Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2015 June

1 Find the set of values of \(x\) for which \(x ^ { 2 } - x - 12 < 0\).

2 Solve the following simultaneous equations. $$x ^ { 2 } + 2 y ^ { 2 } = 36 \quad x + 2 y = 10$$

3 \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_305_825_630_660} The diagram shows a triangle \(A B C\) in which angle \(B = 39 ^ { \circ }\), angle \(C = 28 ^ { \circ } , A B = x \mathrm {~cm}\) and \(A C = ( 2 x - 1 ) \mathrm { cm }\). Find the value of \(x\).

4 A population, \(P\), is modelled by the equation \(P = a \mathrm { e } ^ { b t }\) where \(t\) is time in years, and \(a\) and \(b\) are constants.

1. By considering logarithms, show that a graph of \(\ln P\) against \(t\) is a straight line. State the intercept on the vertical axis and the gradient.
2. Use the graph below to obtain values for \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_657_750_1530_740}

5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).

1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).

6 The functions f and g are given by \(\mathrm { f } ( x ) = \frac { 3 } { x - 1 }\) for all \(x \neq 1\) and \(\mathrm { g } ( x ) = x + 2\) for all real \(x\).

1. Find gf, stating its domain and range.
2. Find \(( \mathrm { gf } ) ^ { - 1 }\), stating any values of \(x\) for which \(( \mathrm { gf } ) ^ { - 1 }\) is not defined.

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following vector equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$

1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of their point of intersection.
2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

8 The complex numbers \(w\) and \(z\) are given by \(w = 3 - \mathrm { i }\) and \(z = 1 + \mathrm { i }\).

1. Express \(\frac { z } { w }\) in the form \(p + \mathrm { i } q\) where \(p\) and \(q\) are real numbers.
2. On the same Argand diagram, mark the points representing \(z , w\) and \(\frac { z } { w }\).
3. Find the value in radians of \(\arg w\).
4. Show that \(z + \frac { 2 } { z }\) is a real number.

9 A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x }\). Find the exact coordinates of the stationary points of the curve.

10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).

11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that $$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$ where \(A\) is an arbitrary constant and \(x > 0\).