CAIE P2 (Pure Mathematics 2) 2015 June

1

1. Use logarithms to solve the equation \(2 ^ { x } = 20 ^ { 5 }\), giving the answer correct to 3 significant figures.
2. Hence determine the number of integers \(n\) satisfying $$20 ^ { - 5 } < 2 ^ { n } < 20 ^ { 5 }$$

2

1. Given that \(( x + 2 )\) is a factor of $$4 x ^ { 3 } + a x ^ { 2 } - ( a + 1 ) x - 18$$ find the value of the constant \(a\).
2. When \(a\) has this value, factorise \(4 x ^ { 3 } + a x ^ { 2 } - ( a + 1 ) x - 18\) completely.

3 It is given that \(\theta\) is an acute angle measured in degrees such that $$2 \sec ^ { 2 } \theta + 3 \tan \theta = 22$$

1. Find the value of \(\tan \theta\).
2. Use an appropriate formula to find the exact value of \(\tan \left( \theta + 135 ^ { \circ } \right)\).

4
\includegraphics[max width=\textwidth, alt={}, center]{cc051d68-7e21-4dc1-b34d-6fb7f12a52fd-2_524_625_1425_758} The diagram shows the curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\).
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0 , x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac { 5 } { 2 }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{cc051d68-7e21-4dc1-b34d-6fb7f12a52fd-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\).
2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

7

1. Find the gradient of the curve $$3 \ln x + 4 \ln y + 6 x y = 6$$ at the point \(( 1,1 )\).
2. The parametric equations of a curve are $$x = \frac { 10 } { t } - t , \quad y = \sqrt { } ( 2 t - 1 ) .$$ Find the gradient of the curve at the point \(( - 3,3 )\).