CAIE P3 (Pure Mathematics 3) 2024 March

1 Find the quotient and remainder when \(x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 12 x + 27\) is divided by \(x ^ { 2 } + 5\).

2

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 x - 5 ) \sqrt { 4 - x }\).
2. State the set of values of \(x\) for which the expansion in part (a) is valid.

3 It is given that \(z = - \sqrt { 3 } + \mathrm { i }\).

1. Express \(z ^ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. The complex number \(\omega\) is such that \(z ^ { 2 } \omega\) is real and \(\left| \frac { z ^ { 2 } } { \omega } \right| = 12\). Find the two possible values of \(\omega\), giving your answers in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\) and \(- \pi < \alpha \leqslant \pi\).

4 The positive numbers \(p\) and \(q\) are such that $$\ln \left( \frac { p } { q } \right) = a \text { and } \ln \left( q ^ { 2 } p \right) = b .$$ Express \(\ln \left( p ^ { 7 } q \right)\) in terms of \(a\) and \(b\).

5

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 2 i | \leqslant 3\) and \(| z | \geqslant | 10 - z |\).
2. Find the greatest value of \(\arg z\) for points in this region.

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

1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 2 \mathrm { x } - 3 \mathrm { y } - 1 } { 4 \mathrm { y } + 3 \mathrm { x } }\).
2. Hence show that the curve does not have a tangent that is parallel to the \(x\)-axis.

7
\includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-10_620_517_260_774} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { 2 \mathrm { x } } - 5 \mathrm { x }\) and its minimum point \(M\), where \(x = \alpha\).

1. Show that \(\alpha\) satisfies the equation \(\alpha = \frac { 1 } { 2 } \ln \left( \frac { 5 } { 1 + 2 \alpha } \right)\).
2. Verify by calculation that \(\alpha\) lies between 0.4 and 0.5.
3. Use an iterative formula based on the equation in part (a) to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8

1. Express \(3 \sin x + 2 \sqrt { 2 } \cos \left( x + \frac { 1 } { 4 } \pi \right)\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
2. Hence solve the equation $$6 \sin \frac { 1 } { 2 } \theta + 4 \sqrt { 2 } \cos \left( \frac { 1 } { 2 } \theta + \frac { 1 } { 4 } \pi \right) = 3$$ for \(- 4 \pi < \theta < 4 \pi\).

9 Relative to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { \mathrm { OA } } = 5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { \mathrm { OB } } = 8 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } \quad \text { and } \quad \overrightarrow { \mathrm { OC } } = 3 \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$

1. Show that \(O A B C\) is a rectangle.
   \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_67_1573_557_324}
   \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_68_1575_648_322}
   \includegraphics[max width=\textwidth, alt={}]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_70_1573_737_324} ....................................................................................................................................... .........................................................................................................................................
2. Use a scalar product to find the acute angle between the diagonals of \(O A B C\).

10 Let \(f ( x ) = \frac { 36 a ^ { 2 } } { ( 2 a + x ) ( 2 a - x ) ( 5 a - 2 x ) }\), where \(a\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find the exact value of \(\int _ { - a } ^ { a } f ( x ) d x\), giving your answer in the form plnq+rlns where \(p\) and \(r\) are integers and \(q\) and \(s\) are prime numbers.

11 The variables \(y\) and \(\theta\) satisfy the differential equation $$( 1 + y ) ( 1 + \cos 2 \theta ) \frac { d y } { d \theta } = e ^ { 3 y }$$ It is given that \(y = 0\) when \(\theta = \frac { 1 } { 4 } \pi\).
Solve the differential equation and find the exact value of \(\tan \theta\) when \(y = 1\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.