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CAIE P3 2002 November Q10
12 marks Standard +0.3
10 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = 4 \mathbf { i } + \mathbf { k } , \quad \overrightarrow { O B } = 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } , \quad \overrightarrow { O D } = - \mathbf { i } - 4 \mathbf { k }$$
  1. Calculate the acute angle between the lines \(A B\) and \(C D\).
  2. Prove that the lines \(A B\) and \(C D\) intersect.
  3. The point \(P\) has position vector \(\mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). Show that the perpendicular distance from \(P\) to the line \(A B\) is equal to \(\sqrt { } 3\).
CAIE P3 2003 November Q1
4 marks Moderate -0.3
1 Solve the inequality \(\left| 2 ^ { x } - 8 \right| < 5\).
CAIE P3 2003 November Q2
4 marks Moderate -0.8
2 Expand \(\left( 2 + x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
CAIE P3 2003 November Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \theta + 3 \cos 2 \theta = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2003 November Q4
6 marks Standard +0.3
4 The equation of a curve is $$\sqrt { } x + \sqrt { } y = \sqrt { } a$$ where \(a\) is a positive constant.
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. The straight line with equation \(y = x\) intersects the curve at the point \(P\). Find the equation of the tangent to the curve at \(P\).
CAIE P3 2003 November Q5
7 marks Standard +0.3
5
  1. By sketching suitable graphs, show that the equation $$\sec x = 3 - x ^ { 2 }$$ has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } ^ { 2 } } \right)$$ converges, then it converges to a root of the equation given in part (i).
  3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to determine the root in the interval \(0 < x < \frac { 1 } { 2 } \pi\) correct to 2 decimal places, showing the result of each iteration.
CAIE P3 2003 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{79efa364-da5a-4888-85a9-dc4de1e0908e-3_543_825_287_660} The diagram shows the curve \(y = ( 3 - x ) \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\). The curve intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Calculate the \(x\)-coordinate of \(M\).
  2. Find the area of the region bounded by \(O A , O B\) and the curve, giving your answer in terms of e.
CAIE P3 2003 November Q7
9 marks Standard +0.8
7 The complex number \(u\) is given by \(u = \frac { 7 + 4 \mathrm { i } } { 3 - 2 \mathrm { i } }\).
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the complex number \(z\) such that \(| z - u | = 2\).
  3. Find the greatest value of \(\arg z\) for points on this locus.
CAIE P3 2003 November Q8
9 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 3 } - x - 2 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form $$A + \frac { B } { x - 1 } + \frac { C x + D } { x ^ { 2 } + 1 }$$ where \(A , B , C\) and \(D\) are constants.
  2. Hence show that \(\int _ { 2 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 1\).
CAIE P3 2003 November Q9
11 marks Standard +0.3
9 Compressed air is escaping from a container. The pressure of the air in the container at time \(t\) is \(P\), and the constant atmospheric pressure of the air outside the container is \(A\). The rate of decrease of \(P\) is proportional to the square root of the pressure difference ( \(P - A\) ). Thus the differential equation connecting \(P\) and \(t\) is $$\frac { \mathrm { d } P } { \mathrm {~d} t } = - k \sqrt { } ( P - A )$$ where \(k\) is a positive constant.
  1. Find, in any form, the general solution of this differential equation.
  2. Given that \(P = 5 A\) when \(t = 0\), and that \(P = 2 A\) when \(t = 2\), show that \(k = \sqrt { } A\).
  3. Find the value of \(t\) when \(P = A\).
  4. Obtain an expression for \(P\) in terms of \(A\) and \(t\).
CAIE P3 2003 November Q10
11 marks Standard +0.3
10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 6 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } + t ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Show that \(l\) and \(m\) intersect, and find the position vector of their point of intersection.
  2. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2004 November Q1
4 marks Moderate -0.8
1 Expand \(\frac { 1 } { ( 2 + x ) ^ { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2004 November Q2
4 marks Moderate -0.3
2 Solve the equation $$\ln ( 1 + x ) = 1 + \ln x$$ giving your answer correct to 2 significant figures.
CAIE P3 2004 November Q3
6 marks Moderate -0.8
3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\). When \(a\) has this value,
  2. factorise \(\mathrm { p } ( x )\),
  3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.
CAIE P3 2004 November Q4
7 marks Standard +0.3
4
  1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$\tan ^ { 2 } x - 6 \tan x + 1 = 0$$
  2. Hence solve the equation \(\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)\), for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P3 2004 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-2_385_476_1653_836} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The point \(N\) on \(O A\) is such that \(B N\) is perpendicular to \(O A\). The area of the triangle \(O N B\) is half the area of the sector \(O A B\).
  1. Show that \(\alpha\) satisfies the equation \(\sin 2 x = x\).
  2. By sketching a suitable pair of graphs, show that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  3. Use the iterative formula $$x _ { n + 1 } = \sin \left( 2 x _ { n } \right)$$ with initial value \(x _ { 1 } = 1\), to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.
CAIE P3 2004 November Q6
8 marks Standard +0.3
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - v\) and \(\frac { u } { v }\).
  2. State the argument of \(\frac { u } { v }\). In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
CAIE P3 2004 November Q7
9 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-3_480_901_973_621} The diagram shows the curve \(y = x ^ { 2 } e ^ { - \frac { 1 } { 2 } x }\).
  1. Find the \(x\)-coordinate of \(M\), the maximum point of the curve.
  2. Find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 1\), giving your answer in terms of e.
CAIE P3 2004 November Q8
9 marks Moderate -0.8
8 An appropriate form for expressing \(\frac { 3 x } { ( x + 1 ) ( x - 2 ) }\) in partial fractions is $$\frac { A } { x + 1 } + \frac { B } { x - 2 }$$ where \(A\) and \(B\) are constants.
  1. Without evaluating any constants, state appropriate forms for expressing the following in partial fractions:
    1. \(\frac { 4 x } { ( x + 4 ) \left( x ^ { 2 } + 3 \right) }\),
    2. \(\frac { 2 x + 1 } { ( x - 2 ) ( x + 2 ) ^ { 2 } }\).
  2. Show that \(\int _ { 3 } ^ { 4 } \frac { 3 x } { ( x + 1 ) ( x - 2 ) } \mathrm { d } x = \ln 5\).
CAIE P3 2004 November Q9
10 marks Challenging +1.3
9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + s ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + t ( - 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Show that \(l\) and \(m\) do not intersect. The point \(P\) lies on \(l\) and the point \(Q\) has position vector \(2 \mathbf { i } - \mathbf { k }\).
  2. Given that the line \(P Q\) is perpendicular to \(l\), find the position vector of \(P\).
  3. Verify that \(Q\) lies on \(m\) and that \(P Q\) is perpendicular to \(m\).
CAIE P3 2004 November Q10
10 marks Standard +0.3
10 A rectangular reservoir has a horizontal base of area \(1000 \mathrm {~m} ^ { 2 }\). At time \(t = 0\), it is empty and water begins to flow into it at a constant rate of \(30 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). At the same time, water begins to flow out at a rate proportional to \(\sqrt { } h\), where \(h \mathrm {~m}\) is the depth of the water at time \(t \mathrm {~s}\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 0.02\).
  1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.01 ( 3 - \sqrt { } h )$$ It is given that, after making the substitution \(x = 3 - \sqrt { } h\), the equation in part (i) becomes $$( x - 3 ) \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.005 x$$
  2. Using the fact that \(x = 3\) when \(t = 0\), solve this differential equation, obtaining an expression for \(t\) in terms of \(x\).
  3. Find the time at which the depth of water reaches 4 m .
CAIE P3 2005 November Q1
4 marks Standard +0.3
1 Given that \(a\) is a positive constant, solve the inequality $$| x - 3 a | > | x - a |$$
CAIE P3 2005 November Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{9275a3ed-8820-481b-9fc8-28c21b81dbed-2_559_789_513_678} Two variable quantities \(x\) and \(y\) are related by the equation \(y = A x ^ { n }\), where \(A\) and \(n\) are constants. The diagram shows the result of plotting \(\ln y\) against \(\ln x\) for four pairs of values of \(x\) and \(y\). Use the diagram to estimate the values of \(A\) and \(n\).
CAIE P3 2005 November Q3
7 marks Standard +0.3
3 The equation of a curve is \(y = x + \cos 2 x\). Find the \(x\)-coordinates of the stationary points of the curve for which \(0 \leqslant x \leqslant \pi\), and determine the nature of each of these stationary points.
CAIE P3 2005 November Q4
7 marks Standard +0.3
4 The equation \(x ^ { 3 } - x - 3 = 0\) has one real root, \(\alpha\).
  1. Show that \(\alpha\) lies between 1 and 2 . Two iterative formulae derived from this equation are as follows: $$\begin{aligned} & x _ { n + 1 } = x _ { n } ^ { 3 } - 3 \\ & x _ { n + 1 } = \left( x _ { n } + 3 \right) ^ { \frac { 1 } { 3 } } \end{aligned}$$ Each formula is used with initial value \(x _ { 1 } = 1.5\).
  2. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.