CAIE P2 (Pure Mathematics 2) 2023 November

1 When the polynomial $$a x ^ { 3 } + 4 a x ^ { 2 } - 7 x - 5$$ is divided by \(( x + 2 )\), the remainder is 33 .
Find the value of the constant \(a\).

2 Solve the equation \(\sec \theta \cos \left( \theta - 60 ^ { \circ } \right) = 4\) for \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-04_652_392_274_872} The diagram shows the curve with equation \(y = 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }\). The points on the curve with \(x\)-coordinates 0 and 2 are denoted by \(A\) and \(B\) respectively. The shaded region is enclosed by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.

1. Find the exact gradient of the curve at \(B\).
2. Find the exact area of the shaded region.

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 - x |\) and \(y = 9 - 2 x\).
2. Solve the inequality \(| 3 - x | > 9 - 2 x\).
3. Use logarithms to solve the inequality \(2 ^ { 3 x - 10 } < 500\). Give your answer in the form \(x < a\), where the value of \(a\) is given correct to 3 significant figures.
4. List the integers that satisfy both of the inequalities \(| 3 - x | > 9 - 2 x\) and \(2 ^ { 3 x - 10 } < 500\).

5

1. Find the quotient when \(6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4\) is divided by ( \(2 x + 1\) ), and show that the remainder is 6 .
2. Hence find $$\int _ { 2 } ^ { 7 } \frac { 6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4 } { 2 x + 1 } d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are integers.

6
\includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-10_803_394_269_863} The diagram shows the curve with parametric equations $$x = 3 \ln ( 2 t - 3 ) , \quad y = 4 t \ln t$$ The curve crosses the \(y\)-axis at the point \(A\). At the point \(B\), the gradient of the curve is 12 .

1. Find the exact gradient of the curve at \(A\).
2. Show that the value of the parameter \(t\) at \(B\) satisfies the equation $$t = \frac { 9 } { 1 + \ln t } + \frac { 3 } { 2 }$$
3. Use an iterative formula, based on the equation in (b), to find the value of \(t\) at \(B\), giving your answer correct to 3 significant figures. Use an initial value of 5 and give the result of each iteration to 5 significant figures.

7

1. Prove that \(\sin 2 x ( \cot x + 3 \tan x ) \equiv 4 - 2 \cos 2 x\).
2. Hence find the exact value of \(\cot \frac { 1 } { 12 } \pi + 3 \tan \frac { 1 } { 12 } \pi\).
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-13_796_789_278_708} The diagram shows the curve with equation \(y = 4 - 2 \cos 2 x\), for \(0 < x < 2 \pi\). At the point \(A\), the gradient of the curve is 4 . The point \(B\) is a minimum point. The \(x\)-coordinates of \(A\) and \(B\) are \(a\) and \(b\) respectively. Show that \(\int _ { a } ^ { b } ( 4 - 2 \cos 2 x ) \mathrm { d } x = 3 \pi + 1\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.