CAIE P2 (Pure Mathematics 2) 2016 March

1 Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 10\) is divided by \(( x + 2 )\).

2 Solve the inequality \(| x - 5 | < | 2 x + 3 |\).

3 It is given that \(k\) is a positive constant. Solve the equation \(2 \ln x = \ln ( 3 k + x ) + \ln ( 2 k - x )\), expressing \(x\) in terms of \(k\).

4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } + 4 x _ { n } ^ { - 3 } \right)$$ with initial value \(x _ { 1 } = 1.5\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

5 Given that \(\int _ { 0 } ^ { a } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x = 65\), find the value of \(a\) correct to 3 decimal places.

6
\includegraphics[max width=\textwidth, alt={}, center]{d53a2d6b-4c5e-4bc6-8aa1-587e97c87920-2_371_839_1409_651} The diagram shows the part of the curve \(y = 3 \mathrm { e } ^ { - x } \sin 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and the stationary point \(M\).

1. Find the equation of the tangent to the curve at the origin.
2. Find the coordinates of \(M\), giving each coordinate correct to 3 decimal places.

7 The equation of a curve is \(2 x ^ { 3 } + y ^ { 3 } = 24\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), and show that the gradient of the curve is never positive.
2. Find the coordinates of the two points on the curve at which the gradient is - 2 .

8

1. Show that \(\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x\).
2. Using the identity in part (i),
   (a) find the least possible value of $$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$ as \(x\) varies,
   (b) find the exact value of \(\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).