CAIE P3 (Pure Mathematics 3) 2021 March

1 Solve the equation \(\ln \left( x ^ { 3 } - 3 \right) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures.

2 The polynomial \(a x ^ { 3 } + 5 x ^ { 2 } - 4 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 2 . Find the values of \(a\) and \(b\).

3 By first expressing the equation \(\tan \left( x + 45 ^ { \circ } \right) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4 The variables \(x\) and \(y\) satisfy the differential equation $$( 1 - \cos x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y \sin x$$ It is given that \(y = 4\) when \(x = \pi\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2 \pi\).

5

1. Express \(\sqrt { 7 } \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(\sqrt { 7 } \sin 2 \theta + 2 \cos 2 \theta = 1\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6 Let \(\mathrm { f } ( x ) = \frac { 5 a } { ( 2 x - a ) ( 3 a - x ) }\), where \(a\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence show that \(\int _ { a } ^ { 2 a } \mathrm { f } ( x ) \mathrm { d } x = \ln 6\).
   \(7 \quad\) Two lines have equations \(\mathbf { r } = \left( \begin{array} { l } 1
   3
   2 \end{array} \right) + s \left( \begin{array} { r } 2
   - 1
   3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 2
   1
   4 \end{array} \right) + t \left( \begin{array} { r } 1
   - 1
   4 \end{array} \right)\).
3. Show that the lines are skew.
4. Find the acute angle between the directions of the two lines.

8 The complex numbers \(u\) and \(v\) are defined by \(u = - 4 + 2 \mathrm { i }\) and \(v = 3 + \mathrm { i }\).

1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Hence express \(\frac { u } { v }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
   In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(2 u + v\) respectively.
3. State fully the geometrical relationship between \(O A\) and \(B C\).
4. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).

9 Let \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { 2 x } + 1 } { \mathrm { e } ^ { 2 x } - 1 }\), for \(x > 0\).

1. The equation \(x = \mathrm { f } ( x )\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5.
2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the exact value of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = - 8\).

10
\includegraphics[max width=\textwidth, alt={}, center]{149a8d28-8d2a-4b01-bed0-f16f1e201f32-18_372_675_264_735} The diagram shows the curve \(y = \sin 2 x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis.
2. Find the exact \(x\)-coordinate of \(M\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.