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

1 Solve the simultaneous equations $$\begin{aligned} x ^ { 2 } + y ^ { 2 } & = 5 \\ y & = 2 x \end{aligned}$$

2 Find the equation of the line perpendicular to the line \(y = 5 x\) which passes through the point \(( 2,11 )\). Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers to be found.

3 The first term of a geometric progression is 50 and the common ratio is 0.9 .

1. Find the fifth term.
2. Find the sum of the first thirty terms.
3. Find the sum to infinity.

4 Solve the equation \(x ^ { 2 } + ( \sqrt { 3 } ) x - 18 = 0\), giving each root in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers.

5 Express \(\frac { 7 - x } { ( x - 1 ) ( x + 2 ) }\) in partial fractions.

6 Describe fully the transformations which, when applied to the graph of \(y = \mathrm { f } ( x )\), will produce the graphs with equations given by

1. \(y = \mathrm { f } ( - x )\),
2. \(y = \mathrm { f } ( x - 3 )\),
3. \(y = \mathrm { f } \left( \frac { x } { 2 } \right)\).

7 Given that \(z\) is a complex number, prove that \(z z ^ { * } = | z | ^ { 2 }\).

8

1. Express \(\sin x - \sqrt { 8 } \cos x\) in the form \(R \sin ( x - \alpha )\) where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\).
2. Hence write down the maximum value of \(\sin x - \sqrt { 8 } \cos x\) and find the smallest positive value of \(x\) for which it occurs.

9 Find \(\int x \sin 2 x \mathrm {~d} x\).

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

11 The functions f and g are defined by \(\mathrm { f } ( x ) = \frac { 1 } { 2 + x } + 5 , x > - 2\) and \(\mathrm { g } ( x ) = | x | , x \in \mathbb { R }\).

1. Given that the range of f is of the form \(\mathrm { f } ( x ) > a\), find \(a\).
2. Find an expression for \(\mathrm { f } ^ { - 1 }\), stating its domain and range.
3. Show that \(\mathrm { gf } ( x ) = \mathrm { f } ( x )\).
4. Find an expression for \(\mathrm { fg } ( x )\). Determine whether fg has an inverse.

12 The diagram shows the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) for \(x > - 1\). \includegraphics[max width=\textwidth, alt={}, center]{806dc286-416e-4785-8d13-0d524f808cb0-3_435_874_897_639}

1. Find the coordinates of the points where the curve crosses the axes.
2. Express \(\frac { x ^ { 2 } - 3 } { x + 1 }\) in the form \(A x + B + \frac { C } { x + 1 }\), where \(A , B\) and \(C\) are constants, and hence show that the exact area enclosed by the \(x\)-axis, the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) and the lines \(x = 2\) and \(x = 4\) is \(4 + \ln \frac { 9 } { 25 }\).

13 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - k ( y - 10 )\), where \(k\) is a constant, given that \(y = 70\) when \(x = 0\) and \(y = 40\) when \(x = 1\). Express your answer in the form \(y = a + b \left( \frac { 1 } { 2 } \right) ^ { x }\) where \(a\) and \(b\) are constants to be found.
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