CAIE P3 (Pure Mathematics 3) 2016 March

1 Solve the equation \(\ln \left( x ^ { 2 } + 4 \right) = 2 \ln x + \ln 4\), giving your answer in an exact form.

2 Express the equation \(\tan \left( \theta + 45 ^ { \circ } \right) - 2 \tan \left( \theta - 45 ^ { \circ } \right) = 4\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

3 The equation \(x ^ { 5 } - 3 x ^ { 3 } + x ^ { 2 } - 4 = 0\) has one positive root.

1. Verify by calculation that this root lies between 1 and 2 .
2. Show that the equation can be rearranged in the form $$\left. x = \sqrt [ 3 ] { ( } 3 x + \frac { 4 } { x ^ { 2 } } - 1 \right)$$
3. Use an iterative formula based on this rearrangement to determine the positive root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

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

5 Let \(I = \int _ { 0 } ^ { 1 } \frac { 9 } { \left( 3 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).

1. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), show that \(I = \sqrt { } 3 \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta\).
2. Hence find the exact value of \(I\).

6 A curve has equation $$\sin y \ln x = x - 2 \sin y$$ for \(- \frac { 1 } { 2 } \pi \leqslant y \leqslant \frac { 1 } { 2 } \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the exact \(x\)-coordinate of the point on the curve at which the tangent is parallel to the \(x\)-axis.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { x + y }$$ and it is given that \(y = 0\) when \(x = 0\).

1. Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).
2. Explain briefly why \(x\) can only take values less than 1 .

8 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\). The plane \(p\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) = 6\).

1. Show that \(l\) is parallel to \(p\).
2. A line \(m\) lies in the plane \(p\) and is perpendicular to \(l\). The line \(m\) passes through the point with coordinates (5, 3, 1). Find a vector equation for \(m\).

9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 3 } + 6 x - 8 } { x \left( x ^ { 2 } + 2 \right) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x } + \frac { C x + D } { x ^ { 2 } + 2 }\).
2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \ln 4\).

10

1. Find the complex number \(z\) satisfying the equation \(z ^ { * } + 1 = 2 \mathrm { i } z\), where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z + 1 - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
   2. Determine the difference between the greatest and least values of \(\arg z\) for points lying in this region.