CAIE P2 (Pure Mathematics 2) 2012 June

1 Solve the equation \(\left| x ^ { 3 } - 14 \right| = 13\), showing all your working.

2
\includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-2_453_771_386_685} 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 \(( 0,2.14 )\) and \(( 5,4.49 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 1 decimal place.

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 3 x ^ { 2 } - 5 x + a + 4 ,$$ where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. When \(a\) has this value,
   (a) factorise \(\mathrm { p } ( x )\) completely,
   (b) find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

4

1. Given that \(35 + \sec ^ { 2 } \theta = 12 \tan \theta\), find the value of \(\tan \theta\).
2. Hence, showing the use of an appropriate formula in each case, find the exact value of
   (a) \(\tan \left( \theta - 45 ^ { \circ } \right)\),
   (b) \(\tan 2 \theta\).

5
\includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

6 A curve has parametric equations $$x = \frac { 1 } { ( 2 t + 1 ) ^ { 2 } } , \quad y = \sqrt { } ( t + 2 )$$ The point \(P\) on the curve has parameter \(p\) and it is given that the gradient of the curve at \(P\) is - 1 .

1. Show that \(p = ( p + 2 ) ^ { \frac { 1 } { 6 } } - \frac { 1 } { 2 }\).
2. Use an iterative process based on the equation in part (i) to find the value of \(p\) correct to 3 decimal places. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places.

7

1. Show that \(( 2 \sin x + \cos x ) ^ { 2 }\) can be written in the form \(\frac { 5 } { 2 } + 2 \sin 2 x - \frac { 3 } { 2 } \cos 2 x\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 2 \sin x + \cos x ) ^ { 2 } \mathrm {~d} x\).