CAIE P3 (Pure Mathematics 3) 2024 June

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

2 Solve the equation \(\ln ( x - 5 ) = 7 - \ln x\). Give your answer correct to 2 decimal places.

3
\includegraphics[max width=\textwidth, alt={}, center]{37f00894-e6b1-4961-bd3c-4852e43173d0-04_597_921_260_573} The variables \(x\) and \(y\) satisfy the equation \(\mathrm { a } ^ { \mathrm { y } } = \mathrm { bx }\), where \(a\) and \(b\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points ( \(0.336,1.00\) ) and ( \(1.31,1.50\) ), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to the nearest integer.

4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).

1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
   The complex number \(v\) is given by \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\).
2. Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

5 The equation of a curve is \(y = \frac { e ^ { \sin x } } { \cos ^ { 2 } x }\) for \(0 \leqslant x \leqslant 2 \pi\).
Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) and hence find the \(x\)-coordinates of the stationary points of the curve.

6

1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } \frac { 1 } { 2 } x = \mathrm { e } ^ { x } - 3\) has exactly one root, denoted by \(\alpha\), in the interval \(0 < x < \pi\).
2. Verify by calculation that \(\alpha\) lies between 1 and 2 .
3. Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula $$x _ { n + 1 } = \ln \left( \operatorname { cosec } \frac { 1 } { 2 } x _ { n } + 3 \right)$$ converges, then it converges to \(\alpha\).
4. Use this iterative formula with an initial value of 1.4 to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
5. State the minimum number of calculated iterations needed with this initial value to determine \(\alpha\) correct to 2 decimal places.

7

1. On a single Argand diagram sketch the loci given by the equations \(| z - 3 + 2 i | = 2\) and \(| w - 3 + 2 \mathrm { i } | = | w + 3 - 4 \mathrm { i } |\) where z and \(w\) are complex numbers.
2. Hence find the least value of \(| \mathbf { z } - \mathbf { w } |\) for points on these loci. Give your answer in an exact form.

8 Use the substitution \(\mathrm { u } = 1 - \sin \mathrm { x }\) to find the exact value of $$\int _ { \pi } ^ { \frac { 3 } { 2 } \pi } \frac { \sin 2 x } { \sqrt { 1 - \sin x } } d x$$ Give your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { 2 }\) where \(a\) and \(b\) are rational numbers to be determined.

9 The equations of two straight lines \(l _ { 1 }\) and \(l _ { 2 }\) are $$l _ { 1 } : \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \quad \mathbf { r } = - \mathbf { i } - \mathbf { j } - \mathbf { k } + \mu ( 3 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) ,$$ where \(a\) is a constant.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.

1. Show that \(a = 4\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) also intersect.
2. Find the position vector of the point of intersection.
   The point \(A\) has position vector \(- 5 \mathbf { i } + \mathbf { j } - 9 \mathbf { k }\).
3. Show that \(A\) lies on \(l _ { 1 }\).
   The point \(B\) is the image of \(A\) after a reflection in the line \(l _ { 2 }\).
4. Find the position vector of \(B\).

10

1. Given that \(2 x = \tan y\), show that \(\frac { d y } { d x } = \frac { 2 } { 1 + 4 x ^ { 2 } }\).
2. Hence find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { \frac { \sqrt { 3 } } { 2 } } x \tan ^ { - 1 } ( 2 x ) \mathrm { d } x\).

11 In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time \(t\) after the first plant is infected there are \(x\) infected plants. The rate of change of \(x\) is proportional to the product of the number of plants infected and the number of plants that are not yet infected. The variables \(x\) and \(t\) are treated as continuous, and it is given that \(\frac { \mathrm { dx } } { \mathrm { dt } } = 0.2\) and \(x = 1\) when \(t = 0\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$1495 \frac { \mathrm { dx } } { \mathrm { dt } } = x ( 300 - x )$$
2. Using partial fractions, solve the differential equation and obtain an expression for \(t\) in terms of a single logarithm involving \(x\).
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