CAIE P3 (Pure Mathematics 3) 2017 June

1 Solve the equation \(\ln \left( x ^ { 2 } + 1 \right) = 1 + 2 \ln x\), giving your answer correct to 3 significant figures.

2 Solve the inequality \(| x - 3 | < 3 x - 4\).

3

1. Express the equation \(\cot \theta - 2 \tan \theta = \sin 2 \theta\) in the form \(a \cos ^ { 4 } \theta + b \cos ^ { 2 } \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be determined.
2. Hence solve the equation \(\cot \theta - 2 \tan \theta = \sin 2 \theta\) for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 The parametric equations of a curve are $$x = t ^ { 2 } + 1 , \quad y = 4 t + \ln ( 2 t - 1 )$$

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the normal to the curve at the point where \(t = 1\). Give your answer in the form \(a x + b y + c = 0\).

5 In a certain chemical process a substance \(A\) reacts with and reduces a substance \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.2 x y\) and \(x = \frac { 10 } { ( 1 + t ) ^ { 2 } }\). At the beginning of the process \(y = 100\).

1. Form a differential equation in \(y\) and \(t\), and solve this differential equation.
2. Find the exact value approached by the mass of \(B\) as \(t\) becomes large. State what happens to the mass of \(A\) as \(t\) becomes large.

6 Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm { i }\) is denoted by \(u\).

1. It is given that \(u\) is a root of the equation \(x ^ { 3 } + a x ^ { 2 } - 3 x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\).
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | < 1\) and \(| z | < | z + \mathrm { i } |\).

7

1. Prove that if \(y = \frac { 1 } { \cos \theta }\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
2. Prove the identity \(\frac { 1 + \sin \theta } { 1 - \sin \theta } \equiv 2 \sec ^ { 2 } \theta + 2 \sec \theta \tan \theta - 1\).
3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \sin \theta } { 1 - \sin \theta } \mathrm { d } \theta\).
   \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 7 x + 4 } { ( 3 x + 2 ) \left( x ^ { 2 } + 5 \right) }\).
4. Express \(\mathrm { f } ( x )\) in partial fractions.
5. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

9 Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 9 \mathbf { i } - \mathbf { j } + 8 \mathbf { k } + \mu ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\).

1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\).
2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(a x + b y + c z = d\).
3. Find the exact value of the perpendicular distance of \(A\) from this plane.

10
\includegraphics[max width=\textwidth, alt={}, center]{83a6d80b-dc74-4936-ac32-858a517a843c-18_353_675_260_735} The diagram shows the curve \(y = x ^ { 2 } \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point at \(M\) where \(x = p\).

1. Show that \(p\) satisfies the equation \(p = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p } \right)\).
2. Use the iterative formula \(p _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis.