Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2016 June

1 Find the equation of the line perpendicular to the line \(y = 5 x + 6\) which passes through the point \(( 1,11 )\). Give your answer in the form \(y = m x + c\).

2 Without using a calculator, simplify the following, giving each answer in the form \(a \sqrt { 5 }\) where \(a\) is an integer. Show all your working.

1. \(4 \sqrt { 10 } \times \sqrt { 2 }\)
2. \(\sqrt { 500 } + \sqrt { 125 }\)

3 Solve \(3 x ^ { 2 } + 11 x - 20 > 0\).

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by \(u _ { n } = 3 n + 5\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(n\) such that \(u _ { n } = 254\).
3. Evaluate \(\sum _ { n = 1 } ^ { 500 } u _ { n }\).

5 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 5 . Find the coordinates of the centre of the circle and the value of \(k\).

6

1. Find the coordinates of the stationary points of the curve with equation $$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$ and determine their nature.
2. Sketch the graph of \(y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }\) and hence state the set of values of \(k\) for which the equation \(3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k\) has exactly four distinct real roots.

7 The functions f and g are defined for all real numbers by $$\mathrm { f } ( x ) = x ^ { 2 } + 2 \quad \text { and } \quad \mathrm { g } ( x ) = 4 x + 3$$

1. State the range of each of the functions f and g .
2. Find the values of \(x\) for which \(\mathrm { fg } ( x ) = \mathrm { gf } ( x )\).
3. The function h , given by \(\mathrm { h } ( x ) = x ^ { 2 } + 2\), has the same range as f but is such that \(\mathrm { h } ^ { - 1 } ( x )\) exists. State a possible domain for h and find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

8

1. Evaluate exactly \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x } \mathrm {~d} x\).
2. Find \(\int \frac { x - 1 } { x + 1 } \mathrm {~d} x\).

9 Determine whether the lines whose equations are $$\mathbf { r } = ( 4 + 2 \mu ) \mathbf { i } + ( 7 + 3 \mu ) \mathbf { j } + ( 3 + 7 \mu ) \mathbf { k } \quad \text { and } \quad \mathbf { r } = ( 35 - 5 \lambda ) \mathbf { i } + ( 6 + 2 \lambda ) \mathbf { j } + ( 14 + 3 \lambda ) \mathbf { k }$$ intersect, are parallel or are skew.

10 The diagram shows the curve with equation $$x = ( y - 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the \(y\)-axis at \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{afc8561d-94ae-42c0-bc6c-e9b091938368-3_588_780_1087_680}

1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Find the exact gradient of the curve at each of the points \(A\) and \(B\).

11

1. Prove that $$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
2. Hence solve the equation $$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$

12 A patch of disease on a leaf is being chemically treated. At time \(t\) days after treatment starts, its length is \(x \mathrm {~cm}\) and the rate of decrease of its length is observed to be inversely proportional to the square root of its length. At time \(t = 3 , x = 4\) and, at this instant, the length is decreasing at 0.05 cm per day. Write down and solve a differential equation to model this situation. Hence find the time it takes for the length to decrease to 0.01 cm .
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