CAIE P3 (Pure Mathematics 3) 2024 June

1 Solve the equation \(8 ^ { 3 - 6 x } = 4 \times 5 ^ { - 2 x }\). Give your answer correct to 3 decimal places.
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2 Find the exact coordinates of the stationary point of the curve \(y = \mathrm { e } ^ { 2 x } \sin 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

3 The square roots of 24-7i can be expressed in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7 \mathrm { i }\) in exact Cartesian form.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-04_2717_35_141_2012} The variables \(x\) and \(y\) satisfy the equation \(k y = \mathrm { e } ^ { c x }\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(2.80,0.372\) ) and ( \(5.10,2.21\) ), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures.

5 Express \(\frac { 6 x ^ { 2 } - 2 x + 2 } { ( x - 1 ) ( 2 x + 1 ) }\) in partial fractions.
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6

1. On an Argand diagram shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 4 - 3 i | \leqslant 2\) and \(\arg ( z - 2 - i ) \geqslant \frac { 1 } { 3 } \pi\).
2. Calculate the greatest value of \(\arg z\) for points in this region.

7 Let \(\mathrm { f } ( x ) = 8 x ^ { 3 } + 54 x ^ { 2 } - 17 x - 21\).

1. Show that \(x + 7\) is a factor of \(\mathrm { f } ( x )\).
2. Find the quotient when \(\mathrm { f } ( x )\) is divided by \(x + 7\).
   
3. Hence solve the equation $$8 \cos ^ { 3 } \theta + 54 \cos ^ { 2 } \theta - 17 \cos \theta - 21 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

8

1. Express \(3 \cos 2 x - \sqrt { 3 } \sin 2 x\) in the form \(R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact values of \(R\) and \(\alpha\).
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2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { 3 } { ( 3 \cos 2 x - \sqrt { 3 } \sin 2 x ) ^ { 2 } } \mathrm {~d} x\), simplifying your answer.

9
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-12_431_547_280_758} A container in the shape of a cuboid has a square base of side \(x\) and a height of ( \(10 - x\) ). It is given that \(x\) varies with time, \(t\), where \(t > 0\). The container decreases in volume at a rate which is inversely proportional to \(t\). When \(t = \frac { 1 } { 10 } , x = \frac { 1 } { 2 }\) and the rate of decrease of \(x\) is \(\frac { 20 } { 37 }\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { - 1 } { 2 t \left( 20 x - 3 x ^ { 2 } \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-12_2739_47_123_2001}
2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(x\).

10 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 a \mathbf { k } + \lambda ( 3 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) ,$$ where \(a\) is a constant.

1. Given that the acute angle between the directions of these lines is \(\frac { 1 } { 4 } \pi\), find the possible values of \(a\).
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2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection.

11 Use the substitution \(2 x = \tan \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 12 } { \left( 1 + 4 x ^ { 2 } \right) ^ { 2 } } d x$$ Give your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
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