CAIE P2 (Pure Mathematics 2) 2019 November

1 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - 6 x + 11$$ Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(\left( x ^ { 2 } + 2 \right)\).

2

1. Solve the equation \(| 4 x + 5 | = | x - 7 |\).
2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 2 } + 5 \right| = \left| 2 ^ { y } - 7 \right|\), giving the answer correct to 3 significant figures.

3
\includegraphics[max width=\textwidth, alt={}, center]{9c26457d-4b65-4cd4-a9b9-128aba92dbf4-04_586_734_260_701} The variables \(x\) and \(y\) satisfy the equation \(y = k x ^ { a }\), where \(k\) and \(a\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(k\) and \(a\) correct to 3 significant figures.

4 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$ converges to the value \(\alpha\).

1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).

5 Find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } ( 2 x + 5 )\).

6

1. Show that \(\int _ { 2 } ^ { 18 } \frac { 3 } { 2 x } \mathrm {~d} x = \ln 27\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \left( \frac { 3 } { 2 } x \right) \mathrm { d } x\). Show all necessary working.

7 The parametric equations of a curve are $$x = 3 \sin 2 \theta , \quad y = 1 + 2 \tan 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).

1. Find the exact gradient of the curve at the point for which \(\theta = \frac { 1 } { 6 } \pi\).
2. Find the value of \(\theta\) at the point where the gradient of the curve is 2 , giving the value correct to 3 significant figures.

8

1. Express \(0.5 \cos \theta - 1.2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(0.5 \cos \theta - 1.2 \sin \theta = 0.8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
3. Determine the greatest and least possible values of \(( 3 - \cos \theta + 2.4 \sin \theta ) ^ { 2 }\) as \(\theta\) varies.
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