CAIE P3 (Pure Mathematics 3) 2023 November

1

1. Sketch the graph of \(y = | 4 x - 2 |\).
2. Solve the inequality \(1 + 3 x < | 4 x - 2 |\).

2 The parametric equations of a curve are $$x = ( \ln t ) ^ { 2 } , \quad y = \mathrm { e } ^ { 2 - t ^ { 2 } }$$ for \(t > 0\).
Find the gradient of the curve at the point where \(t = \mathrm { e }\), simplifying your answer.

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 11 x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(( 2 x - 1 )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 12 . Find the values of \(a\) and \(b\).

4

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 3 \mathrm { i } | \leqslant 2\) and \(\operatorname { Re } z \leqslant 3\).
2. Find the greatest value of \(\arg z\) for points in this region.

5 Find the exact value of \(\int _ { 0 } ^ { 6 } \frac { x ( x + 1 ) } { x ^ { 2 } + 4 } \mathrm {~d} x\).

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 2 - \cos x$$ has one root in the interval \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
2. Show by calculation that this root lies between 0.6 and 0.8 .
3. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 - \cos x _ { n } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. By expressing \(3 \theta\) as \(2 \theta + \theta\), prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
2. Hence solve the equation $$\cos 3 \theta + \cos \theta \cos 2 \theta = \cos ^ { 2 } \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8 It is given that \(\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )\), where \(a\) and \(\lambda\) are real constants.

1. Show that \(3 a ^ { 2 } + 4 a - 4 = 0\).
2. Hence find the possible values of \(a\) and the corresponding values of \(\lambda\).

9
\includegraphics[max width=\textwidth, alt={}, center]{39cf66af-095b-404b-a38c-0aa7684c4a27-14_428_787_274_671} The diagram shows the curve \(y = \sin x \cos 2 x\), for \(0 \leqslant x \leqslant \pi\), and a maximum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).

1. Find the value of \(a\) correct to 2 decimal places.
2. Find the exact area of the region \(R\), giving your answer in simplified form.

10 The equations of the lines \(l\) and \(m\) are given by $$l : \mathbf { r } = \left( \begin{array} { r } 3
- 2
1 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
2 \end{array} \right) \quad \text { and } \quad m : \mathbf { r } = \left( \begin{array} { r } 6
- 3
6 \end{array} \right) + \mu \left( \begin{array} { r } - 2
4
c \end{array} \right)$$ where \(c\) is a positive constant. It is given that the angle between \(l\) and \(m\) is \(60 ^ { \circ }\).

1. Find the value of \(c\).
2. Show that the length of the perpendicular from \(( 6 , - 3,6 )\) to \(l\) is \(\sqrt { 11 }\).

11 The variables \(x\) and \(y\) satisfy the differential equation $$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + y = 0$$ It is given that \(x = 1\) when \(y = 1\).

1. Solve the differential equation to obtain an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) when \(x\) tends to infinity. Give your answer in an 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.