CAIE P2 (Pure Mathematics 2) 2019 June

1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + ( k + 1 ) x ^ { 2 } - m x + 3 k$$ where \(k\) and \(m\) are constants. Given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), express \(m\) in terms of \(k\).

2

1. Solve the equation \(| 4 + 2 x | = | 3 - 5 x |\).
2. Hence solve the equation \(\left| 4 + 2 e ^ { 3 y } \right| = \left| 3 - 5 e ^ { 3 y } \right|\), giving the answer correct to 3 significant figures.

3 Find the exact coordinates of the stationary point of the curve with equation \(y = \frac { 3 x } { \ln x }\).

4

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 4 \sin 2 x + 2 \cos ^ { 2 } x \right) \mathrm { d } x\). Show all necessary working.
2. Use the trapezium rule with two intervals to find an approximation to \(\int _ { 2 } ^ { 8 } \sqrt { } ( \ln ( 1 + x ) ) \mathrm { d } x\)

5

1. Find the quotient and remainder when \(2 x ^ { 3 } + x ^ { 2 } - 8 x\) is divided by ( \(2 x + 1\) ).
2. Hence find the exact value of \(\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x\), giving the answer in the form \(\ln \left( k \mathrm { e } ^ { a } \right)\) where \(k\) and \(a\) are constants.

6
\includegraphics[max width=\textwidth, alt={}, center]{f5e0b088-73db-405b-a832-aa01d9fcba64-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .

1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.

7

1. 1. Express \(4 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
   2. Hence find the smallest positive value of \(\theta\) satisfying the equation \(4 \sin \theta + 4 \cos \theta = 5\).
2. Solve the equation $$4 \cot 2 x = 5 + \tan x$$ for \(0 < x < \pi\), showing all necessary working and giving the answers correct to 2 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.