Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2014 June

1

1. Express \(x ^ { 2 } - 8 x + 10\) in the form \(( x - a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found.
2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 10\) and the corresponding value of \(x\).

2 Sketch the curve with equation \(y = \tan x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
On the same diagram, sketch the curve with equation \(y = \tan ^ { - 1 } x\) for all \(x\).
State the geometrical relationship between the curves.

3 Solve the inequality \(| 2 x - 1 | < 3\).

4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-2_949_1127_1041_507} Draw the graphs of

1. \(\mathrm { f } ( x + 2 ) + 1\),
2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).

5 A root of the equation \(z ^ { 2 } + p z + q = 0\) is \(3 + \mathrm { i }\), where \(p\) and \(q\) are real. Write down the other root of the equation and hence calculate the values of \(p\) and \(q\).

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-3_648_679_342_733}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

7 Taking \(x = 2\) as a first approximation, use the Newton-Raphson process to find a root of the equation \(\frac { 1 } { x ^ { 2 } } - 0.119 - 0.018 x = 0\). Give your answer correct to 3 significant figures.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Show that \(t = - \ln 2\) at the point on the curve where the gradient is 3 .

9 It is given that \(x , 6\) and \(x + 5\) are consecutive terms of a geometric progression.

1. Show that \(x ^ { 2 } + 5 x - 36 = 0\) and find the possible values of \(x\).
2. Hence find the possible values of the common ratio. Furthermore, \(x , 6\) and \(x + 5\) are the second, third and fourth terms of a geometric progression for which the sum to infinity exists.
3. Find the first term and the sum to infinity.

10

1. Show that \(\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln \left( \frac { 3 } { \sqrt { 5 } } \right)\).
2. Find \(\int x \sqrt { x - 2 } \mathrm {~d} x\).

11 A differential equation is given by \(2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ( 1 - y )\).

1. Express \(\frac { 2 } { y ( 1 - y ) }\) in partial fractions.
2. Hence show by integration that \(\frac { y ^ { 2 } } { ( 1 - y ) ^ { 2 } } = A \mathrm { e } ^ { x }\).
3. Given that \(x = 0\) when \(y = 2\), find the value of \(A\) and express \(y\) in terms of \(x\).

12

1. Use the identity \(\tan 2 x \equiv \frac { 2 \tan x } { 1 - \tan ^ { 2 } x }\) to show that \(\tan 4 x \equiv \frac { 4 \left( 1 - \tan ^ { 2 } x \right) \tan x } { 1 - 6 \tan ^ { 2 } x + \tan ^ { 4 } x }\).
2. Hence, given that \(x = \frac { 1 } { 16 } \pi\) is a root of the equation \(\tan ^ { 4 } x + p \tan ^ { 3 } x - 6 \tan ^ { 2 } x - p \tan x + 1 = 0\) where \(p\) is a positive constant, find the value of \(p\).