CAIE P3 (Pure Mathematics 3) 2020 March

1

1. Sketch the graph of \(y = | x - 2 |\).
2. Solve the inequality \(| x - 2 | < 3 x - 4\).

2 Solve the equation \(\ln 3 + \ln ( 2 x + 5 ) = 2 \ln ( x + 2 )\). Give your answer in a simplified exact form.

3

1. By sketching a suitable pair of graphs, show that the equation \(\sec x = 2 - \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.8 and 1 .
3. Use the iterative formula \(x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { 4 - x _ { n } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\). Give your answer in a simplified exact form.

5

1. Show that \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 2 \cot 2 x\).
2. Hence solve the equation \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 4\), for \(0 < x < \pi\).

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$ It is given that \(y = 0\) when \(x = 1\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) as \(x\) tends to infinity.

7 The equation of a curve is \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 5\).

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

8
\includegraphics[max width=\textwidth, alt={}, center]{8f81a526-783c-4321-b540-c9deccfee17b-12_639_713_262_715} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 3\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The point \(M\) on \(A B\) is such that \(M B = 2 A M\). The midpoint of \(F G\) is \(N\).

1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Find the position vector of \(P\), the foot of the perpendicular from \(D\) to the line through \(M\) and \(N\). [4]

9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

10

1. The complex numbers \(v\) and \(w\) satisfy the equations $$v + \mathrm { i } w = 5 \quad \text { and } \quad ( 1 + 2 \mathrm { i } ) v - w = 3 \mathrm { i } .$$ Solve the equations for \(v\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z - 2 - 3 \mathrm { i } | = 1\).
   2. Calculate the least value of \(\arg z\) for points on this locus.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.