CAIE P3 (Pure Mathematics 3) 2020 November

1 Solve the inequality \(2 - 5 x > 2 | x - 3 |\).

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \geqslant 2\) and \(| z - 1 + \mathrm { i } | \leqslant 1\).

3 The parametric equations of a curve are $$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).

4 Solve the equation $$\log _ { 10 } ( 2 x + 1 ) = 2 \log _ { 10 } ( x + 1 ) - 1$$ Give your answers correct to 3 decimal places.

5

1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } x = 1 + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) has exactly two roots in the interval \(0 < x < \pi\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \pi - \sin ^ { - 1 } \left( \frac { 1 } { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } + 1 } \right)$$ with initial value \(x _ { 1 } = 2\), converges to one of these roots.
   Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. Express \(\sqrt { 6 } \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(\sqrt { 6 } \cos \frac { 1 } { 3 } x + 3 \sin \frac { 1 } { 3 } x = 2.5\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

7

1. Verify that \(- 1 + \sqrt { 5 } \mathrm { i }\) is a root of the equation \(2 x ^ { 3 } + x ^ { 2 } + 6 x - 18 = 0\).
2. Find the other roots of this equation.

8 The coordinates \(( x , y )\) of a general point of a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 1 - 2 x ^ { 2 } \right) y$$ for \(x > 0\). It is given that \(y = 1\) when \(x = 1\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

9 Let \(\mathrm { f } ( x ) = \frac { 8 + 5 x + 12 x ^ { 2 } } { ( 1 - x ) ( 2 + 3 x ) ^ { 2 } }\).

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 \(x ^ { 2 }\).

10
\includegraphics[max width=\textwidth, alt={}, center]{5f80ae11-34c3-4d2f-89f8-71b4ac021c7d-16_426_908_262_616} The diagram shows the curve \(y = ( 2 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of e.

11 Two lines have equations \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant.

1. Given that the two lines intersect, find the value of \(a\) and the position vector of the point of intersection.
2. Given instead that the acute angle between the directions of the two lines is \(\cos ^ { - 1 } \left( \frac { 1 } { 6 } \right)\), find the two possible values of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.