CAIE P2 (Pure Mathematics 2) 2015 June

1

1. Solve the equation \(| 3 x + 4 | = | 3 x - 11 |\).
2. Hence, using logarithms, solve the equation \(\left| 3 \times 2 ^ { y } + 4 \right| = \left| 3 \times 2 ^ { y } - 11 \right|\), giving the answer correct to 3 significant figures.

2
\includegraphics[max width=\textwidth, alt={}, center]{595e38f4-c52e-4509-8b16-f08e30dec96b-2_456_716_529_712} The variables \(x\) and \(y\) satisfy the equation $$y = A \mathrm { e } ^ { p ( x - 1 ) } ,$$ where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 2,1.60 )\) and \(( 5,2.92 )\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 significant figures.

3 The equation of a curve is $$y = 6 \sin x - 2 \cos 2 x$$ Find the equation of the tangent to the curve at the point \(\left( \frac { 1 } { 6 } \pi , 2 \right)\). Give the answer in the form \(y = m x + c\), where the values of \(m\) and \(c\) are correct to 3 significant figures.

4 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\). It is also given that, when \(\mathrm { g } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the greatest possible value of \(\mathrm { g } ( x ) - \mathrm { f } ( x )\) as \(x\) varies.

5

1. Given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 1 \right) \mathrm { d } x = 10\), show that the positive constant \(a\) satisfies the equation $$a = 2 \ln \left( \frac { 16 - a } { 6 } \right)$$
2. Use the iterative formula \(a _ { n + 1 } = 2 \ln \left( \frac { 16 - a _ { n } } { 6 } \right)\) with \(a _ { 1 } = 2\) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6

1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
2. Hence
   (a) solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
   (b) find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).

7 The equation of a curve is $$y ^ { 3 } + 4 x y = 16$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 y } { 3 y ^ { 2 } + 4 x }\).
2. Show that the curve has no stationary points.
3. Find the coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.