1.02t Solve modulus equations: graphically with modulus function

1 The diagram below shows the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-02_579_1150_351_482}

1. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
   (3 marks)
2. Hence, or otherwise, solve the inequality $$| 2 x - 3 | \geqslant | x |$$ (2 marks)

1. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by

$$\begin{aligned} & d _ { A } = | 5 t - 31 | \\ & d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right| \end{aligned}$$ respectively, where \(t\) is the time in seconds after motion begins.

1. Use algebra to find the range of time for which particle \(A\) is further away from \(O\) than particle \(B\) is from \(O\). It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
2. Use this information to assess the validity of the model.

2

1. Given that \(| n | = 5\), find the greatest value of \(| 2 n - 3 |\), justifying your answer.
2. Solve the equation \(| 3 x - 6 | = | x - 6 |\).

3

1. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\). \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-4_423_682_1110_694} On the diagram in your Printed Answer Booklet, give the coordinates of the points of intersection of the graphs with the coordinate axes.
2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).

4

1. Sketch the graph of \(y = | 8 - 2 x |\).
2. Solve the equation \(| 8 - 2 x | = 4\).
3. Solve the inequality \(| 8 - 2 x | > 4\).

2

1. Sketch, on the axes below, the curve with equation \(y = 4 - | 2 x + 1 |\), indicating the coordinates where the curve crosses the axes.
2. Solve the equation \(x = 4 - | 2 x + 1 |\).
3. Solve the inequality \(x < 4 - | 2 x + 1 |\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = | 2 x + 1 |\) onto the graph of \(y = 4 - | 2 x + 1 |\).
   [0pt] [4 marks] \section*{Answer space for question 2}
   1. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-04_851_1459_1000_319}

9 A curve \(C\) has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the coordinates of the points where \(C\) intersects the \(x\)-axis. (2 marks)
   2. Write down the equations of all the asymptotes of \(C\).
2. 1. Show that, if the line \(y = k\) intersects \(C\), then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
   2. Given that there is only one stationary point on \(C\), find the coordinates of this stationary point.
      (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

4

1. On Figure 1 add a sketch of the graph of $$y = | 3 x - 6 |$$ 4
2. Find the coordinates of the points of intersection of the two graphs.
   Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-05_2488_1716_219_153}

10

1. Write down the equations of the asymptotes of \(C\) 10
2. The line \(L\) has equation $$y = - \frac { 2 } { 5 } x + 2$$ 10 (b) (i) Draw the line \(L\) on Figure 1 10 (b) (ii) Hence, or otherwise, solve the inequality $$\frac { 2 x - 10 } { 3 x - 5 } \leq - \frac { 2 } { 5 } x + 2$$

Solve the inequality \(|5x + 7| > |2x - 3|\). [4]

Solve the equation \(|x - 2| = |\frac{1}{3}x|\). [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = e^{-x} - 1\).

1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac{1}{2}|x - 1|\). Show the coordinates of the points where the graph meets the axes. [2]

The \(x\)-coordinate of the point of intersection of the graphs is \(\alpha\).

1. Show that \(x = \alpha\) is a root of the equation \(x + 2e^{-x} - 3 = 0\). [3]
2. Show that \(-1 < \alpha < 0\). [2]

The iterative formula \(x_{n+1} = -\ln[\frac{1}{2}(3 - x_n)]\) is used to solve the equation \(x + 2e^{-x} - 3 = 0\).

1. Starting with \(x_0 = -1\), find the values of \(x_1\) and \(x_2\). [2]
2. Show that, to 2 decimal places, \(\alpha = -0.58\). [2]

Find the set of values of \(x\) for which $$|x^2 - 4| > 3x.$$ [5]

Solve the inequality \(|2x - 3| < |x + 1|\). [5]

Solve the inequality \(|4x - 3| < |2x + 1|\). [5]

By sketching the graphs of \(y = |2x + 1|\) and \(y = -x\) on the same axes, show that the equation \(|2x + 1| = -x\) has two roots. Find these roots. [4]