Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2013 June

2 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).

3 A sector, \(P O Q\), of a circle centre \(O\) has radius 7 cm and angle 1.7 radians (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{41d9ff74-82de-4ac5-928f-f6ab008319d2-2_469_723_662_712}

1. Find the length of the line \(P Q\).
2. Hence find the perimeter of the shaded area.

4 Solve the equation \(2 ^ { 5 x } = 15\), giving the value of \(x\) correct to 3 significant figures.

5

1. Find \(\int \left( 3 x ^ { 2 } - 4 x + 8 \right) \mathrm { d } x\).
2. Hence find \(\int _ { 1 } ^ { 3 } \left( 3 x ^ { 2 } - 4 x + 8 \right) \mathrm { d } x\).

6

1. Sketch the graph of \(y = \cos 2 x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

7 The complex number \(z\) is given by \(- 20 + 21 \mathrm { i }\). Showing all your working,

1. find the value of \(| z |\),
2. calculate the value of \(\arg z\) correct to 3 significant figures,
3. express \(\frac { 1 } { z }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

8

1. Let \(\mathrm { f } ( x ) = x ^ { 3 } - x - 1\). Use a sign change method to show that the equation \(x ^ { 3 } - x - 1 = 0\) has a root between \(x = 1\) and \(x = 2\).
2. By taking \(x = 1\) as a first approximation to this root, use the Newton-Raphson formula to find this root correct to 3 decimal places.

9

1. Show that \(\sin \theta + \sqrt { 3 } \cos \theta\) can be expressed in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the values of \(R\) and \(\alpha\).
2. Hence find the value of \(\theta\), where \(0 < \theta < \pi\), such that \(\sin \theta + \sqrt { 3 } \cos \theta = 0.8\).

10 Two intersecting straight lines have equations $$\frac { x - 5 } { 4 } = \frac { y - 11 } { 3 } = \frac { z - 7 } { - 5 } \quad \text { and } \quad \frac { x - 9 } { - 2 } = \frac { y - 4 } { 1 } = \frac { z + 4 } { 4 } .$$ Find the coordinates of their point of intersection.

11 A curve has parametric equations given by $$x = 2 \sin \theta , \quad y = \cos 2 \theta$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin \theta\).
2. Hence find the equation of the tangent to the curve at \(\theta = \frac { 1 } { 2 } \pi\).
3. Find the cartesian equation of the curve.

12

1. Prove the identity \(\frac { 1 } { ( x + h ) ^ { 2 } } - \frac { 1 } { x ^ { 2 } } \equiv \frac { - 2 h x - h ^ { 2 } } { x ^ { 2 } ( x + h ) ^ { 2 } }\).
2. Given that \(\mathrm { f } ( x ) = x ^ { - 2 }\), use differentiation from first principles to find an expression for \(\mathrm { f } ^ { \prime } ( x )\).

13 By first factorising completely \(x ^ { 3 } + x ^ { 2 } - 5 x + 3\), find \(\int \frac { 2 x ^ { 2 } + x + 1 } { x ^ { 3 } + x ^ { 2 } - 5 x + 3 } \mathrm {~d} x\).