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CAIE P3 2005 November Q5
7 marks Moderate -0.3
5 By expressing \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), solve the equation $$8 \sin \theta - 6 \cos \theta = 7$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2005 November Q6
8 marks Standard +0.3
6
  1. Use the substitution \(x = \sin ^ { 2 } \theta\) to show that $$\int \sqrt { } \left( \frac { x } { 1 - x } \right) \mathrm { d } x = \int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
  2. Hence find the exact value of $$\left. \int _ { 0 } ^ { \frac { 1 } { 4 } } \sqrt { ( } \frac { x } { 1 - x } \right) \mathrm { d } x$$
CAIE P3 2005 November Q7
8 marks Moderate -0.3
7 The equation \(2 x ^ { 3 } + x ^ { 2 } + 25 = 0\) has one real root and two complex roots.
  1. Verify that \(1 + 2 \mathrm { i }\) is one of the complex roots.
  2. Write down the other complex root of the equation.
  3. Sketch an Argand diagram showing the point representing the complex number \(1 + 2 \mathrm { i }\). Show on the same diagram the set of points representing the complex numbers \(z\) which satisfy $$| z | = | z - 1 - 2 \mathrm { i } |$$
CAIE P3 2005 November Q8
8 marks Standard +0.3
8 In a certain chemical reaction the amount, \(x\) grams, of a substance present is decreasing. The rate of decrease of \(x\) is proportional to the product of \(x\) and the time, \(t\) seconds, since the start of the reaction. Thus \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - k x t$$ where \(k\) is a positive constant. At the start of the reaction, when \(t = 0 , x = 100\).
  1. Solve this differential equation, obtaining a relation between \(x , k\) and \(t\).
  2. 20 seconds after the start of the reaction the amount of substance present is 90 grams. Find the time after the start of the reaction at which the amount of substance present is 50 grams.
CAIE P3 2005 November Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 3 x ^ { 2 } + x } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2005 November Q10
11 marks Standard +0.3
10 The straight line \(l\) passes through the points \(A\) and \(B\) with position vectors $$2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$$ respectively. This line intersects the plane \(p\) with equation \(x - 2 y + 2 z = 6\) at the point \(C\).
  1. Find the position vector of \(C\).
  2. Find the acute angle between \(l\) and \(p\).
  3. Show that the perpendicular distance from \(A\) to \(p\) is equal to 2 .
CAIE P3 2006 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(\left| 3 ^ { x } - 8 \right| < 0.5\), giving 3 significant figures in your answer.
CAIE P3 2006 November Q2
4 marks Standard +0.3
2 Solve the equation $$\tan x \tan 2 x = 1 ,$$ giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2006 November Q3
6 marks Moderate -0.8
3 The curve with equation \(y = 6 \mathrm { e } ^ { x } - \mathrm { e } ^ { 3 x }\) has one stationary point.
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2006 November Q4
6 marks Moderate -0.3
4 Given that \(y = 2\) when \(x = 0\), solve the differential equation $$y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + y ^ { 2 }$$ obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
CAIE P3 2006 November Q5
6 marks Standard +0.8
5
  1. Simplify \(( \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) ) ( \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) )\), showing your working, and deduce that $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) } = \frac { \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) } { 2 x }$$
  2. Using this result, or otherwise, obtain the expansion of $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2006 November Q6
9 marks Standard +0.3
6 The equation of a curve is \(x ^ { 3 } + 2 y ^ { 3 } = 3 x y\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { 2 y ^ { 2 } - x }\).
  2. Find the coordinates of the point, other than the origin, where the curve has a tangent which is parallel to the \(x\)-axis.
CAIE P3 2006 November Q7
9 marks
7 The line \(l\) has equation \(\mathbf { r } = \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation \(x + 2 y + 3 z = 5\).
  1. Show that the line \(l\) lies in the plane \(p\).
  2. A second plane is perpendicular to the plane \(p\), parallel to the line \(l\) and contains the point with position vector \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2006 November Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 7 x + 4 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 2 + \ln \frac { 5 } { 3 }\).
CAIE P3 2006 November Q9
10 marks Standard +0.3
9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Find the modulus and argument of \(u\).
  3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
  4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.
CAIE P3 2006 November Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{9e3b4c96-0989-4ffb-bd74-0e73b79ca45a-3_430_807_1375_667} The diagram shows the curve \(y = x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The point \(M\) is a maximum point.
  1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(1 = 2 x \tan 2 x\).
  2. The equation in part (i) can be rearranged in the form \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\). Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right) ,$$ with initial value \(x _ { 1 } = 0.4\), to calculate the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  3. Use integration by parts to find the exact area of the region enclosed between the curve and the \(x\)-axis from 0 to \(\frac { 1 } { 4 } \pi\).
CAIE P3 2007 November Q1
4 marks Moderate -0.3
1 Find the exact value of the constant \(k\) for which \(\int _ { 1 } ^ { k } \frac { 1 } { 2 x - 1 } \mathrm {~d} x = 1\).
CAIE P3 2007 November Q2
4 marks Standard +0.3
2 The polynomial \(x ^ { 4 } + 3 x ^ { 2 } + a\), where \(a\) is a constant, 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 \(\mathrm { p } ( x )\).
CAIE P3 2007 November Q3
4 marks Standard +0.3
3 Use integration by parts to show that $$\int _ { 2 } ^ { 4 } \ln x \mathrm {~d} x = 6 \ln 2 - 2$$
CAIE P3 2007 November Q4
6 marks Standard +0.3
4 The curve with equation \(y = \mathrm { e } ^ { - x } \sin x\) has one stationary point for which \(0 \leqslant x \leqslant \pi\).
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2007 November Q5
7 marks Moderate -0.3
5
  1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$ can be written in the form $$\tan ^ { 2 } x + 2 \tan x - 1 = 0$$
  2. Hence solve the equation $$\tan \left( 45 ^ { \circ } + x \right) - \tan x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P3 2007 November Q6
8 marks Moderate -0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$2 - x = \ln x$$ has only one root.
  2. Verify by calculation that this root lies between 1.4 and 1.7.
  3. Show that this root also satisfies the equation $$x = \frac { 1 } { 3 } ( 4 + x - 2 \ln x )$$
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( 4 + x _ { n } - 2 \ln x _ { n } \right)$$ with initial value \(x _ { 1 } = 1.5\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2007 November Q7
10 marks Standard +0.3
7 The number of insects in a population \(t\) days after the start of observations is denoted by \(N\). The variation in the number of insects is modelled by a differential equation of the form $$\frac { \mathrm { d } N } { \mathrm {~d} t } = k N \cos ( 0.02 t )$$ where \(k\) is a constant and \(N\) is taken to be a continuous variable. It is given that \(N = 125\) when \(t = 0\).
  1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
  2. Given also that \(N = 166\) when \(t = 30\), find the value of \(k\).
  3. Obtain an expression for \(N\) in terms of \(t\), and find the least value of \(N\) predicted by this model.
CAIE P3 2007 November Q8
10 marks Standard +0.3
8
  1. The complex number \(z\) is given by \(z = \frac { 4 - 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\).
    1. Express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Find the modulus and argument of \(z\).
  2. Find the two square roots of the complex number 5-12i, giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2007 November Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 2 - x + 8 x ^ { 2 } } { ( 1 - x ) ( 1 + 2 x ) ( 2 + x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).