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CAIE P3 2009 November Q5
8 marks Standard +0.3
5 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). The result of differentiating \(\mathrm { p } ( x )\) with respect to \(x\) is denoted by \(\mathrm { p } ^ { \prime } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and of \(\mathrm { p } ^ { \prime } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2009 November Q6
8 marks Standard +0.3
6
  1. Use the substitution \(x = 2 \tan \theta\) to show that $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$$
CAIE P3 2009 November Q7
9 marks Moderate -0.3
7 The complex numbers \(- 2 + \mathrm { i }\) and \(3 + \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find, in the form \(x + \mathrm { i } y\), the complex numbers
    (a) \(u + v\),
    (b) \(\frac { u } { v }\), showing all your working.
  2. State the argument of \(\frac { u } { v }\). In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u + v\) respectively.
  3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
  4. State fully the geometrical relationship between the line segments \(O A\) and \(B C\).
CAIE P3 2009 November Q8
10 marks Standard +0.3
8
  1. Express \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2009 November Q9
9 marks Moderate -0.8
9 The temperature of a quantity of liquid at time \(t\) is \(\theta\). The liquid is cooling in an atmosphere whose temperature is constant and equal to \(A\). The rate of decrease of \(\theta\) is proportional to the temperature difference \(( \theta - A )\). Thus \(\theta\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - A )$$ where \(k\) is a positive constant.
  1. Find, in any form, the solution of this differential equation, given that \(\theta = 4 A\) when \(t = 0\).
  2. Given also that \(\theta = 3 A\) when \(t = 1\), show that \(k = \ln \frac { 3 } { 2 }\).
  3. Find \(\theta\) in terms of \(A\) when \(t = 2\), expressing your answer in its simplest form.
CAIE P3 2009 November Q10
10 marks Standard +0.3
10 The plane \(p\) has equation \(2 x - 3 y + 6 z = 16\). The plane \(q\) is parallel to \(p\) and contains the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\).
  1. Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the perpendicular distance between \(p\) and \(q\).
  3. The line \(l\) is parallel to the plane \(p\) and also parallel to the plane with equation \(x - 2 y + 2 z = 5\). Given that \(l\) passes through the origin, find a vector equation for \(l\).
CAIE P3 2010 November Q1
4 marks Standard +0.3
1 Solve the inequality \(2 | x - 3 | > | 3 x + 1 |\).
CAIE P3 2010 November Q2
4 marks Standard +0.3
2 Solve the equation $$\ln \left( 1 + x ^ { 2 } \right) = 1 + 2 \ln x$$ giving your answer correct to 3 significant figures.
CAIE P3 2010 November Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \left( \theta + 60 ^ { \circ } \right) = 2 \sin \theta$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2010 November Q4
7 marks Standard +0.3
4
  1. By sketching suitable graphs, show that the equation $$4 x ^ { 2 } - 1 = \cot x$$ has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.6 and 1 .
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { } \left( 1 + \cot x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 November Q5
7 marks Standard +0.8
5 Let \(I = \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x\).
  1. Using the substitution \(x = 2 \sin \theta\), show that $$I = \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of \(I\).
CAIE P3 2010 November Q6
9 marks Moderate -0.8
6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$
  1. Find the modulus and argument of \(z\).
  2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
    (a) \(2 z + z ^ { * }\),
    (b) \(\frac { \mathrm { i } z ^ { * } } { z }\).
  3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).
CAIE P3 2010 November Q7
9 marks Standard +0.8
7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line \(A B\) and \(O P\) is perpendicular to \(A B\).
  1. Find a vector equation for the line \(A B\).
  2. Find the position vector of \(P\).
  3. Find the equation of the plane which contains \(A B\) and which is perpendicular to the plane \(O A B\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2010 November Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 3 x } { ( 1 + x ) \left( 1 + 2 x ^ { 2 } \right) }\).
  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 \(x ^ { 3 }\).
CAIE P3 2010 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).
CAIE P3 2010 November Q10
10 marks Moderate -0.3
10 A certain substance is formed in a chemical reaction. The mass of substance formed \(t\) seconds after the start of the reaction is \(x\) grams. At any time the rate of formation of the substance is proportional to \(( 20 - x )\). When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.05 ( 20 - x ) .$$
  2. Find, in any form, the solution of this differential equation.
  3. Find \(x\) when \(t = 10\), giving your answer correct to 1 decimal place.
  4. State what happens to the value of \(x\) as \(t\) becomes very large. \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 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{822f851a-7fae-43b8-9ebc-94588f01e51c-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).
CAIE P3 2010 November Q1
3 marks Easy -1.2
1 Expand \(( 1 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2010 November Q2
5 marks Moderate -0.8
2 The parametric equations of a curve are $$x = \frac { t } { 2 t + 3 } , \quad y = \mathrm { e } ^ { - 2 t }$$ Find the gradient of the curve at the point for which \(t = 0\).
CAIE P3 2010 November Q3
6 marks Moderate -0.3
3 The complex number \(w\) is defined by \(w = 2 + \mathrm { i }\).
  1. Showing your working, express \(w ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real. Find the modulus of \(w ^ { 2 }\).
  2. Shade on an Argand diagram the region whose points represent the complex numbers \(z\) which satisfy $$\left| z - w ^ { 2 } \right| \leqslant \left| w ^ { 2 } \right|$$
CAIE P3 2010 November Q4
6 marks Moderate -0.3
4 It is given that \(\mathrm { f } ( x ) = 4 \cos ^ { 2 } 3 x\).
  1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 9 } \pi \right)\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
CAIE P3 2010 November Q5
7 marks Standard +0.3
5 Show that \(\int _ { 0 } ^ { 7 } \frac { 2 x + 7 } { ( 2 x + 1 ) ( x + 2 ) } \mathrm { d } x = \ln 50\).
CAIE P3 2010 November Q6
8 marks Standard +0.3
6 The straight line \(l\) passes through the points with coordinates \(( - 5,3,6 )\) and \(( 5,8,1 )\). The plane \(p\) has equation \(2 x - y + 4 z = 9\).
  1. Find the coordinates of the point of intersection of \(l\) and \(p\).
  2. Find the acute angle between \(l\) and \(p\).
CAIE P3 2010 November Q7
8 marks Standard +0.3
7
  1. Given that \(\int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = \frac { 2 } { 5 }\), show that \(a = \frac { 5 } { 3 } ( 1 + \ln a )\).
  2. Use an iteration formula based on the equation \(a = \frac { 5 } { 3 } ( 1 + \ln a )\) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.
CAIE P3 2010 November Q8
9 marks Standard +0.3
8
  1. Express \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence, in each of the following cases, find the smallest positive angle \(\theta\) which satisfies the equation
    (a) \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta = - 4\),
    (b) \(( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3\).