CAIE P2 (Pure Mathematics 2) 2019 March

1 Solve the equation \(\sec ^ { 2 } \theta + \tan ^ { 2 } \theta = 5 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). Show all necessary working.

2 Given that \(x\) satisfies the equation \(| 2 x + 3 | = | 2 x - 1 |\), find the value of $$| 4 x - 3 | - | 6 x |$$

3
\includegraphics[max width=\textwidth, alt={}, center]{772c14a1-f79a-4147-a293-0ff34f930e20-04_577_569_260_788} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { p x + p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 1,2.835 )\) and \(( 6,6.585 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

4

1. Find the quotient when \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9\) is divided by ( \(2 x + 1\) ), and show that the remainder is 5 .
2. Show that the equation \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0\) has exactly one real root.

5 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 x + 1 }\) and the point \(P\) on the curve has \(y\)-coordinate 10 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 40 x + 10 )\).
2. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 40 x _ { n } + 10 \right)\) with \(x _ { 1 } = 2.3\) to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
3. Find the gradient of the curve at \(P\), giving the answer correct to 3 significant figures.

6

1. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 2 } { x } + \frac { 2 } { 2 x + 1 } \right) \mathrm { d } x = \ln 48\).
2. Find \(\int \sin 2 x ( \cot x + 2 \operatorname { cosec } x ) \mathrm { d } x\).

7 The parametric equations of a curve are $$x = 2 t - \sin 2 t , \quad y = 5 t + \cos 2 t$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). At the point \(P\) on the curve, the gradient of the curve is 2 .

1. Show that the value of the parameter at \(P\) satisfies the equation \(2 \sin 2 t - 4 \cos 2 t = 1\).
2. By first expressing \(2 \sin 2 t - 4 \cos 2 t\) in the form \(R \sin ( 2 t - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the coordinates of \(P\). Give each coordinate correct to 3 significant figures.
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