CAIE P3 (Pure Mathematics 3) 2023 June

1 Solve the equation $$3 \mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { - 2 x } = 5$$ Give the answer correct to 3 decimal places.

2

1. Sketch the graph of \(y = | 2 x + 3 |\).
2. Solve the inequality \(3 x + 8 > | 2 x + 3 |\).

3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 3 + x ) \sqrt { 1 + 4 x }\).

4

1. Show that the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) can be expressed in the form $$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
2. Hence solve the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5 The equation of a curve is \(x ^ { 2 } y - a y ^ { 2 } = 4 a ^ { 3 }\), where \(a\) is a non-zero constant.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x y } { 2 a y - x ^ { 2 } }\).
2. Hence find the coordinates of the points where the tangent to the curve is parallel to the \(y\)-axis. [4]

6 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ - 2 \\ - 4 \end{array} \right) .$$ The quadrilateral \(A B C D\) is a parallelogram.

1. Find the position vector of \(D\).
2. The angle between \(B A\) and \(B C\) is \(\theta\). Find the exact value of \(\cos \theta\).
3. Hence find the area of \(A B C D\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\cos 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 4 \tan 2 x } { \sin ^ { 2 } 3 y }$$ where \(0 \leqslant x < \frac { 1 } { 4 } \pi\). It is given that \(y = 0\) when \(x = \frac { 1 } { 6 } \pi\).
Solve the differential equation to obtain the value of \(x\) when \(y = \frac { 1 } { 6 } \pi\). Give your answer correct to 3 decimal places.

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

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find the exact value of \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.

9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = \frac { 1 } { 8 }\).

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

10 The polynomial \(x ^ { 3 } + 5 x ^ { 2 } + 31 x + 75\) is denoted by \(\mathrm { p } ( x )\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).
2. Show that \(z = - 1 + 2 \sqrt { 6 } \mathrm { i }\) is a root of \(\mathrm { p } ( z ) = 0\).
3. Hence find the complex numbers \(z\) which are roots of \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).
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