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CAIE P3 2007 November Q10
12 marks Standard +0.3
10 The straight line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 6 \mathbf { j } - 3 \mathbf { k } + s ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\). The plane \(p\) has equation \(( \mathbf { r } - 3 \mathbf { i } ) \cdot ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } ) = 0\). 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 a vector equation for the line which lies in \(p\), passes through \(A\) and is perpendicular to \(l\).
CAIE P3 2008 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 2 ) = 2 + \ln x$$ giving your answer correct to 3 decimal places.
CAIE P3 2008 November Q2
4 marks Moderate -0.3
2 Expand \(( 1 + x ) \sqrt { } ( 1 - 2 x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2008 November Q3
5 marks Standard +0.3
3 The curve \(y = \frac { \mathrm { e } ^ { x } } { \cos x }\), for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), has one stationary point. Find the \(x\)-coordinate of this point.
CAIE P3 2008 November Q4
5 marks Moderate -0.5
4 The parametric equations of a curve are $$x = a ( 2 \theta - \sin 2 \theta ) , \quad y = a ( 1 - \cos 2 \theta )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).
CAIE P3 2008 November Q5
6 marks Standard +0.3
5 The polynomial \(4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(2 x ^ { 2 } - 3 x + 3\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\), justifying your answer.
CAIE P3 2008 November Q6
8 marks Standard +0.3
6
  1. Express \(5 \sin x + 12 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \sin 2 \theta + 12 \cos 2 \theta = 11$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2008 November Q7
10 marks Standard +0.3
7 Two planes have equations \(2 x - y - 3 z = 7\) and \(x + 2 y + 2 z = 0\).
  1. Find the acute angle between the planes.
  2. Find a vector equation for their line of intersection.
CAIE P3 2008 November Q8
10 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{c687888e-bef0-4ea9-b5b3-e614028cc07c-3_654_805_274_671} An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is empty. At time \(t\) hours after filling begins, the volume of liquid is \(V \mathrm {~m} ^ { 3 }\) and the depth of liquid is \(h \mathrm {~m}\). It is given that \(V = \frac { 4 } { 3 } h ^ { 3 }\). The liquid is poured in at a rate of \(20 \mathrm {~m} ^ { 3 }\) per hour, but owing to leakage, liquid is lost at a rate proportional to \(h ^ { 2 }\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 4.95\).
  1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 5 } { h ^ { 2 } } - \frac { 1 } { 20 } .$$
  2. Verify that \(\frac { 20 h ^ { 2 } } { 100 - h ^ { 2 } } \equiv - 20 + \frac { 2000 } { ( 10 - h ) ( 10 + h ) }\).
  3. Hence solve the differential equation in part (i), obtaining an expression for \(t\) in terms of \(h\).
CAIE P3 2008 November Q9
12 marks Challenging +1.2
9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6\).
  1. Show that \(a\) satisfies the equation $$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
  2. By sketching a suitable pair of graphs, show that this equation has only one root.
  3. Verify by calculation that this root lies between 2 and 2.5.
  4. Use an iterative formula based on the equation in part (i) to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2008 November Q10
12 marks Standard +0.8
10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).
  1. Find the modulus and argument of \(w\).
  2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
  3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
  4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
CAIE P3 2009 November Q1
4 marks Moderate -0.3
1 Solve the inequality \(2 - 3 x < | x - 3 |\).
CAIE P3 2009 November Q2
4 marks Moderate -0.8
2 Solve the equation \(3 ^ { x + 2 } = 3 ^ { x } + 3 ^ { 2 }\), giving your answer correct to 3 significant figures.
CAIE P3 2009 November Q3
5 marks Standard +0.3
3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 15 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P3 2009 November Q4
6 marks Standard +0.8
4 A curve has equation \(y = \mathrm { e } ^ { - 3 x } \tan x\). Find the \(x\)-coordinates of the stationary points on the curve in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\). Give your answers correct to 3 decimal places.
CAIE P3 2009 November Q5
8 marks Standard +0.3
5
  1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta\).
  2. Using this result find, in simplified form, the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$
CAIE P3 2009 November Q6
8 marks Standard +0.3
6 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$ The mid-point of \(A B\) is \(M\). The point \(N\) lies on \(A C\) between \(A\) and \(C\) and is such that \(A N = 2 N C\).
  1. Find a vector equation of the line \(M N\).
  2. It is given that \(M N\) intersects \(B C\) at the point \(P\). Find the position vector of \(P\).
CAIE P3 2009 November Q7
10 marks Standard +0.3
7 The complex number \(- 2 + \mathrm { i }\) is denoted by \(u\).
  1. Given that \(u\) is a root of the equation \(x ^ { 3 } - 11 x - k = 0\), where \(k\) is real, find the value of \(k\).
  2. Write down the other complex root of this equation.
  3. Find the modulus and argument of \(u\).
  4. Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities $$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$
CAIE P3 2009 November Q8
10 marks Standard +0.3
8
  1. Express \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2009 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{8d134c65-af23-4508-acef-49b6ab49e374-3_504_910_625_614} The diagram shows the curve \(y = \frac { \ln x } { \sqrt { } x }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at the point \(A\).
  1. State the coordinates of \(A\).
  2. Find the exact value of the \(x\)-coordinate of \(M\).
  3. Using integration by parts, show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 4\) is equal to \(8 \ln 2 - 4\).
CAIE P3 2009 November Q10
10 marks Standard +0.3
10 In a model of the expansion of a sphere of radius \(r \mathrm {~cm}\), it is assumed that, at time \(t\) seconds after the start, the rate of increase of the surface area of the sphere is proportional to its volume. When \(t = 0\), \(r = 5\) and \(\frac { \mathrm { d } r } { \mathrm {~d} t } = 2\).
  1. Show that \(r\) satisfies the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = 0.08 r ^ { 2 }$$ [The surface area \(A\) and volume \(V\) of a sphere of radius \(r\) are given by the formulae \(A = 4 \pi r ^ { 2 }\), \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]
  2. Solve this differential equation, obtaining an expression for \(r\) in terms of \(t\).
  3. Deduce from your answer to part (ii) the set of values that \(t\) can take, according to this model.
CAIE P3 2009 November Q1
4 marks Moderate -0.5
1 Solve the equation $$\ln ( 5 - x ) = \ln 5 - \ln x$$ giving your answers correct to 3 significant figures.
CAIE P3 2009 November Q2
5 marks Moderate -0.8
2 The equation \(x ^ { 3 } - 8 x - 13 = 0\) has one real root.
  1. Find the two consecutive integers between which this root lies.
  2. Use the iterative formula $$x _ { n + 1 } = \left( 8 x _ { n } + 13 \right) ^ { \frac { 1 } { 3 } }$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2009 November Q3
6 marks Standard +0.3
3 The equation of a curve is \(x ^ { 3 } - x ^ { 2 } y - y ^ { 3 } = 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the equation of the tangent to the curve at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\).
CAIE P3 2009 November Q4
6 marks Standard +0.8
4 The angles \(\alpha\) and \(\beta\) lie in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\), and are such that $$\tan \alpha = 2 \tan \beta \quad \text { and } \quad \tan ( \alpha + \beta ) = 3 .$$ Find the possible values of \(\alpha\) and \(\beta\).