CAIE P3 (Pure Mathematics 3) 2020 June

1 Solve the inequality \(| 2 x - 1 | > 3 | x + 2 |\).

2 Find the exact value of \(\int _ { 0 } ^ { 1 } ( 2 - x ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).

3

1. Show that the equation $$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$ can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
2. Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.

4 The equation of a curve is \(y = x \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right)\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The tangent to the curve at the point where \(x = 2\) meets the \(y\)-axis at the point with coordinates \(( 0 , p )\). Find \(p\).

5 By first expressing the equation $$\tan \theta \tan \left( \theta + 45 ^ { \circ } \right) = 2 \cot 2 \theta$$ as a quadratic equation in \(\tan \theta\), solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

6

1. By sketching a suitable pair of graphs, show that the equation \(x ^ { 5 } = 2 + x\) has exactly one real root.
2. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 5 } + 2 } { 5 x _ { n } ^ { 4 } - 1 }$$ converges, then it converges to the root of the equation in part (a).
3. Use the iterative formula with initial value \(x _ { 1 } = 1.5\) to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. \(7 \quad\) Let \(\mathrm { f } ( x ) = \frac { 2 } { ( 2 x - 1 ) ( 2 x + 1 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
3. Hence show that \(\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)\).

8 Relative to the origin \(O\), the points \(A , B\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + 5 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O D } = 3 \mathbf { i } + 2 \mathbf { k }$$ A fourth point \(C\) is such that \(A B C D\) is a parallelogram.

1. Find the position vector of \(C\) and verify that the parallelogram is not a rhombus.
2. Find angle \(B A D\), giving your answer in degrees.
3. Find the area of the parallelogram correct to 3 significant figures.

9

1. The complex numbers \(u\) and \(w\) are such that $$u - w = 2 \mathrm { i } \quad \text { and } \quad u w = 6$$ Find \(u\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities $$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
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   A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is \(A\) and the radius is \(r\), as shown in the diagram. The depth of water at time \(t\) is \(h\). At time \(t = 0\) the tank is full and the depth of the water is \(r\). At this instant a tap at \(A\) is opened and water begins to flow out at a rate proportional to \(\sqrt { h }\). The tank becomes empty at time \(t = 14\). The volume of water in the tank is \(V\) when the depth is \(h\). It is given that \(V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)\).
   1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { B } { 2 r h ^ { \frac { 1 } { 2 } } - h ^ { \frac { 3 } { 2 } } } ,$$ where \(B\) is a positive constant.
   2. Solve the differential equation and obtain an expression for \(t\) in terms of \(h\) and \(r\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.