CAIE P2 (Pure Mathematics 2) 2019 November

1

1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).

2 Find the exact value of \(\int _ { 1 } ^ { 2 } \left( 2 \mathrm { e } ^ { 2 x } - 1 \right) ^ { 2 } \mathrm {~d} x\). Show all necessary working.

3 A curve has equation \(y = \frac { 3 + 2 \ln x } { 1 + \ln x }\). Find the exact gradient of the curve at the point for which \(y = 4\).

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + a x ^ { 2 } - 15 x - 18$$ where \(a\) is a constant. It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
3. Hence solve the equation \(\mathrm { p } \left( \mathrm { e } ^ { \sqrt { } y } \right) = 0\), giving the answer correct to 2 significant figures.

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } + 4 \cos 2 x - \sin x \right) \mathrm { d } x = 2\), where \(a\) is a constant.

1. Show that \(a = \sqrt [ 3 ] { } ( 3 - 2 \sin 2 a - \cos a )\).
2. Using the equation in part (i), show by calculation that \(0.5 < a < 0.75\).
3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6

1. Showing all necessary working, solve the equation $$\sec \alpha \operatorname { cosec } \alpha = 7$$ for \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Showing all necessary working, solve the equation $$\sin \left( \beta + 20 ^ { \circ } \right) + \sin \left( \beta - 20 ^ { \circ } \right) = 6 \cos \beta$$ for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\).

7 The equation of a curve is \(x ^ { 2 } - 4 x y - 2 y ^ { 2 } = 1\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the gradient of the curve at the point \(( - 1,2 )\) is \(- \frac { 5 } { 2 }\). [5]
2. Show that the curve has no stationary points.
3. Find the \(x\)-coordinate of each of the points on the curve at which the tangent is parallel to the \(y\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.