CAIE P3 (Pure Mathematics 3) 2016 November

1 Solve the equation \(\frac { 3 ^ { x } + 2 } { 3 ^ { x } - 2 } = 8\), giving your answer correct to 3 decimal places.

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

3 Express the equation \(\sec \theta = 3 \cos \theta + \tan \theta\) as a quadratic equation in \(\sin \theta\). Hence solve this equation for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

4 The equation of a curve is \(x y ( x - 6 y ) = 9 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.

5

1. Prove the identity \(\tan 2 \theta - \tan \theta \equiv \tan \theta \sec 2 \theta\).
2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan \theta \sec 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).

6

1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } \frac { 1 } { 2 } x = \frac { 1 } { 3 } x + 1$$ has one root in the interval \(0 < x \leqslant \pi\).
2. Show by calculation that this root lies between 1.4 and 1.6.
3. Show that, if a sequence of values in the interval \(0 < x \leqslant \pi\) given by the iterative formula $$x _ { n + 1 } = 2 \sin ^ { - 1 } \left( \frac { 3 } { x _ { n } + 3 } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

7
\includegraphics[max width=\textwidth, alt={}, center]{f4614578-f5f6-4283-8185-8b5598ad91d5-3_416_679_258_731} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).

1. Find the exact \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.

8 Two planes have equations \(3 x + y - z = 2\) and \(x - y + 2 z = 3\).

1. Show that the planes are perpendicular.
2. Find a vector equation for the line of intersection of the two planes.

1. Solve the equation \(( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 1 - \mathrm { i } | \leqslant 2\) and \(- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).

10 A large field of area \(4 \mathrm {~km} ^ { 2 }\) is becoming infected with a soil disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of growth of the infected area is given by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 4 - x )\), where \(k\) is a positive constant. It is given that when \(t = 0 , x = 0.4\) and that when \(t = 2 , x = 2\).

1. Solve the differential equation and show that \(k = \frac { 1 } { 4 } \ln 3\).
2. Find the value of \(t\) when \(90 \%\) of the area of the field is infected.