CAIE P3 (Pure Mathematics 3) 2019 November

1 Given that \(\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x\), express \(y\) in terms of \(x\).

2 Solve the inequality \(| 2 x - 3 | > 4 | x + 1 |\).

3 The parametric equations of a curve are $$x = 2 t + \sin 2 t , \quad y = \ln ( 1 - \cos 2 t )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } 2 t\).

4 The number of insects in a population \(t\) weeks after the start of observations is denoted by \(N\). The population is decreasing at a rate proportional to \(N \mathrm { e } ^ { - 0.02 t }\). The variables \(N\) and \(t\) are treated as continuous, and it is given that when \(t = 0 , N = 1000\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = - 10\).

1. Show that \(N\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = - 0.01 \mathrm { e } ^ { - 0.02 t } N .$$
2. Solve the differential equation and find the value of \(t\) when \(N = 800\).
3. State what happens to the value of \(N\) as \(t\) becomes large.

5 The curve with equation \(y = \mathrm { e } ^ { - 2 x } \ln ( x - 1 )\) has a stationary point when \(x = p\).

1. Show that \(p\) satisfies the equation \(x = 1 + \exp \left( \frac { 1 } { 2 ( x - 1 ) } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
2. Verify by calculation that \(p\) lies between 2.2 and 2.6.
3. Use an iterative formula based on the equation in part (i) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6

1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
2. Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } ( \pi + \ln 4 )\).
   \(7 \quad\) Two lines \(l\) and \(m\) have equations \(\mathbf { r } = a \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) respectively, where \(a\) is a constant. It is given that the lines intersect.

1. Find the value of \(a\).
2. When \(a\) has this value, find the equation of the plane containing \(l\) and \(m\).

8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 6 } { x ^ { 2 } ( x + 2 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence, showing full working, show that the exact value of \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\) is \(\frac { 9 } { 4 }\).

9

1. By first expanding \(\cos ( 2 x + x )\), show that \(\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x\).
2. Hence solve the equation \(\cos 3 x + 3 \cos x + 1 = 0\), for \(0 \leqslant x \leqslant \pi\).
3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x\).

10

1. The complex number \(u\) is given by \(u = - 3 - ( 2 \sqrt { } 10 )\) i. Showing all necessary working and without using a calculator, find the square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
2. On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of the complex number \(z\).
   [0pt] [5] If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.