CAIE P2 (Pure Mathematics 2) 2017 November

1 Use logarithms to solve the equation \(5 ^ { 3 x - 1 } = 2 ^ { 4 x }\), giving your answer correct to 3 significant figures.

2 It is given that \(x\) satisfies the equation \(| x + 1 | = 4\). Find the possible values of $$| x + 4 | - | x - 4 | .$$

3 The equation of a curve is \(y = \tan \frac { 1 } { 2 } x + 3 \sin \frac { 1 } { 2 } x\). The curve has a stationary point \(M\) in the interval \(\pi < x < 2 \pi\). Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.

4 The polynomials \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are defined by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } + a x - 15 \quad \text { and } \quad \mathrm { q } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + b x + 21 ,$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\) and also of \(\mathrm { q } ( x )\).

1. Find the values of \(a\) and \(b\).
2. Show that the equation \(\mathrm { q } ( x ) - \mathrm { p } ( x ) = 0\) has only one real root.
   \includegraphics[max width=\textwidth, alt={}, center]{e2b16207-2cb7-412b-ba7f-758e4d3f1ffb-06_631_643_260_749} The diagram shows the curve \(y = 4 e ^ { - 2 x }\) and a straight line. The curve crosses the \(y\)-axis at the point \(P\). The straight line crosses the \(y\)-axis at the point \(( 0,9 )\) and its gradient is equal to the gradient of the curve at \(P\). The straight line meets the curve at two points, one of which is \(Q\) as shown.

1. Show that the \(x\)-coordinate of \(Q\) satisfies the equation \(x = \frac { 9 } { 8 } - \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
2. Use an iterative formula based on the equation in part (i) to find the \(x\)-coordinate of \(Q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin x ( 4 \sin x + 6 \cos x ) \mathrm { d } x\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 6 } { 3 x + 2 } \mathrm {~d} x = \ln 49\), find the value of the positive constant \(a\).

7 The equation of a curve is \(x ^ { 2 } + 4 x y + 2 y ^ { 2 } = 7\).

1. Find the equation of the tangent to the curve at the point \(( - 1,3 )\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
2. Show that there is no point on the curve at which the gradient is \(\frac { 1 } { 2 }\).