CAIE P3 (Pure Mathematics 3) 2010 November

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

2 Solve the equation $$\ln \left( 1 + x ^ { 2 } \right) = 1 + 2 \ln x$$ giving your answer correct to 3 significant figures.

3 Solve the equation $$\cos \left( \theta + 60 ^ { \circ } \right) = 2 \sin \theta$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

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

5 Let \(I = \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x\).

1. Using the substitution \(x = 2 \sin \theta\), show that $$I = \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
2. Hence find the exact value of \(I\).

6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$

1. Find the modulus and argument of \(z\).
2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
   1. \(2 z + z ^ { * }\),
   2. \(\frac { \mathrm { i } z ^ { * } } { z }\).
   3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).

7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line \(A B\) and \(O P\) is perpendicular to \(A B\).

1. Find a vector equation for the line \(A B\).
2. Find the position vector of \(P\).
3. Find the equation of the plane which contains \(A B\) and which is perpendicular to the plane \(O A B\), giving your answer in the form \(a x + b y + c z = d\).

8 Let \(\mathrm { f } ( x ) = \frac { 3 x } { ( 1 + x ) \left( 1 + 2 x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).

10 A certain substance is formed in a chemical reaction. The mass of substance formed \(t\) seconds after the start of the reaction is \(x\) grams. At any time the rate of formation of the substance is proportional to \(( 20 - x )\). When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.05 ( 20 - x ) .$$
2. Find, in any form, the solution of this differential equation.
3. Find \(x\) when \(t = 10\), giving your answer correct to 1 decimal place.
4. State what happens to the value of \(x\) as \(t\) becomes very large.