CAIE P3 (Pure Mathematics 3) 2003 June

1

1. Show that the equation $$\sin \left( x - 60 ^ { \circ } \right) - \cos \left( 30 ^ { \circ } - x \right) = 1$$ can be written in the form \(\cos x = k\), where \(k\) is a constant.
2. Hence solve the equation, for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

2 Find the exact value of \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\).

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

4 The polynomial \(x ^ { 4 } - 2 x ^ { 3 } - 2 x ^ { 2 } + a\) is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x )\) is divisible by \(x ^ { 2 } - 4 x + 4\).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(\mathrm { f } ( x )\) is never negative.

5 The complex number 2 i is denoted by \(u\). The complex number with modulus 1 and argument \(\frac { 2 } { 3 } \pi\) is denoted by \(w\).

1. Find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(w , u w\) and \(\frac { u } { w }\).
2. Sketch an Argand diagram showing the points \(U , A\) and \(B\) representing the complex numbers \(u\), \(u w\) and \(\frac { u } { w }\) respectively.
3. Prove that triangle \(U A B\) is equilateral.

6 Let \(\mathrm { f } ( x ) = \frac { 9 x ^ { 2 } + 4 } { ( 2 x + 1 ) ( x - 2 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that, when \(x\) is sufficiently small for \(x ^ { 3 }\) and higher powers to be neglected, $$f ( x ) = 1 - x + 5 x ^ { 2 }$$

7 In a chemical reaction a compound \(X\) is formed from a compound \(Y\). The masses in grams of \(X\) and \(Y\) present at time \(t\) seconds after the start of the reaction are \(x\) and \(y\) respectively. The sum of the two masses is equal to 100 grams throughout the reaction. At any time, the rate of formation of \(X\) is proportional to the mass of \(Y\) at that time. When \(t = 0 , x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1.9\).

1. Show that \(x\) satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.02 ( 100 - x ) .$$
2. Solve this differential equation, obtaining an expression for \(x\) in terms of \(t\).
3. State what happens to the value of \(x\) as \(t\) becomes very large.

8 The equation of a curve is \(y = \ln x + \frac { 2 } { x }\), where \(x > 0\).

1. Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\). State an equation satisfied by \(\alpha\), and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 3\).
3. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.

9 Two planes have equations \(x + 2 y - 2 z = 2\) and \(2 x - 3 y + 6 z = 3\). The planes intersect in the straight line \(l\).

1. Calculate the acute angle between the two planes.
2. Find a vector equation for the line \(l\).

10

1. Prove the identity $$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x$$
2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \cot x \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \operatorname { cosec } 2 x \mathrm {~d} x\), giving your answer in the form \(a \ln b\).