CAIE P3 (Pure Mathematics 3) 2024 November

1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).

1. On the Argand diagram below, sketch the locus of the points representing \(z\).
2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351}
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}

2 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 4\).

1. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { \frac { 4 } { 5 - 2 x _ { n } } }$$ converges, then it converges to a root of the equation \(\mathrm { f } ( x ) = 0\).
2. The equation has a root close to 1.2 . Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

3
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).

1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour.
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}

4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

5

1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).

6 The lines \(l\) and \(m\) have vector equations $$l : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { k } ) \quad \text { and } \quad m : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) .$$ Lines \(l\) and \(m\) intersect at the point \(P\).

1. State the coordinates of \(P\).
2. Find the exact value of the cosine of the acute angle between \(l\) and \(m\).
3. The point \(A\) on line \(I\) has coordinates ( \(0,1,1\) ). The point \(B\) on line \(m\) has coordinates ( \(0,2 , - 8\) ). Find the exact area of triangle \(A P B\).

7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\).
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009}
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.

8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in partial fractions.
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009}
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
3. State the set of values of \(x\) for which the expansion in part (b) is valid.

9

1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).

10 A water tank is in the shape of a cuboid with base area \(40000 \mathrm {~cm} ^ { 2 }\). At time \(t\) minutes the depth of water in the tank is \(h \mathrm {~cm}\). Water is pumped into the tank at a rate of \(50000 \mathrm {~cm} ^ { 3 }\) per minute. Water is leaking out of the tank through a hole in the bottom at a rate of \(600 \mathrm {~cm} ^ { 3 }\) per minute.

1. Show that \(200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 250 - 3 h\).
   
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-17_2723_33_99_22}
2. It is given that when \(t = 0 , h = 50\). Find the time taken for the depth of water in the tank to reach 80 cm . Give your answer correct to 2 significant figures.

11
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_565_634_260_717} The diagram shows the curve \(y = 2 \sin x \sqrt { 2 + \cos x }\), for \(0 \leqslant x \leqslant 2 \pi\), and its minimum point \(M\), where \(x = a\).

1. Find the value of \(a\) correct to 2 decimal places.
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_2716_38_109_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-19_2726_33_97_22}
2. Use the substitution \(u = 2 + \cos x\) to find the exact area of the shaded region \(R\).
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