CAIE P2 (Pure Mathematics 2) 2004 November

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

2 Solve the equation \(x ^ { 3.9 } = 11 x ^ { 3.2 }\), where \(x \neq 0\).

3 Find the values of \(x\) satisfying the equation $$3 \sin 2 x = \cos x$$ for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).

4 The cubic polynomial \(2 x ^ { 3 } - 5 x ^ { 2 } + a x + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 6 . Find the values of \(a\) and \(b\).

5 The curve with equation \(y = x ^ { 2 } \ln x\), where \(x > 0\), has one stationary point.

1. Find the \(x\)-coordinate of this point, giving your answer in terms of e .
2. Determine whether this point is a maximum or a minimum point.

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\cot x = x$$
2. Verify by calculation that this root lies between 0.8 and 0.9 radians.
3. Show that this value of \(x\) is also a root of the equation $$x = \tan ^ { - 1 } \left( \frac { 1 } { x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { x _ { n } } \right)$$ to determine this root correct to 2 decimal places, showing the result of each iteration.

7 \includegraphics[max width=\textwidth, alt={}, center]{25dffd43-9456-449b-be77-8402109ee603-3_608_672_283_733} The diagram shows the curve \(y = 2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - 2 x }\). The curve cuts the \(y\)-axis at \(A\).

1. Write down the coordinates of \(A\).
2. Find the equation of the tangent to the curve at \(A\), and state the coordinates of the point where this tangent meets the \(x\)-axis.
3. Calculate the area of the region bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = 1\), giving your answer correct to 2 significant figures.

8

1. Express \(\cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
2. Hence show that $$\frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } = \frac { 1 } { 2 } \sec ^ { 2 } \left( \theta - \frac { 1 } { 4 } \pi \right)$$
3. By differentiating \(\frac { \sin x } { \cos x }\), show that if \(y = \tan x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x\).
4. Using the results of parts (ii) and (iii), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$