CAIE P3 (Pure Mathematics 3) 2023 November

1 Find the exact coordinates of the points on the curve \(y = \frac { x ^ { 2 } } { 1 - 3 x }\) at which the gradient of the tangent is equal to 8 .

2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathrm { i } | \leqslant | z + 2 - \mathrm { i } |\) and \(0 \leqslant \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi\).

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\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-04_860_451_264_833} The variables \(x\) and \(y\) are related by the equation \(y = a b ^ { x }\), where \(a\) and \(b\) are constants. The diagram shows the result of plotting \(\ln y\) against \(x\) for two pairs of values of \(x\) and \(y\). The coordinates of these points are \(( 1,3.7 )\) and \(( 2.2,6.46 )\). Use this information to find the values of \(a\) and \(b\).

4 The complex number \(u\) is defined by \(u = \frac { 3 + 2 \mathrm { i } } { a - 5 \mathrm { i } }\), where \(a\) is real.

1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
2. Given that \(\arg u = \frac { 1 } { 4 } \pi\), find the value of \(a\).

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1. Given that $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ find the exact value of \(\tan x\).
2. Hence find the exact roots of the equation $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ for \(0 \leqslant x \leqslant 2 \pi\).

6 The parametric equations of a curve are $$x = \sqrt { t } + 3 , \quad y = \ln t$$ for \(t > 0\).

1. Obtain a simplified expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Hence find the exact coordinates of the point on the curve at which the gradient of the normal is - 2 .

7 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { x } { \tan \theta } \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = x ^ { 2 } + 3$$ It is given that \(x = 1\) when \(\theta = 0\).
Solve the differential equation, obtaining an expression for \(x ^ { 2 }\) in terms of \(\theta\).

8

1. By sketching a suitable pair of graphs, show that the equation $$\sqrt { x } = \mathrm { e } ^ { x } - 3$$ has only one root.
2. Show by calculation that this root lies between 1 and 2 .
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \ln \left( 3 + \sqrt { x _ { n } } \right)$$ converges, then it converges to the root of the equation in (a).
4. Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-12_375_645_274_742} The diagram shows the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 4 } x ^ { 2 } }\), for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Using the substitution \(x = \sqrt { u }\), or otherwise, find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\).

10 Let \(\mathrm { f } ( x ) = \frac { 24 x + 13 } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid.

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\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-16_593_780_264_685} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 3\) units, \(O C = 2\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O D\) and \(O C\) respectively. \(M\) is the midpoint of \(E F\).

1. Find the position vector of \(M\).
   The position vector of \(P\) is \(\mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
2. Calculate angle PAM.
3. Find the exact length of the perpendicular from \(P\) to the line passing through \(O\) and \(M\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.