CAIE P3 (Pure Mathematics 3) 2023 June

1 Solve the inequality \(| 5 x - 3 | < 2 | 3 x - 7 |\).

2 Solve the equation \(\ln \left( 2 x ^ { 2 } - 3 \right) = 2 \ln x - \ln 2\), giving your answer in an exact form.

3

1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z + 3 - 2 \mathrm { i } | = 2\).
2. Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

4 Solve the equation \(2 \cos x - \cos \frac { 1 } { 2 } x = 1\) for \(0 \leqslant x \leqslant 2 \pi\).

5 The complex number \(2 + y \mathrm { i }\) is denoted by \(a\), where \(y\) is a real number and \(y < 0\). It is given that \(\mathrm { f } ( a ) = a ^ { 3 } - a ^ { 2 } - 2 a\).

1. Find a simplified expression for \(\mathrm { f } ( a )\) in terms of \(y\).
2. Given that \(\operatorname { Re } ( \mathrm { f } ( a ) ) = - 20\), find \(\arg a\).

6 The equation \(\cot \frac { 1 } { 2 } x = 3 x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).

1. Show by calculation that \(\alpha\) lies between 0.5 and 1 .
2. Show that, if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \tan ^ { - 1 } \left( \frac { 1 } { 3 x _ { n } } \right) \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 The equation of a curve is \(3 x ^ { 2 } + 4 x y + 3 y ^ { 2 } = 5\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x + 2 y } { 2 x + 3 y }\).
2. Hence find the exact coordinates of the two points on the curve at which the tangent is parallel to \(y + 2 x = 0\).

8

1. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 + 9 y ^ { 2 } } { \mathrm { e } ^ { 2 x + 1 } } .$$ It is given that \(y = 0\) when \(x = 1\).
   Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) as \(x\) tends to infinity. Give your answer in an exact form.

9 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 17 x - 17 } { ( 1 + 2 x ) ( 2 - x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 5 } { 2 } - \ln 72\).
   \includegraphics[max width=\textwidth, alt={}, center]{60bb482b-fa41-42ea-a112-62851e5a19aa-16_524_725_269_696} The diagram shows the curve \(y = ( x + 5 ) \sqrt { 3 - 2 x }\) and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Using the substitution \(u = 3 - 2 x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt { 13 }\), where \(a\) is a rational number. [5]

11 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(2 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.