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CAIE P3 2005 June Q6
8 marks Standard +0.3
6
  1. Prove the identity $$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
  2. Hence solve the equation $$\cos 4 \theta + 4 \cos 2 \theta = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2005 June Q7
8 marks Standard +0.3
7
  1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify, by calculation, that this root lies between 0.5 and 1 .
  3. Show that this root also satisfies the equation $$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
  4. Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$ with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2005 June Q8
9 marks Standard +0.3
8
  1. Using partial fractions, find $$\int \frac { 1 } { y ( 4 - y ) } \mathrm { d } y$$
  2. Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 4 - y ) ,$$ obtaining an expression for \(y\) in terms of \(x\).
  3. State what happens to the value of \(y\) if \(x\) becomes very large and positive.
CAIE P3 2005 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-4_429_748_264_699} The diagram shows part of the curve \(y = \frac { x } { x ^ { 2 } + 1 }\) and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and by the lines \(y = 0\) and \(x = p\).
  1. Calculate the \(x\)-coordinate of \(M\).
  2. Find the area of \(R\) in terms of \(p\).
  3. Hence calculate the value of \(p\) for which the area of \(R\) is 1 , giving your answer correct to 3 significant figures.
CAIE P3 2005 June Q10
11 marks Standard +0.3
10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by $$\overrightarrow { O A } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$ The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } + s ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).
  1. Prove that the line \(I\) does not intersect the line through \(A\) and \(B\).
  2. Find the equation of the plane containing \(l\) and the point \(A\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2007 June Q1
4 marks Moderate -0.8
1 Expand \(( 2 + 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2007 June Q2
4 marks Easy -1.2
2 The polynomial \(x ^ { 3 } - 2 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the quadratic factor of \(\mathrm { p } ( x )\).
CAIE P3 2007 June Q3
4 marks Moderate -0.5
3 The equation of a curve is \(y = x \sin 2 x\), where \(x\) is in radians. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 4 } \pi\).
CAIE P3 2007 June Q4
6 marks Standard +0.3
4 Using the substitution \(u = 3 ^ { x }\), or otherwise, solve, correct to 3 significant figures, the equation $$3 ^ { x } = 2 + 3 ^ { - x }$$
CAIE P3 2007 June Q5
7 marks Standard +0.3
5
  1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + ( \sqrt { } 3 ) \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { 1 } { \sqrt { } 3 }\).
CAIE P3 2007 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-2_355_601_1562_772} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \pi\). The area of triangle \(A O B\) is half the area of the sector.
  1. Show that \(\alpha\) satisfies the equation $$x = 2 \sin x$$
  2. Verify by calculation that \(\alpha\) lies between \(\frac { 1 } { 2 } \pi\) and \(\frac { 2 } { 3 } \pi\).
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \sin x _ { n } \right)$$ converges, then it converges to a root of the equation in part (i).
  4. Use this iterative formula, with initial value \(x _ { 1 } = 1.8\), to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2007 June Q7
9 marks Standard +0.3
7 Let \(I = \int _ { 1 } ^ { 4 } \frac { 1 } { x ( 4 - \sqrt { } x ) } \mathrm { d } x\).
  1. Use the substitution \(u = \sqrt { } x\) to show that \(I = \int _ { 1 } ^ { 2 } \frac { 2 } { u ( 4 - u ) } \mathrm { d } u\).
  2. Hence show that \(I = \frac { 1 } { 2 } \ln 3\).
CAIE P3 2007 June Q8
10 marks Standard +0.3
8 The complex number \(\frac { 2 } { - 1 + \mathrm { i } }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\) and \(u ^ { 2 }\).
  2. Sketch an Argand diagram showing the points representing the complex numbers \(u\) and \(u ^ { 2 }\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(\left| z - u ^ { 2 } \right| < | z - u |\).
CAIE P3 2007 June Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-3_693_537_1206_804} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and three points \(A , B\) and \(C\) with position vectors \(\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 0 \\ 0 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the acute angle between the planes \(A B C\) and \(O A B\).
CAIE P3 2007 June Q10
12 marks Moderate -0.8
10 A model for the height, \(h\) metres, of a certain type of tree at time \(t\) years after being planted assumes that, while the tree is growing, the rate of increase in height is proportional to \(( 9 - h ) ^ { \frac { 1 } { 3 } }\). It is given that, when \(t = 0 , h = 1\) and \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.2\).
  1. Show that \(h\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.1 ( 9 - h ) ^ { \frac { 1 } { 3 } } .$$
  2. Solve this differential equation, and obtain an expression for \(h\) in terms of \(t\).
  3. Find the maximum height of the tree and the time taken to reach this height after planting.
  4. Calculate the time taken to reach half the maximum height.
CAIE P3 2008 June Q1
4 marks Standard +0.8
1 Solve the inequality \(| x - 2 | > 3 | 2 x + 1 |\).
CAIE P3 2008 June Q2
5 marks Standard +0.3
2 Solve, correct to 3 significant figures, the equation $$\mathrm { e } ^ { x } + \mathrm { e } ^ { 2 x } = \mathrm { e } ^ { 3 x }$$
CAIE P3 2008 June Q3
6 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-2_337_828_657_657} In the diagram, \(A B C D\) is a rectangle with \(A B = 3 a\) and \(A D = a\). A circular arc, with centre \(A\) and radius \(r\), joins points \(M\) and \(N\) on \(A B\) and \(C D\) respectively. The angle \(M A N\) is \(x\) radians. The perimeter of the sector \(A M N\) is equal to half the perimeter of the rectangle.
  1. Show that \(x\) satisfies the equation $$\sin x = \frac { 1 } { 4 } ( 2 + x ) \text {. }$$
  2. This equation has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 + x _ { n } } { 4 } \right) ,$$ with initial value \(x _ { 1 } = 0.8\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2008 June Q4
7 marks Standard +0.3
4
  1. Show that the equation \(\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right)\) can be written in the form $$\tan ^ { 2 } \theta + ( 6 \sqrt { } 3 ) \tan \theta - 5 = 0$$
  2. Hence, or otherwise, solve the equation $$\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right) ,$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2008 June Q5
7 marks Standard +0.8
5 The variable complex number \(z\) is given by $$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$ where \(\theta\) takes all values in the interval \(- \pi < \theta \leqslant \pi\).
  1. Show that \(| z - \mathrm { i } | = 2\), for all values of \(\theta\). Hence sketch, in an Argand diagram, the locus of the point representing \(z\).
  2. Prove that the real part of \(\frac { 1 } { z + 2 - \mathrm { i } }\) is constant for \(- \pi < \theta < \pi\).
CAIE P3 2008 June Q6
8 marks Standard +0.8
6 The equation of a curve is \(x y ( x + y ) = 2 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.
CAIE P3 2008 June Q7
9 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) \equiv \frac { x ^ { 2 } + 3 x + 3 } { ( x + 1 ) ( x + 3 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \frac { 1 } { 2 } \ln 2\).
CAIE P3 2008 June Q8
9 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-3_597_951_1471_598} In the diagram the tangent to a curve at a general point \(P\) with coordinates \(( x , y )\) meets the \(x\)-axis at \(T\). The point \(N\) on the \(x\)-axis is such that \(P N\) is perpendicular to the \(x\)-axis. The curve is such that, for all values of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\), the area of triangle \(P T N\) is equal to \(\tan x\), where \(x\) is in radians.
  1. Using the fact that the gradient of the curve at \(P\) is \(\frac { P N } { T N }\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } y ^ { 2 } \cot x .$$
  2. Given that \(y = 2\) when \(x = \frac { 1 } { 6 } \pi\), solve this differential equation to find the equation of the curve, expressing \(y\) in terms of \(x\).
CAIE P3 2008 June Q9
10 marks Challenging +1.2
9 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-4_547_1401_264_370} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \sqrt { } ( 1 + 2 x )\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.
CAIE P3 2008 June Q10
10 marks Standard +0.8
10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } .$$ The line \(l\) has vector equation $$\mathbf { r } = ( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }$$
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. The point \(P\) lies on \(l\) and is such that angle \(P A B\) is equal to \(60 ^ { \circ }\). Given that the position vector of \(P\) is \(( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }\), show that \(3 t ^ { 2 } + 7 t + 2 = 0\). Hence find the only possible position vector of \(P\).