CAIE P3 (Pure Mathematics 3) 2012 November

1 Find the set of values of \(x\) satisfying the inequality \(3 | x - 1 | < | 2 x + 1 |\).

2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.

3 Solve the equation $$\sin \left( \theta + 45 ^ { \circ } \right) = 2 \cos \left( \theta - 30 ^ { \circ } \right)$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 When \(( 1 + a x ) ^ { - 2 }\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 3 }\) are equal.

1. Find the exact value of \(a\).
2. When \(a\) has this value, obtain the expansion up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

5

1. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
2. Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
3. Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
4. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).

6 The variables \(x\) and \(y\) are related by the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 - y ^ { 2 }$$ When \(x = 2 , y = 0\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

7 The equation of a curve is \(\ln ( x y ) - y ^ { 3 } = 1\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x \left( 3 y ^ { 3 } - 1 \right) }\).
2. Find the coordinates of the point where the tangent to the curve is parallel to the \(y\)-axis, giving each coordinate correct to 3 significant figures.

8 \includegraphics[max width=\textwidth, alt={}, center]{7fe27759-d014-4bc6-8391-342d9df8280e-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).

1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right) .$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).