1.02l Modulus function: notation, relations, equations and inequalities

3. a. "If \(p\) and \(q\) are irrational numbers, where \(p \neq q , q \neq 0\), then \(\frac { p } { q }\) is also irrational." Disprove this statement by means of a counter example.
b. (i) Sketch the graph of \(y = | x | - 2\).
(ii) Explain why \(| x - 2 | \geq | x | - 2\) for all real values of \(x\).

1. Given that \(a\) is a positive constant,
   a. Sketch the graph with equation

$$y = | a - 2 x |$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and \(y\)-axis.
b. Solve the inequality \(| a - 2 x | > x + 2 a\)

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation $$y = 3 | x - 2 | + 5$$ The vertex of the graph is at the point \(P\), shown in Figure 1.

1. Find the coordinates of \(P\).
2. Solve the equation $$16 - 4 x = 3 | x - 2 | + 5$$ A line \(l\) has equation \(y = k x + 4\) where \(k\) is a constant.
   Given that \(l\) intersects \(y = 3 | x - 2 | + 5\) at 2 distinct points,
3. find the range of values of \(k\).

1. (a) "If \(m\) and \(n\) are irrational numbers, where \(m \neq n\), then \(m n\) is also irrational."

Disprove this statement by means of a counter example.
(b) (i) Sketch the graph of \(y = | x | + 3\) (ii) Explain why \(| x | + 3 \geqslant | x + 3 |\) for all real values of \(x\).

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 3 - 2 x |\) Solve $$| 3 - 2 x | = 7 + x$$

9. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 4 shows a sketch of a Ferris wheel.
The height above the ground, \(H \mathrm {~m}\), of a passenger on the Ferris wheel, \(t\) seconds after the wheel starts turning, is modelled by the equation $$H = \left| A \sin ( b t + \alpha ) ^ { \circ } \right|$$ where \(A\), \(b\) and \(\alpha\) are constants.
Figure 5 shows a sketch of the graph of \(H\) against \(t\), for one revolution of the wheel.
Given that

• the maximum height of the passenger above the ground is 50 m
• the passenger is 1 m above the ground when the wheel starts turning
• the wheel takes 720 seconds to complete one revolution
  1. find a complete equation for the model, giving the exact value of \(A\), the exact value of \(b\) and the value of \(\alpha\) to 3 significant figures.
  2. Explain why an equation of the form

$$H = \left| A \sin ( b t + \alpha ) ^ { \circ } \right| + d$$ where \(d\) is a positive constant, would be a more appropriate model.

12. \begin{figure}[h] \end{figure} The number of subscribers to two different music streaming companies is being monitored. The number of subscribers, \(N _ { \mathrm { A } }\), in thousands, to company \(\mathbf { A }\) is modelled by the equation $$N _ { \mathrm { A } } = | t - 3 | + 4 \quad t \geqslant 0$$ where \(t\) is the time in years since monitoring began.
The number of subscribers, \(N _ { \mathrm { B } }\), in thousands, to company B is modelled by the equation $$N _ { \mathrm { B } } = 8 - | 2 t - 6 | \quad t \geqslant 0$$ where \(t\) is the time in years since monitoring began.
Figure 2 shows a sketch of the graph of \(N _ { \mathrm { A } }\) and the graph of \(N _ { \mathrm { B } }\) over a 5-year period.
Use the equations of the models to answer parts (a), (b), (c) and (d).

1. Find the initial difference between the number of subscribers to company \(\mathbf { A }\) and the number of subscribers to company B. When \(t = T\) company A reduced its subscription prices and the number of subscribers increased.
2. Suggest a value for \(T\), giving a reason for your answer.
3. Find the range of values of \(t\) for which \(N _ { \mathrm { A } } > N _ { \mathrm { B } }\) giving your answer in set notation.
4. State a limitation of the model used for company B.

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point \(P\), shown in Figure 2.

1. Find the coordinates of \(P\).
2. Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line \(l\) has equation \(y = a x\), where \(a\) is a constant.
   Given that \(l\) intersects \(y = 2 | x + 4 | - 5\) at least once,
3. find the range of possible values of \(a\), writing your answer in set notation.

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = k - | 2 x - 3 k |$$ stating
   $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 f \left( \frac { 1 } { 2 } x \right)$$

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$

1. State the range of f
2. Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots, (c) state the set of possible values for \(k\).

1. (a) Sketch the graph with equation

$$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
(b) Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$ (c) Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.

3 Solve the inequality \(| 2 x - 1 | \geq 4\).

2 Solve the inequality \(| 2 x + 1 | < 5\).

2 The straight line \(y = 5 - 2 x\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-04_705_773_881_239}

1. On the copy of the diagram in the Printed Answer Booklet, sketch the graph of \(y = | 5 - 2 x |\).
2. Solve the inequality \(| 5 - 2 x | < 3\).

2 The graph of \(y = | 1 - x | - 2\) is shown in Fig. 2. \begin{figure}[h] \end{figure} Determine the set of values of \(x\) for which \(| 1 - x | > 2\).

7

1. Sketch the graph of \(y = | 2 x |\).
2. On a separate diagram, sketch the graph of \(y = 4 - | 2 x |\), indicating the coordinates of the points where the graph crosses the coordinate axes.
3. Solve \(4 - | 2 x | = x\).
4. Hence, or otherwise, solve the inequality \(4 - | 2 x | > x\).

7

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 4 x ^ { 2 } - 5\).
2. Sketch the graph of \(y = \left| 4 x ^ { 2 } - 5 \right|\), indicating the coordinates of the point where the curve crosses the \(y\)-axis.
3. 1. Solve the equation \(\left| 4 x ^ { 2 } - 5 \right| = 4\).
   2. Hence, or otherwise, solve the inequality \(\left| 4 x ^ { 2 } - 5 \right| \geqslant 4\).

4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 \cos \frac { 1 } { 2 } x , & \text { for } 0 \leqslant x \leqslant 2 \pi \\ \mathrm {~g} ( x ) = | x | , & \text { for all real values of } x \end{array}$$

1. Find the range of f .
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 1\), giving your answer in an exact form.
3. 1. Write down an expression for \(\mathrm { gf } ( x )\).
   2. Sketch the graph of \(y = \operatorname { gf } ( x )\) for \(0 \leqslant x \leqslant 2 \pi\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos \frac { 1 } { 2 } x\).

5

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x - \mathrm { e } )\).
2. Sketch, on the axes given below, the graph of \(y = | 4 \ln ( x - \mathrm { e } ) |\), indicating the exact value of the \(x\)-coordinate where the curve meets the \(x\)-axis.
3. 1. Solve the equation \(| 4 \ln ( x - e ) | = 4\).
   2. Hence, or otherwise, solve the inequality \(| 4 \ln ( x - e ) | \geqslant 4\). \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-3_655_1428_2023_315}

4 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-5_629_1113_370_461}

1. On the axes below, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x - 1 )\).

5 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 4 } { 3 } , \text { for real values of } x , \text { where } \boldsymbol { x } \leqslant \mathbf { 0 }$$

1. State the range of f.
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Write down the domain of \(\mathrm { f } ^ { - 1 }\).
   2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. The function g is defined by $$\mathrm { g } ( x ) = \ln | 3 x - 1 | , \quad \text { for real values of } x , \text { where } x \neq \frac { 1 } { 3 }$$ The curve with equation \(y = \mathrm { g } ( x )\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-6_469_819_1254_612}
   1. The curve \(y = \mathrm { g } ( x )\) intersects the \(x\)-axis at the origin and at the point \(P\). Find the \(x\)-coordinate of \(P\).
   2. State whether the function \(g\) has an inverse. Give a reason for your answer.
   3. Show that \(\operatorname { gf } ( x ) = \ln \left| x ^ { 2 } - k \right|\), stating the value of the constant \(k\).
   4. Solve the equation \(\mathrm { gf } ( x ) = 0\).

6

1. 1. Sketch the graph of \(y = 4 - x ^ { 2 }\), indicating the coordinates of the points where the graph crosses the coordinate axes.
   2. The region between the graph and the \(x\)-axis from \(x = 0\) to \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume of the solid generated.
2. 1. Sketch the graph of \(y = \left| 4 - x ^ { 2 } \right|\).
   2. Solve \(\left| 4 - x ^ { 2 } \right| = 3\).
   3. Hence, or otherwise, solve the inequality \(\left| 4 - x ^ { 2 } \right| < 3\).

4

1. Sketch and label on the same set of axes the graphs of:
   1. \(y = | x |\);
   2. \(y = | 2 x - 4 |\).
2. 1. Solve the equation \(| x | = | 2 x - 4 |\).
   2. Hence, or otherwise, solve the inequality \(| x | > | 2 x - 4 |\).

4

1. Sketch the graph of \(y = \left| 50 - x ^ { 2 } \right|\), indicating the coordinates of the point where the graph crosses the \(y\)-axis.
2. Solve the equation \(\left| 50 - x ^ { 2 } \right| = 14\).
3. Hence, or otherwise, solve the inequality \(\left| 50 - x ^ { 2 } \right| > 14\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 50 - x ^ { 2 }\).