CAIE P3 (Pure Mathematics 3) 2018 November

1 Find the set of values of \(x\) satisfying the inequality \(2 | 2 x - a | < | x + 3 a |\), where \(a\) is a positive constant. [4]

2 Showing all necessary working, solve the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } = 4\), giving your answer correct to 2 decimal places.

3

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

4 The parametric equations of a curve are $$x = 2 \sin \theta + \sin 2 \theta , \quad y = 2 \cos \theta + \cos 2 \theta$$ where \(0 < \theta < \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis.

5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).

6

1. Show that the equation ( \(\sqrt { } 2\) ) \(\operatorname { cosec } x + \cot x = \sqrt { } 3\) can be expressed in the form \(R \sin ( x - \alpha ) = \sqrt { } 2\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(( \sqrt { } 2 ) \operatorname { cosec } x + \cot x = \sqrt { } 3\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{c861e691-66da-4269-9057-4a343be9835e-12_357_565_260_790} The diagram shows the curve \(y = 5 \sin ^ { 2 } x \cos ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]

8

1. Showing all necessary working, express the complex number \(\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures.
2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - 3 + 2 i | = 1\). Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

9 Let \(\mathrm { f } ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).

10 The planes \(m\) and \(n\) have equations \(3 x + y - 2 z = 10\) and \(x - 2 y + 2 z = 5\) respectively. The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).

1. Show that \(l\) is parallel to \(m\).
2. Calculate the acute angle between the planes \(m\) and \(n\).
3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2 . Find the position vectors of the two possible positions of \(P\).
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