CAIE P3 (Pure Mathematics 3) 2004 November

1 Expand \(\frac { 1 } { ( 2 + x ) ^ { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2 Solve the equation $$\ln ( 1 + x ) = 1 + \ln x$$ giving your answer correct to 2 significant figures.

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

1. Find the value of \(a\). When \(a\) has this value,
2. factorise \(\mathrm { p } ( x )\),
3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

4

1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$\tan ^ { 2 } x - 6 \tan x + 1 = 0$$
2. Hence solve the equation \(\tan \left( 45 ^ { \circ } + x \right) = 2 \tan \left( 45 ^ { \circ } - x \right)\), for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-2_385_476_1653_836} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). The angle \(A O B\) is \(\alpha\) radians, where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The point \(N\) on \(O A\) is such that \(B N\) is perpendicular to \(O A\). The area of the triangle \(O N B\) is half the area of the sector \(O A B\).

1. Show that \(\alpha\) satisfies the equation \(\sin 2 x = x\).
2. By sketching a suitable pair of graphs, show that this equation has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
3. Use the iterative formula $$x _ { n + 1 } = \sin \left( 2 x _ { n } \right)$$ with initial value \(x _ { 1 } = 1\), to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.

6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.

1. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - v\) and \(\frac { u } { v }\).
2. State the argument of \(\frac { u } { v }\). In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the numbers \(u , v\) and \(u - v\) respectively.
3. State fully the geometrical relationship between \(O C\) and \(B A\).
4. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.

7
\includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-3_480_901_973_621} The diagram shows the curve \(y = x ^ { 2 } e ^ { - \frac { 1 } { 2 } x }\).

1. Find the \(x\)-coordinate of \(M\), the maximum point of the curve.
2. Find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 1\), giving your answer in terms of e.

8 An appropriate form for expressing \(\frac { 3 x } { ( x + 1 ) ( x - 2 ) }\) in partial fractions is $$\frac { A } { x + 1 } + \frac { B } { x - 2 }$$ where \(A\) and \(B\) are constants.

1. Without evaluating any constants, state appropriate forms for expressing the following in partial fractions:
   1. \(\frac { 4 x } { ( x + 4 ) \left( x ^ { 2 } + 3 \right) }\),
   2. \(\frac { 2 x + 1 } { ( x - 2 ) ( x + 2 ) ^ { 2 } }\).
2. Show that \(\int _ { 3 } ^ { 4 } \frac { 3 x } { ( x + 1 ) ( x - 2 ) } \mathrm { d } x = \ln 5\).

9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + s ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + t ( - 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )$$ respectively.

1. Show that \(l\) and \(m\) do not intersect. The point \(P\) lies on \(l\) and the point \(Q\) has position vector \(2 \mathbf { i } - \mathbf { k }\).
2. Given that the line \(P Q\) is perpendicular to \(l\), find the position vector of \(P\).
3. Verify that \(Q\) lies on \(m\) and that \(P Q\) is perpendicular to \(m\).

10 A rectangular reservoir has a horizontal base of area \(1000 \mathrm {~m} ^ { 2 }\). At time \(t = 0\), it is empty and water begins to flow into it at a constant rate of \(30 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). At the same time, water begins to flow out at a rate proportional to \(\sqrt { } h\), where \(h \mathrm {~m}\) is the depth of the water at time \(t \mathrm {~s}\). When \(h = 1 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = 0.02\).

1. Show that \(h\) satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = 0.01 ( 3 - \sqrt { } h )$$ It is given that, after making the substitution \(x = 3 - \sqrt { } h\), the equation in part (i) becomes $$( x - 3 ) \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.005 x$$
2. Using the fact that \(x = 3\) when \(t = 0\), solve this differential equation, obtaining an expression for \(t\) in terms of \(x\).
3. Find the time at which the depth of water reaches 4 m .