CAIE P2 (Pure Mathematics 2) 2012 November

1 Solve the inequality \(| 2 x + 1 | < | 2 x - 5 |\).

2 The curve with equation \(y = \frac { \sin 2 x } { \mathrm { e } ^ { 2 x } }\) has one stationary point in the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the exact \(x\)-coordinate of this point.

3 The polynomial \(x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4\) is denoted by \(\mathrm { p } ( x )\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - 3 x + 2\).
2. Hence solve the equation \(\mathrm { p } ( x ) = 0\).

4
\includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-2_499_787_922_678} The diagram shows the part of the curve \(y = \sqrt { } ( 2 - \sin x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 2 - \sin x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
2. The line \(y = x\) intersects the curve \(y = \sqrt { } ( 2 - \sin x )\) at the point \(P\). Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \sin x _ { n } \right)$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5
\includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-3_512_732_251_705} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { - x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 1,2.9 )\) and \(( 3.5,1.4 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 2 decimal places.

6

1. Find \(\int 4 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
2. Show that \(\int _ { 1 } ^ { 3 } \frac { 6 } { 3 x - 1 } \mathrm {~d} x = \ln 16\).

7 The equation of a curve is $$3 x ^ { 2 } - 4 x y + 2 y ^ { 2 } - 6 = 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x - 2 y } { 2 x - 2 y }\).
2. Find the coordinates of each of the points on the curve where the tangent is parallel to the \(x\)-axis.

8

1. Given that \(\tan A = t\) and \(\tan ( A + B ) = 4\), find \(\tan B\) in terms of \(t\).
2. Solve the equation $$2 \tan \left( 45 ^ { \circ } - x \right) = 3 \tan x$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).