CAIE P2 (Pure Mathematics 2) 2010 November

1 Solve the inequality \(| 3 x + 1 | > 8\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 7 x _ { n } } { 8 } + \frac { 5 } { 2 x _ { n } ^ { 4 } }$$ with initial value \(x _ { 1 } = 1.7\), converges to \(\alpha\).

1. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence show that \(\alpha = \sqrt [ 5 ] { 20 }\).

3 The polynomial \(x ^ { 3 } + 4 x ^ { 2 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) is equal to the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and find the quotient when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\).

4

1. Find \(\int \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
2. Express \(\sin ^ { 2 } 3 x\) in terms of \(\cos 6 x\) and hence find \(\int \sin ^ { 2 } 3 x \mathrm {~d} x\).

5
\includegraphics[max width=\textwidth, alt={}, center]{e814d76c-8757-4cc4-a69c-e3636b4cab16-2_604_887_1667_628} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.4,0.8\) ) and ( \(2.2,1.2\) ), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.

6

1. Express \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$2 \sin \theta - \cos \theta = - 0.4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{e814d76c-8757-4cc4-a69c-e3636b4cab16-3_611_1084_648_532} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 1 } ^ { 4 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.

8 The equation of a curve is $$x ^ { 2 } + 2 x y - y ^ { 2 } + 8 = 0$$

1. Show that the tangent to the curve at the point \(( - 2,2 )\) is parallel to the \(x\)-axis.
2. Find the equation of the tangent to the curve at the other point on the curve for which \(x = - 2\), giving your answer in the form \(y = m x + c\).