CAIE P2 (Pure Mathematics 2) 2023 March

1 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).

2 Solve the equation \(\tan \left( \theta - 60 ^ { \circ } \right) = 3 \cot \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that the remainder is 35 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) and show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

4

1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 11 |\) and \(y = 3 x - 3\).
2. Solve the inequality \(| 2 x - 11 | < 3 x - 3\).
3. Find the smallest integer \(N\) satisfying the inequality \(| 2 \ln N - 11 | < 3 \ln N - 3\).

5 It is given that \(\int _ { 1 } ^ { a } \left( \frac { 4 } { 1 + 2 x } + \frac { 3 } { x } \right) \mathrm { d } x = \ln 10\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt [ 3 ] { 90 ( 1 + 2 a ) ^ { - 2 } }\).
2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-10_798_495_269_810} The diagram shows the curve with equation \(y = \frac { 4 \mathrm { e } ^ { 2 x } + 9 } { \mathrm { e } ^ { x } + 2 }\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).
3. Find the exact value of the gradient of the curve at \(P\).
4. Find the exact coordinates of \(M\).

7
\includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-12_446_613_274_760} The diagram shows the curve with parametric equations $$x = k \tan t , \quad y = 3 \sin 2 t - 4 \sin t ,$$ for \(0 < t < \frac { 1 } { 2 } \pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).

1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction.
2. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(k\) and \(\cos t\).
3. Given that the normal to the curve at \(P\) has gradient \(\frac { 9 } { 10 }\), find the value of \(k\), giving your answer as an exact fraction.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.