CAIE P3 (Pure Mathematics 3) 2012 June

1 Solve the equation $$\ln ( 3 x + 4 ) = 2 \ln ( x + 1 )$$ giving your answer correct to 3 significant figures.

2
\includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689} In the diagram, \(A B C\) is a triangle in which angle \(A B C\) is a right angle and \(B C = a\). A circular arc, with centre \(C\) and radius \(a\), joins \(B\) and the point \(M\) on \(A C\). The angle \(A C B\) is \(\theta\) radians. The area of the sector \(C M B\) is equal to one third of the area of the triangle \(A B C\).

1. Show that \(\theta\) satisfies the equation $$\tan \theta = 3 \theta .$$
2. This equation has one root in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). Use the iterative formula $$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

4 Solve the equation $$\operatorname { cosec } 2 \theta = \sec \theta + \cot \theta$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 x + y }$$ and \(y = 0\) when \(x = 0\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

6 The equation of a curve is \(y = 3 \sin x + 4 \cos ^ { 3 } x\).

1. Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\).
2. Determine the nature of the stationary point in this interval for which \(x\) is least.

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing the complex numbers \(u , 1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\) respectively.
3. By considering the arguments of \(1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\), show that $$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$

8 Let \(I = \int _ { 2 } ^ { 5 } \frac { 5 } { x + \sqrt { } ( 6 - x ) } \mathrm { d } x\).

1. Using the substitution \(u = \sqrt { } ( 6 - x )\), show that $$I = \int _ { 1 } ^ { 2 } \frac { 10 u } { ( 3 - u ) ( 2 + u ) } \mathrm { d } u$$
2. Hence show that \(I = 2 \ln \left( \frac { 9 } { 2 } \right)\).

9
\includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689} The diagram shows the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\). The shaded region between the curve, the \(x\)-axis and the line \(x = \mathrm { e }\) is denoted by \(R\).

1. Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = m x + c\).
2. Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

10 Two planes, \(m\) and \(n\), have equations \(x + 2 y - 2 z = 1\) and \(2 x - 2 y + z = 7\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).

1. Show that \(l\) is parallel to \(m\).
2. Find the position vector of the point of intersection of \(l\) and \(n\).
3. A point \(P\) lying on \(l\) is such that its perpendicular distances from \(m\) and \(n\) are equal. Find the position vectors of the two possible positions for \(P\) and calculate the distance between them.
   [0pt] [The perpendicular distance of a point with position vector \(x _ { 1 } \mathbf { i } + y _ { 1 } \mathbf { j } + z _ { 1 } \mathbf { k }\) from the plane \(a x + b y + c z = d\) is \(\frac { \left| a x _ { 1 } + b y _ { 1 } + c z _ { 1 } - d \right| } { \sqrt { } \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) }\).] \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. }