CAIE P3 (Pure Mathematics 3) 2014 November

1 Solve the inequality \(| 3 x - 1 | < | 2 x + 5 |\).

2 A curve is defined for \(0 < \theta < \frac { 1 } { 2 } \pi\) by the parametric equations $$x = \tan \theta , \quad y = 2 \cos ^ { 2 } \theta \sin \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 5 } \theta - 4 \cos ^ { 3 } \theta\).

3 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by \(\left( x ^ { 2 } + 1 \right)\).

4

1. Show that \(\cos \left( \theta - 60 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right) \equiv \cos \theta\).
2. Given that \(\frac { \cos \left( 2 x - 60 ^ { \circ } \right) + \cos \left( 2 x + 60 ^ { \circ } \right) } { \cos \left( x - 60 ^ { \circ } \right) + \cos \left( x + 60 ^ { \circ } \right) } = 3\), find the exact value of \(\cos x\).

5 The complex numbers \(w\) and \(z\) are defined by \(w = 5 + 3 \mathrm { i }\) and \(z = 4 + \mathrm { i }\).

1. Express \(\frac { \mathrm { i } w } { z }\) in the form \(x + \mathrm { i } y\), showing all your working and giving the exact values of \(x\) and \(y\).
2. Find \(w z\) and hence, by considering arguments, show that $$\tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right) = \frac { 1 } { 4 } \pi$$

6 It is given that \(I = \int _ { 0 } ^ { 0.3 } \left( 1 + 3 x ^ { 2 } \right) ^ { - 2 } \mathrm {~d} x\).

1. Use the trapezium rule with 3 intervals to find an approximation to \(I\), giving the answer correct to 3 decimal places.
2. For small values of \(x , \left( 1 + 3 x ^ { 2 } \right) ^ { - 2 } \approx 1 + a x ^ { 2 } + b x ^ { 4 }\). Find the values of the constants \(a\) and \(b\). Hence, by evaluating \(\int _ { 0 } ^ { 0.3 } \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \mathrm { d } x\), find a second approximation to \(I\), giving the answer correct to 3 decimal places.

7 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = a \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 3 a \mathbf { k } )$$ where \(a\) is a constant.

1. Show that the lines intersect for all values of \(a\).
2. Given that the point of intersection is at a distance of 9 units from the origin, find the possible values of \(a\).

8 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 5 } x y ^ { \frac { 1 } { 2 } } \sin \left( \frac { 1 } { 3 } x \right)$$

1. Find the general solution, giving \(y\) in terms of \(x\).
2. Given that \(y = 100\) when \(x = 0\), find the value of \(y\) when \(x = 25\).

9

1. Sketch the curve \(y = \ln ( x + 1 )\) and hence, by sketching a second curve, show that the equation $$x ^ { 3 } + \ln ( x + 1 ) = 40$$ has exactly one real root. State the equation of the second curve.
2. Verify by calculation that the root lies between 3 and 4 .
3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 40 - \ln \left( x _ { n } + 1 \right) \right)$$ with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
4. Deduce the root of the equation $$\left( \mathrm { e } ^ { y } - 1 \right) ^ { 3 } + y = 40$$ giving the answer correct to 2 decimal places.

10 By first using the substitution \(u = \mathrm { e } ^ { x }\), show that $$\int _ { 0 } ^ { \ln 4 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } + 3 \mathrm { e } ^ { x } + 2 } \mathrm {~d} x = \ln \left( \frac { 8 } { 5 } \right)$$