CAIE P3 (Pure Mathematics 3) 2023 November

1 Find the set of values of \(x\) satisfying the inequality \(\left| 2 ^ { x + 1 } - 2 \right| < 0.5\), giving your answer to 3 significant figures.

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

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 38 and when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is \(\frac { 19 } { 2 }\). Find the values of \(a\) and \(b\).

4 Solve the quadratic equation \(( 3 + \mathrm { i } ) w ^ { 2 } - 2 w + 3 - \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

5 Find the exact coordinates of the stationary points of the curve \(y = \frac { \mathrm { e } ^ { 3 x ^ { 2 } - 1 } } { 1 - x ^ { 2 } }\).

6

1. Show that the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\) can be written in the form $$4 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta - 1 = 0$$
2. Hence solve the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x ^ { 2 } + 6 x } { 2 y + 3 }\).
2. Hence find the coordinates of the points on the curve at which the tangent is parallel to the \(x\)-axis. [5]

8 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 4 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \cos ^ { 2 } 3 y .$$ It is given that \(y = 0\) when \(x = 2\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

9 Let \(\mathrm { f } ( x ) = \frac { 17 x ^ { 2 } - 7 x + 16 } { \left( 2 + 3 x ^ { 2 } \right) ( 2 - x ) }\).

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 \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.

10
\includegraphics[max width=\textwidth, alt={}, center]{a49b720b-f8d2-42ff-b147-5d676993aa4c-16_611_689_274_721} The diagram shows the curve \(y = x \cos 2 x\), for \(x \geqslant 0\).

1. Find the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 } \pi\).
2. Find the exact area of the shaded region shown in the diagram, bounded by the curve and the \(x\)-axis.

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

1. Find a unit vector in the direction of \(l\).
   The line \(m\) passes through the points \(A\) and \(B\).
2. Find a vector equation for \(m\).
3. Determine whether lines \(l\) and \(m\) are parallel, intersect or are skew.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.