CAIE P2 (Pure Mathematics 2) 2014 June

1 Solve the inequality \(| 3 x - 2 | \geqslant | x + 4 |\).

2 Find the gradient of each of the following curves at the point for which \(x = 0\).

1. \(y = 3 \sin x + \tan 2 x\)
2. \(y = \frac { 6 } { 1 + \mathrm { e } ^ { 2 x } }\)

3

1. Find the quotient when \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 15\) is divided by ( \(x ^ { 2 } - 4\) ), and confirm that the remainder is 7 .
2. Hence solve the equation \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).

4

1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
2. Show by calculation that the root lies between 2.0 and 2.5.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

1. Prove that \(\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }\).
2. Hence
   (a) find the exact value of \(\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi\),
   (b) evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta\).

6

1. Show that \(\int _ { 6 } ^ { 16 } \frac { 6 } { 2 x - 7 } \mathrm {~d} x = \ln 125\).
2. Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 17 } \log _ { 10 } x d x$$ giving your answer correct to 3 significant figures.

7 The equation of a curve is $$2 x ^ { 2 } + 3 x y + y ^ { 2 } = 3$$

1. Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
2. Show that the curve has no stationary points.