CAIE P3 (Pure Mathematics 3) 2021 June

1 Expand \(( 1 + 3 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.

2 Solve the equation \(4 ^ { x } = 3 + 4 ^ { - x }\). Give your answer correct to 3 decimal places.

3 The parametric equations of a curve are $$x = t + \ln ( t + 2 ) , \quad y = ( t - 1 ) \mathrm { e } ^ { - 2 t }$$ where \(t > - 2\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
2. Find the exact \(y\)-coordinate of the stationary point of the curve.

4 Let \(f ( x ) = \frac { 15 - 6 x } { ( 1 + 2 x ) ( 4 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(\ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

5

1. By first expanding \(\tan ( 2 \theta + 2 \theta )\), show that the equation \(\tan 4 \theta = \frac { 1 } { 2 } \tan \theta\) may be expressed as \(\tan ^ { 4 } \theta + 2 \tan ^ { 2 } \theta - 7 = 0\).
2. Hence solve the equation \(\tan 4 \theta = \frac { 1 } { 2 } \tan \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6

1. By sketching a suitable pair of graphs, show that the equation \(\cot \frac { 1 } { 2 } x = 1 + \mathrm { e } ^ { - x }\) has exactly one root in the interval \(0 < x \leqslant \pi\).
2. Verify by calculation that this root lies between 1 and 1.5.
3. Use the iterative formula \(x _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 1 } { 1 + \mathrm { e } ^ { - x _ { n } } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7
\includegraphics[max width=\textwidth, alt={}, center]{1990cbac-d96f-4484-be4b-67dab35b3147-10_647_519_260_813} For the curve shown in the diagram, the normal to the curve at the point \(P\) with coordinates \(( x , y )\) meets the \(x\)-axis at \(N\). The point \(M\) is the foot of the perpendicular from \(P\) to the \(x\)-axis. The curve is such that for all values of \(x\) in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\), the area of triangle \(P M N\) is equal to \(\tan x\).

1. 1. Show that \(\frac { M N } { y } = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that \(x\) and \(y\) satisfy the differential equation \(\frac { 1 } { 2 } y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \tan x\).
2. Given that \(y = 1\) when \(x = 0\), solve this differential equation to find the equation of the curve, expressing \(y\) in terms of \(x\).

8
\includegraphics[max width=\textwidth, alt={}, center]{1990cbac-d96f-4484-be4b-67dab35b3147-12_458_725_262_708} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 4 } }\) and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. By using integration by parts, show that for all \(a > 1 , \int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 4 } } \mathrm {~d} x < \frac { 1 } { 9 }\).

9 The quadrilateral \(A B C D\) is a trapezium in which \(A B\) and \(D C\) are parallel. With respect to the origin \(O\), the position vectors of \(A , B\) and \(C\) are given by \(\overrightarrow { O A } = - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \overrightarrow { O B } = \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\).

1. Given that \(\overrightarrow { D C } = 3 \overrightarrow { A B }\), find the position vector of \(D\).
2. State a vector equation for the line through \(A\) and \(B\).
3. Find the distance between the parallel sides and hence find the area of the trapezium.

10

1. Verify that \(- 1 + \sqrt { 2 } \mathrm { i }\) is a root of the equation \(z ^ { 4 } + 3 z ^ { 2 } + 2 z + 12 = 0\).
2. Find the other roots of this equation.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.