CAIE P3 (Pure Mathematics 3) 2008 November

1 Solve the equation $$\ln ( x + 2 ) = 2 + \ln x$$ giving your answer correct to 3 decimal places.

2 Expand \(( 1 + x ) \sqrt { } ( 1 - 2 x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

3 The curve \(y = \frac { \mathrm { e } ^ { x } } { \cos x }\), for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), has one stationary point. Find the \(x\)-coordinate of this point.

4 The parametric equations of a curve are $$x = a ( 2 \theta - \sin 2 \theta ) , \quad y = a ( 1 - \cos 2 \theta )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).

5 The polynomial \(4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(2 x ^ { 2 } - 3 x + 3\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\), justifying your answer.

6

1. Express \(5 \sin x + 12 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$5 \sin 2 \theta + 12 \cos 2 \theta = 11$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

7 Two planes have equations \(2 x - y - 3 z = 7\) and \(x + 2 y + 2 z = 0\).

1. Find the acute angle between the planes.
2. Find a vector equation for their line of intersection.

8
\includegraphics[max width=\textwidth, alt={}, center]{c687888e-bef0-4ea9-b5b3-e614028cc07c-3_654_805_274_671} An underground storage tank is being filled with liquid as shown in the diagram. Initially the tank is empty. At time \(t\) hours after filling begins, the volume of liquid is \(V \mathrm {~m} ^ { 3 }\) and the depth of liquid is \(h \mathrm {~m}\). It is given that \(V = \frac { 4 } { 3 } h ^ { 3 }\). The liquid is poured in at a rate of \(20 \mathrm {~m} ^ { 3 }\) per hour, but owing to leakage, liquid is lost at a rate proportional to \(h ^ { 2 }\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 4.95\).

1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 5 } { h ^ { 2 } } - \frac { 1 } { 20 } .$$
2. Verify that \(\frac { 20 h ^ { 2 } } { 100 - h ^ { 2 } } \equiv - 20 + \frac { 2000 } { ( 10 - h ) ( 10 + h ) }\).
3. Hence solve the differential equation in part (i), obtaining an expression for \(t\) in terms of \(h\).

9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6\).

1. Show that \(a\) satisfies the equation $$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
2. By sketching a suitable pair of graphs, show that this equation has only one root.
3. Verify by calculation that this root lies between 2 and 2.5.
4. Use an iterative formula based on the equation in part (i) to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).

1. Find the modulus and argument of \(w\).
2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.