CAIE P3 (Pure Mathematics 3) 2022 March

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

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 - 3 \mathrm { i } | \leqslant 2\) and \(\arg z \leqslant \frac { 3 } { 4 } \pi\).

3
\includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-05_666_800_260_667} The variables \(x\) and \(y\) satisfy the equation \(x ^ { n } y ^ { 2 } = C\), where \(n\) and \(C\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0.31,1.21 )\) and \(( 1.06,0.91 )\), as shown in the diagram. Find the value of \(n\) and find the value of \(C\) correct to 2 decimal places.

4 The parametric equations of a curve are $$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).

5 The angles \(\alpha\) and \(\beta\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) and are such that $$\tan ( \alpha + \beta ) = 2 \quad \text { and } \quad \tan \alpha = 3 \tan \beta .$$ Find the possible values of \(\alpha\) and \(\beta\).

6 Find the complex numbers \(w\) which satisfy the equation \(w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1\) and are such that \(\operatorname { Re } w \leqslant 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

7

1. By sketching a suitable pair of graphs, show that the equation \(4 - x ^ { 2 } = \sec \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \pi\).
2. Verify by calculation that this root lies between 1 and 2 .
3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 4 - \sec \frac { 1 } { 2 } x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8

1. Find the quotient and remainder when \(8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7\) is divided by \(4 x ^ { 2 } + 1\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\).

9 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\).
Solve the differential equation and find the exact value of \(y\) when \(x = 3\), giving your answer in a simplified form.

10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) respectively. The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\).

1. Find a vector equation for the line through \(A\) and \(B\).
2. Find the acute angle between the directions of \(A B\) and \(l\), giving your answer in degrees.
3. Show that the line through \(A\) and \(B\) does not intersect the line \(l\).

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\includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-16_556_698_274_712} The diagram shows the curve \(y = \sin x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 significant figures.
2. Using the substitution \(u = \cos x\), find the area of the shaded region enclosed by the curve and the \(x\)-axis in the first quadrant, giving your answer in a simplified exact form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.