CAIE P3 (Pure Mathematics 3) 2015 June

1 Use logarithms to solve the equation \(2 ^ { 5 x } = 3 ^ { 2 x + 1 }\), giving the answer correct to 3 significant figures.

2 Use the trapezium rule with three intervals to find an approximation to $$\int _ { 0 } ^ { 3 } \left| 3 ^ { x } - 10 \right| \mathrm { d } x$$

3 Show that, for small values of \(x ^ { 2 }\), $$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$ where the value of the constant \(k\) is to be determined.

4 The equation of a curve is $$y = 3 \cos 2 x + 7 \sin x + 2$$ Find the \(x\)-coordinates of the stationary points in the interval \(0 \leqslant x \leqslant \pi\). Give each answer correct to 3 significant figures.

5

1. Find \(\int \left( 4 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \sin \left( x + \frac { 1 } { 6 } \pi \right) } { \sin x } \mathrm {~d} x\).

6 The straight line \(l _ { 1 }\) passes through the points \(( 0,1,5 )\) and \(( 2 , - 2,1 )\). The straight line \(l _ { 2 }\) has equation \(\mathbf { r } = 7 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\).

1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the acute angle between the direction of the line \(l _ { 2 }\) and the direction of the \(x\)-axis.

7 Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x \left( 3 y ^ { 2 } + 10 y + 3 \right)$$ obtaining an expression for \(y\) in terms of \(x\).

8 The complex number \(w\) is defined by \(w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }\).

1. Without using a calculator, show that \(w = 2 + 4 \mathrm { i }\).
2. It is given that \(p\) is a real number such that \(\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi\). Find the set of possible values of \(p\).
3. The complex conjugate of \(w\) is denoted by \(w ^ { * }\). The complex numbers \(w\) and \(w ^ { * }\) are represented in an Argand diagram by the points \(S\) and \(T\) respectively. Find, in the form \(| z - a | = k\), the equation of the circle passing through \(S , T\) and the origin.

9
\includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { 2 - x }\) and its maximum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is 2 .
2. Find the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm { e } ^ { 2 - x } \mathrm {~d} x\).

10
\includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_515_508_1105_815} The diagram shows part of the curve with parametric equations $$x = 2 \ln ( t + 2 ) , \quad y = t ^ { 3 } + 2 t + 3$$

1. Find the gradient of the curve at the origin.
2. At the point \(P\) on the curve, the value of the parameter is \(p\). It is given that the gradient of the curve at \(P\) is \(\frac { 1 } { 2 }\).
   (a) Show that \(p = \frac { 1 } { 3 p ^ { 2 } + 2 } - 2\).
   (b) By first using an iterative formula based on the equation in part (a), determine the coordinates of the point \(P\). Give the result of each iteration to 5 decimal places and each coordinate of \(P\) correct to 2 decimal places.