks Moderate -0.8
5 The polynomial \(2 x ^ { 3 } + 5 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 9 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2010 June Q6
8 marks Moderate -0.3
6 The equation of a curve is $$x \ln y = 2 x + 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { y } { x ^ { 2 } }\).
  2. Find the equation of the tangent to the curve at the point where \(y = 1\), giving your answer in the form \(a x + b y + c = 0\).
CAIE P3 2010 June Q7
8 marks Standard +0.3
7 The variables \(x\) and \(t\) are related by the differential equation $$\mathrm { e } ^ { 2 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } x$$ where \(t \geqslant 0\). When \(t = 0 , x = 0\).
  1. Solve the differential equation, obtaining an expression for \(x\) in terms of \(t\).
  2. State what happens to the value of \(x\) when \(t\) becomes very large.
  3. Explain why \(x\) increases as \(t\) increases.
CAIE P3 2010 June Q8
9 marks Standard +0.8
8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
  1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
  2. Prove that the real part of \(\frac { 1 } { z }\) is constant.
CAIE P3 2010 June Q9
9 marks Standard +0.3
9 The plane \(p\) has equation \(3 x + 2 y + 4 z = 13\). A second plane \(q\) is perpendicular to \(p\) and has equation \(a x + y + z = 4\), where \(a\) is a constant.
  1. Find the value of \(a\).
  2. The line with equation \(\mathbf { r } = \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) meets the plane \(p\) at the point \(A\) and the plane \(q\) at the point \(B\). Find the length of \(A B\).
CAIE P3 2010 June Q10
10 marks Standard +0.3
10
  1. Find the values of the constants \(A , B , C\) and \(D\) such that $$\frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \equiv A + \frac { B } { x } + \frac { C } { x ^ { 2 } } + \frac { D } { 2 x - 1 }$$
  2. Hence show that $$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \mathrm { d } x = \frac { 3 } { 2 } + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 27 } \right)$$
CAIE P3 2010 June Q1
4 marks Standard +0.8
1 Solve the inequality \(| x - 3 | > 2 | x + 1 |\).
CAIE P3 2010 June Q2
4 marks Moderate -0.8
2 The variables \(x\) and \(y\) satisfy the equation \(y ^ { 3 } = A \mathrm { e } ^ { 2 x }\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.
  1. Find the gradient of this line.
  2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places.
CAIE P3 2010 June Q3
5 marks Standard +0.3
3 Solve the equation $$\tan \left( 45 ^ { \circ } - x \right) = 2 \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2010 June Q4
7 marks Moderate -0.3
4 Given that \(x = 1\) when \(t = 0\), solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { x } - \frac { x } { 4 } ,$$ obtaining an expression for \(x ^ { 2 }\) in terms of \(t\).
CAIE P3 2010 June Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{c5ec981d-7ff7-4698-82c4-eb0506b635a3-2_515_1031_1384_555} The diagram shows the curve \(y = \mathrm { e } ^ { - x } - \mathrm { e } ^ { - 2 x }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).
  1. Find the exact value of \(p\).
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac { 1 } { 8 }\).
CAIE P3 2010 June Q6
8 marks Moderate -0.3
6 The curve \(y = \frac { \ln x } { x + 1 }\) has one stationary point.
  1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac { x + 1 } { \ln x }$$ and that this \(x\)-coordinate lies between 3 and 4 .
  2. Use the iterative formula $$x _ { n + 1 } = \frac { x _ { n } + 1 } { \ln x _ { n } }$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 June Q7
8 marks Standard +0.3
7
  1. Prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Using this result, find the exact value of $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } \theta \mathrm {~d} \theta$$
CAIE P3 2010 June Q8
9 marks Standard +0.3
8
  1. The equation \(2 x ^ { 3 } - x ^ { 2 } + 2 x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i \sqrt { } 3\) is one of the complex roots. State the other complex root.
  2. On a sketch of an Argand diagram, show the point representing the complex number \(1 + \mathrm { i } \sqrt { } 3\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 - i \sqrt { } 3 | \leqslant 1\) and \(\arg z \leqslant \frac { 1 } { 3 } \pi\).
    1. Express \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in partial fractions.
    2. Hence obtain the expansion of \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2010 June Q10
12 marks Standard +0.3
10 The straight line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(3 x - y + 2 z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).
  1. Find the position vector of \(A\).
  2. Find the acute angle between \(l\) and \(p\).
  3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).