CAIE P2 (Pure Mathematics 2) 2002 November

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

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

3

1. Express \(9 ^ { x }\) in terms of \(y\), where \(y = 3 ^ { x }\).
2. Hence solve the equation $$2 \left( 9 ^ { x } \right) - 7 \left( 3 ^ { x } \right) + 3 = 0 ,$$ expressing your answers for \(x\) in terms of logarithms where appropriate.

4

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 $$\sin x = \frac { 1 } { x ^ { 2 } }$$
2. Verify by calculation that this root lies between 1 and 1.5.
3. Show that this value of \(x\) is also a root of the equation $$x = \sqrt { } ( \operatorname { cosec } x )$$
4. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \operatorname { cosec } x _ { n } \right)$$ to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate.

5 The angle \(x\), measured in degrees, satisfies the equation $$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$

1. By expanding each side, show that the equation may be simplified to $$( 2 \sqrt { } 3 ) \cos x = \sin x$$
2. Find the two possible values of \(x\) lying between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
3. Find the exact value of \(\cos 2 x\), giving your answer as a fraction.

6

1. Find the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \sin 2 x + \cos x ) \mathrm { d } x\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{9894d97f-3b7b-4dbe-b94a-2c8415442038-3_517_880_422_669} The diagram shows part of the curve \(y = \frac { 1 } { x + 1 }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 1 , y = 0\) and \(x = p\).
   1. Find, in terms of \(p\), the area of \(R\).
   2. Hence find, correct to 1 decimal place, the value of \(p\) for which the area of \(R\) is equal to 2 .

7 The equation of a curve is $$2 x ^ { 2 } + 3 y ^ { 2 } - 2 x y = 10 .$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y - x }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.