CAIE P3 (Pure Mathematics 3) 2022 June

1 Solve the equation \(\ln \left( \mathrm { e } ^ { 2 x } + 3 \right) = 2 x + \ln 3\), giving your answer correct to 3 decimal places.

2 Solve the equation \(3 \cos 2 \theta = 3 \cos \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3 The polynomial \(a x ^ { 3 } + x ^ { 2 } + b x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by ( \(2 x - 1\) ) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).

4 The equation of a curve is \(y = \cos ^ { 3 } x \sqrt { \sin x }\). It is given that the curve has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this stationary point, giving your answer correct to 3 significant figures.

5

1. By sketching a suitable pair of graphs, show that the equation \(\ln x = 3 x - x ^ { 2 }\) has one real root.
2. Verify by calculation that the root lies between 2 and 2.8.
3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { y - x } ,$$ and \(y = 0\) when \(x = 0\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. Find the value of \(y\) when \(x = 1\), giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.

7 The equation of a curve is \(x ^ { 3 } + 3 x ^ { 2 } y - y ^ { 3 } = 3\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + 2 x y } { y ^ { 2 } - x ^ { 2 } }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.

8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( 3 x - 1 ) \left( x ^ { 2 } + 3 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int _ { 1 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in a simplified exact form.

9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = - \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 5 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } + \mathbf { k } )$$ respectively, where \(a\) and \(b\) are constants.

1. Given that \(l\) and \(m\) intersect, show that \(2 b - a = 4\).
2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).

10 The complex number \(- 1 + \sqrt { 7 } \mathrm { i }\) is denoted by \(u\). It is given that \(u\) is a root of the equation $$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$ where \(k\) is a real constant.

1. Find the value of \(k\).
2. Find the other two roots of the equation.
3. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - u | = 2\).
4. Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.