CAIE P2 (Pure Mathematics 2) 2021 June

1 Solve the inequality \(| 3 x - 7 | < | 4 x + 5 |\).

2 By first expanding \(\sin \left( \theta + 30 ^ { \circ } \right)\), solve the equation \(\sin \left( \theta + 30 ^ { \circ } \right) \operatorname { cosec } \theta = 2\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3

1. Show that \(( \sec x + \cos x ) ^ { 2 }\) can be expressed as \(\sec ^ { 2 } x + a + b \cos 2 x\), where \(a\) and \(b\) are constants to be determined.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x\).

4 A curve has parametric equations $$x = \ln ( 2 t + 6 ) - \ln t , \quad y = t \ln t$$

1. Find the value of \(t\) at the point \(P\) on the curve for which \(x = \ln 4\).
2. Find the exact gradient of the curve at \(P\).

5
\includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-08_575_618_262_762} The diagram shows the curve with equation \(y = \frac { 3 x + 2 } { \ln x }\). The curve has a minimum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { 3 x + 2 } { 3 \ln x }\). [3]
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 3 and 4.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.

6

1. Use the trapezium rule with three intervals to find an approximation to \(\int _ { 1 } ^ { 4 } \frac { 6 } { 1 + \sqrt { x } } \mathrm {~d} x\). Give your answer correct to 5 significant figures.
2. Find the exact value of \(\int _ { 1 } ^ { 4 } 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 } \mathrm {~d} x\).
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-11_556_805_262_705} The diagram shows the curves \(y = \frac { 6 } { 1 + \sqrt { x } }\) and \(y = 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 }\) which meet at a point with \(x\)-coordinate 4. The shaded region is bounded by the two curves and the line \(x = 1\). Use your answers to parts (a) and (b) to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
4. State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the exact area of the shaded region.

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$ where \(a\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
3. Hence find the exact values of \(y\) that satisfy the equation \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.