CAIE P2 (Pure Mathematics 2) 2015 November

1

1. Solve the equation \(| 3 x - 2 | = 5\).
2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.

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

1. Determine the value of \(\alpha\) correct to 3 decimal places, giving the result of each iteration to 5 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

3
\includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-2_456_725_1082_712} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { m }\), where \(K\) and \(m\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(K\) and \(m\) correct to 2 significant figures.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + 11 x ^ { 2 } + a x + a$$ where \(a\) is a constant. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).

1. Use the factor theorem to show that \(a = - 4\).
2. When \(a = - 4\),
   (a) factorise \(\mathrm { p } ( x )\) completely,
   (b) solve the equation \(6 \sec ^ { 3 } \theta + 11 \sec ^ { 2 } \theta + a \sec \theta + a = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

5 Find the \(x\)-coordinates of the stationary points of the following curves:

1. \(y = 4 x \mathrm { e } ^ { - 3 x }\);
2. \(y = \frac { 4 x ^ { 2 } } { x + 1 }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_453_650_258_744} The diagram shows the curve with parametric equations $$x = 3 \cos t , \quad y = 2 \cos \left( t - \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant t < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 } ( \sqrt { } 3 - \cot t )\).
2. Find the equation of the tangent to the curve at the point where the curve crosses the positive \(y\)-axis. Give the answer in the form \(y = m x + c\).

7

1. Show that the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 2 } x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x\) is \(\frac { 1 } { 6 } \pi + \frac { 9 } { 8 } \sqrt { } 3\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_444_495_1523_865} The diagram shows the curve \(y = \cos x + \frac { 1 } { \cos x }\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). The shaded region is bounded by the curve and the lines \(x = 0 , x = \frac { 1 } { 3 } \pi\) and \(y = 0\). Find the exact volume of the solid obtained when the shaded region is rotated completely about the \(x\)-axis.