Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2012 Specimen

1

1. Express each of the following as a single logarithm.
   1. \(\log _ { a } 5 + \log _ { a } 3\)
   2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.

2 The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).

3 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - x y + y ^ { 2 } = 7 .$$

4 Find

1. \(\quad \int ( 2 x + 3 ) ^ { 4 } \mathrm {~d} x\)
2. \(\quad \int \left( 1 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\)

5 When \(x ^ { 4 } - 4 x ^ { 3 } + 5 x ^ { 2 } + x + a\) is divided by \(x ^ { 2 } - x + 1\), the quotient is \(x ^ { 2 } + b x + 1\) and the remainder is \(c x - 3\). Find the values of the constants \(a , b\) and \(c\).

6 The complex number \(5 - 3 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(\quad 6 z - z ^ { * }\),
2. \(\quad ( z - \mathrm { i } ) ^ { 2 }\),
3. \(\frac { 5 } { z }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-3_456_606_182_735} The diagram shows the region \(R\) bounded by the curve \(y = \frac { 1 } { \sqrt { 5 x + 3 } }\) and the lines \(x = 0\), \(x = 3\) and \(y = 0\). Find the exact volume of the solid formed when the region \(R\) is rotated completely about the \(x\)-axis, simplifying your answer.

8

1. Express \(\frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\) where \(A\) and \(B\) are constants.
2. Hence find the exact value of \(\int _ { 6 } ^ { 10 } \frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.

9 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 2 t$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the exact value of \(t\) at the point on the curve where the gradient is 2 .

10 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } = ( 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + s ( 6 \mathbf { i } + 9 \mathbf { j } - 3 \mathbf { k } ) , \\ & L _ { 2 } = ( 2 \mathbf { i } + 3 \mathbf { j } ) + t ( - 3 \mathbf { i } - 8 \mathbf { j } + 6 \mathbf { k } ) , \\ & L _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) + u ( - 2 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$ In each of the following cases, find the value of \(c\).

1. \(\quad L _ { 1 }\) and \(L _ { 3 }\) are parallel.
2. \(\quad L _ { 2 }\) and \(L _ { 3 }\) intersect.

11 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\). \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-4_631_901_532_571} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed.
2. In the first model the equation is $$y = \mathrm { e } ^ { - x } \cos 12 x$$ Show that this model has a maximum point close to \(A\) and a minimum point close to \(B\), and state the coordinates of these maximum and minimum points and also the \(y\) value when \(x = 0.3\).
3. In an alternative model the equation is $$y = f \cos ( \lambda x ) + g$$ where the constants \(f , \lambda\) and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). Find suitable values for \(f , \lambda\) and \(g\).
4. Using the alternative model, state the value of \(y\) when \(x = 0.3\) and hence comment on how accurate each model is in fitting the three given points.