CAIE P2 (Pure Mathematics 2) 2009 November

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

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

3 The polynomial \(4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

4

1. Show that the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\) can be written in the form \(\tan x = k\), where \(k\) is a constant.
2. Hence solve the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5

1. Express \(\cos ^ { 2 } 2 x\) in terms of \(\cos 4 x\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } \cos ^ { 2 } 2 x \mathrm {~d} x\).

6 The curve with equation \(y = x \ln x\) has one stationary point.

1. Find the exact coordinates of this point, giving your answers in terms of e .
2. Determine whether this point is a maximum or a minimum point.

7
\includegraphics[max width=\textwidth, alt={}, center]{67a12825-d7ce-4853-ada4-b8d3009331b5-3_531_759_262_694} The diagram shows the curve \(y = \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and the lines \(y = 1\) and \(x = p\), where \(p\) is a constant.

1. Find the area of \(R\) in terms of \(p\).
2. Show that if the area of \(R\) is equal to 1 then $$p = 2 - \mathrm { e } ^ { - p }$$
3. Use the iterative formula $$p _ { n + 1 } = 2 - \mathrm { e } ^ { - p _ { n } }$$ with initial value \(p _ { 1 } = 2\), to calculate the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8 The equation of a curve is \(y ^ { 2 } + 2 x y - x ^ { 2 } = 2\).

1. Find the coordinates of the two points on the curve where \(x = 1\).
2. Show by differentiation that at one of these points the tangent to the curve is parallel to the \(x\)-axis. Find the equation of the tangent to the curve at the other point, giving your answer in the form \(a x + b y + c = 0\).