CAIE P3 (Pure Mathematics 3) 2024 November

1 Expand \(( 9 - 3 x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2

1. By sketching a suitable pair of graphs, show that the equation \(\cot 2 x = \sec x\) has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Show that if a sequence of real values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \cos x _ { n } \right)$$ converges, then it converges to the root in part (a).

3 The square roots of 6-8i 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 \(6 - 8 \mathrm { i }\) in exact Cartesian form.
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4 Solve the equation \(5 ^ { x } = 5 ^ { x + 2 } - 10\). Give your answer correct to 3 decimal places.

5

1. The complex number \(u\) is given by $$u = \frac { \left( \cos \frac { 1 } { 7 } \pi + i \sin \frac { 1 } { 7 } \pi \right) ^ { 4 } } { \cos \frac { 1 } { 7 } \pi - i \sin \frac { 1 } { 7 } \pi }$$ Find the exact value of \(\arg u\).
2. The complex numbers \(u\) and \(u ^ { * }\) are plotted on an Argand diagram. Describe the single geometrical transformation that maps \(u\) onto \(u ^ { * }\) and state the exact value of \(\arg u ^ { * }\).
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   \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-07_588_869_255_603} The variables \(x\) and \(y\) satisfy the equation \(a y = b ^ { x }\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0.50,2.24\) ) and ( \(3.40,8.27\) ), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure.

7

1. Show that the equation \(\tan ^ { 3 } x + 2 \tan 2 x - \tan x = 0\) may be expressed as $$\tan ^ { 4 } x - 2 \tan ^ { 2 } x - 3 = 0$$ for \(\tan x \neq 0\).
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2. Hence solve the equation \(\tan ^ { 3 } 2 \theta + 2 \tan 4 \theta - \tan 2 \theta = 0\) for \(0 < \theta < \pi\). Give your answers in exact form.

8 The parametric equations of a curve are $$x = \tan ^ { 2 } 2 t , \quad y = \cos 2 t$$ for \(0 < t < \frac { 1 } { 4 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \cos ^ { 3 } 2 t\).
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2. Hence find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 8 } \pi\). Give your answer in the form \(y = m x + c\).

9 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
1
- 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 0
4
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 3
- 2
2 \end{array} \right)$$

1. The point \(D\) is such that \(A B C D\) is a trapezium with \(\overrightarrow { D C } = 3 \overrightarrow { A B }\). Find the position vector of \(D\).
2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\).
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3. Using a scalar product, calculate angle \(A B C\).

10 A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40 \pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8 \pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { 50 - r } { 5 r ^ { 2 } }$$
2. Find the quotient and remainder when \(5 r ^ { 2 }\) is divided by \(50 - r\).
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\).
4. Find the value of \(t\) when the radius of the balloon is 12 .

11 Let \(\mathrm { f } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } - 3 \mathrm { e } ^ { x } + 2 }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
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2. Use the substitution \(u = e ^ { x }\) and partial fractions to find the exact value of \(\int _ { \ln 3 } ^ { \ln 5 } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
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