Modulus function

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

1. In this question you must show detailed reasoning. Solve the inequality \(|x - 2| \leqslant |2x - 6|\). [4]
2. Give full details of a sequence of two transformations needed to transform the graph of \(y = |x - 2|\) to the graph of \(y = |2x - 6|\). [3]

1. Given that \(|t| = 3\), find the possible values of \(|2t - 1|\). [3]
2. Solve the inequality \(|x - t^2| > |x + 3\sqrt{2}|\). [4]

1 Solve the inequality \(| x + 1 | > | x |\).

1 Solve the inequality \(| x + 1 | > | x |\).

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

Solve the equation \(|3 - 2x| = 4|x|\). [4]

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

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

2 Given that \(\mathrm { f } ( x ) = | x |\) and \(\mathrm { g } ( x ) = x + 1\), sketch the graphs of the composite functions \(y = \mathrm { fg } ( x )\) and \(y = \operatorname { gf } ( x )\), indicating clearly which is which.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the origin \(O\) and the points \(A ( 5,4 )\) and \(B ( - 5 , - 4 )\).
In separate diagrams, sketch the graph with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( | x | )\),
3. \(y = 2 f ( x + 1 )\). On each sketch, show the coordinates of the points corresponding to \(A\) and \(B\).

3. \includegraphics[max width=\textwidth, alt={}, center]{14ec6709-e1cb-42d7-af99-91365e50e4fc-1_535_810_877_406} The diagram shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at \(( - 3,2 )\) and a minimum point at \(( 2 , - 4 )\).

1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 3 \mathrm { f } ( 2 x )\).
2. Write down the values of the constants \(a\) and \(b\) such that the curve with equation \(y = a + \mathrm { f } ( x + b )\) has a minimum point at the origin \(O\).

3 Solve $$x ^ { 2 } \geqslant | 5 x - 6 |$$ [5 marks]

3. Use algebra to obtain the set of values of \(x\) for which $$\left| x ^ { 2 } + x - 2 \right| < \frac { 1 } { 2 } ( x + 5 )$$

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

Solutions relying entirely on calculator technology are not acceptable.
Use algebra to determine the set of values of \(x\) for which $$\frac { x ^ { 2 } - 9 } { | x + 8 | } > 6 - 2 x$$

4

1. Solve the equation \(| 4 x - 1 | = | x - 3 |\).
2. Hence solve the equation \(\left| 4 ^ { y + 1 } - 1 \right| = \left| 4 ^ { y } - 3 \right|\) correct to 3 significant figures.

1

1. Solve the equation \(| 3 x - 2 | = 5\).
2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = 2 x + 7\).
2. Solve the equation \(| 3 x - 5 | = 2 x + 7\).
3. Hence solve the equation \(\left| 3 ^ { y + 1 } - 5 \right| = 2 \times 3 ^ { y } + 7\), giving your answer correct to 3 significant figures.

2. (a) Sketch, on the same axes,

1. \(y = | 2 x - 3 |\)
2. \(y = 4 - x ^ { 2 }\) (b) Find the set of values of \(x\) for which $$4 - x ^ { 2 } > | 2 x - 3 |$$

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph of \(C _ { 1 }\) with equation $$y = 5 - | 3 x - 22 |$$

1. Write down the coordinates of
   1. the vertex of \(C _ { 1 }\)
   2. the intersection of \(C _ { 1 }\) with the \(y\)-axis.
2. Find the \(x\) coordinates of the intersections of \(C _ { 1 }\) with the \(x\)-axis. Diagram 1, shown on page 21, is a copy of Figure 3.
3. On Diagram 1, sketch the curve \(C _ { 2 }\) with equation $$y = \frac { 1 } { 9 } x ^ { 2 } - 9$$ Identify clearly the coordinates of any points of intersection of \(C _ { 2 }\) with the coordinate axes.
4. Find the coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying entirely on calculator technology are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-21_629_803_1137_573} \section*{Diagram 1} Solutions relying entirely on calculator technology are not acceptable.

1. (a) Sketch the graph of \(y = \left| x ^ { 2 } - a ^ { 2 } \right|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes.
   (b) Solve \(\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1\).
   (c) Find the set of values of \(x\) for which \(\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1\).

Solve the equation \(|2x - 1| = |x|\). [4]

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

Find the exact solutions of the equation \(|6x - 1| = |x - 1|\). [4]

Solve the inequality \(2x > |x - 1|\). [4]

4

1. Sketch the graph of \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-04_933_1093_349_475} 4
2. Solve the inequality $$4 - | 2 x - 6 | > 2$$

The function f is defined by $$f(x) = \left|\sin x + \frac{1}{2}\right| \quad (0 \leq x \leq 2\pi)$$ Find the set of values of \(x\) for which $$f(x) \geq \frac{1}{2}$$ Give your answer in set notation. [5 marks]

Identify the graph of \(y = 1 - |x + 2|\) from the options below. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_1}

The graph of \(y = f(x)\) is shown below. \includegraphics{figure_1} One of the four equations listed below is the equation of the graph \(y = f(x)\) Identify which one is the correct equation of the graph. Tick (\(\checkmark\)) one box. [1 mark] \(y = |x + 2| + 3\) \(y = |x + 2| - 3\) \(y = |x - 2| + 3\) \(y = |x - 2| - 3\)

8. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The graph shown in Figure 2 has equation $$y = a - | 2 x - b |$$ where \(a\) and \(b\) are positive constants, \(a > b\)

1. Find, giving your answer in terms of \(a\) and \(b\),
   1. the coordinates of the maximum point of the graph,
   2. the coordinates of the point of intersection of the graph with the \(y\)-axis,
   3. the coordinates of the points of intersection of the graph with the \(x\)-axis. On page 24 there is a copy of Figure 2 called Diagram 1.
2. On Diagram 1, sketch the graph with equation $$y = | x | - 1$$ Given that the graphs \(y = | x | - 1\) and \(y = a - | 2 x - b |\) intersect at \(x = - 3\) and \(x = 5\)
3. find the value of \(a\) and the value of \(b\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{76989f19-2624-4e86-a8ee-4978dd1014c2-24_675_652_1959_712} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

1 Solve the inequality \(| 2 x - 7 | > 3\).

1 Solve the inequality \(| 2 x - 7 | > 3\).

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 \(| 1 - x | > 2\).

1 Solve the equation \(| 3 x + 4 a | = 5 a\), where \(a\) is a positive constant.

1 Solve the equation \(| 3 x + 4 a | = 5 a\), where \(a\) is a positive constant.

1 Solve the inequality \(| x + 3 a | > 2 | x - 2 a |\), where \(a\) is a positive constant.

1. a. Sketch the graph of the function with equation

$$y = 11 - 2 | 2 - x |$$ Stating the coordinates of the maximum point and any points where the graph cuts the \(y\)-axis.
b. Solve the equation $$4 x = 11 - 2 | 2 - x |$$ A straight line \(l\) has equation \(y = k x + 13\), where \(k\) is a constant.
Given that \(l\) does not meet or intersect \(y = 11 - 2 | 2 - x |\) c. find the range of possible value of \(k\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 | x - 5 | + 10$$ The point \(P\), shown in Figure 1, is the vertex of the graph.

1. State the coordinates of \(P\)
2. Use algebra to solve $$2 | x - 5 | + 10 > 6 x$$ (Solutions relying on calculator technology are not acceptable.)
3. Find the point to which \(P\) is mapped, when the graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = 3 \mathrm { f } ( x - 2 )\)

8. \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.
   [0pt]

1. Sketch the graph of \(y = |3x - 1|\). [1]
2. Hence, solve \(5x + 3 < |3x - 1|\). [3]

2

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

5. (a) Sketch the graph with equation $$y = | 4 x - 3 |$$ stating the coordinates of any points where the graph cuts or meets the axes. Find the complete set of values of \(x\) for which
(b) $$| 4 x - 3 | > 2 - 2 x$$ (c) $$| 4 x - 3 | > \frac { 3 } { 2 } - 2 x$$

1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [6]
2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | 3 x + a | + a$$ and where \(a\) is a positive constant. The graph has a vertex at the point \(P\), as shown in Figure 2 .

1. Find, in terms of \(a\), the coordinates of \(P\).
2. Sketch the graph with equation \(y = g ( x )\), where $$g ( x ) = | x + 5 a |$$ On your sketch, show the coordinates, in terms of \(a\), of each point where the graph cuts or meets the coordinate axes. The graph with equation \(y = \mathrm { g } ( x )\) intersects the graph with equation \(y = \mathrm { f } ( x )\) at two points.
3. Find, in terms of \(a\), the coordinates of the two points. \includegraphics[max width=\textwidth, alt={}, center]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-11_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

5

1. Sketch, on the same diagram, the graphs of \(y = | x + 2 k |\) and \(y = | 2 x - 3 k |\), where \(k\) is a positive constant. Give, in terms of \(k\), the coordinates of the points where each graph meets the axes.
2. Find, in terms of \(k\), the coordinates of each of the two points where the graphs intersect.
3. Find, in terms of \(k\), the largest value of \(t\) satisfying the inequality $$\left| 2 ^ { t } + 2 k \right| \geqslant \left| 2 ^ { t + 1 } - 3 k \right| .$$

Solve the equation \(|3x + 2| = 1\).

2 Solve the equation \(| x + 3 | = 5\).

1 Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. 1

1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).

1. Sketch the graph of \(y = |x - 2a|\), given that \(a > 0\). [2]
2. Solve \(|x - 2a| > 2x + a\), where \(a > 0\). [3]

1

1. Sketch the graph of \(\mathrm { y } = | \mathrm { x } - 2 \mathrm { a } |\), where \(a\) is a positive constant.
2. Solve the inequality \(2 \mathrm { x } - 3 \mathrm { a } < | \mathrm { x } - 2 \mathrm { a } |\).

12.

1. Sketch the graph with equation $$y = | 3 x - 2 a |$$ where \(a\) is a positive constant.
   State the coordinates of each point where the graph cuts or meets the coordinate axes.
2. Solve, in terms of \(a\), the inequality $$| 3 x - 2 a | \leqslant x + a$$ Given that \(| 3 x - 2 a | \leqslant x + a\)
3. find, in terms of \(a\), the range of possible values of \(\mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = 5 a - \left| \frac { 1 } { 2 } a - x \right|$$

10. (a) Sketch, on the same axes, the graphs with equation \(y = | 2 x - 3 |\), and the line with equation \(y = 5 x - 1\).
(b) Solve the inequality \(| 2 x - 3 | < 5 x - 1\).

4 Sketch the graph of \(y = | 2 x - 3 |\).

\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$

1. State the range of \(f\) [1]
2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
3. state the set of possible values for \(k\). [2]

3. (a) Find the set of values of \(x\) for which $$x + 4 > \frac { 2 } { x + 3 }$$ (b) Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \frac { 2 } { | x + 3 | }$$

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

1 Solve the equation \(| 3 x - 2 | = x\).

2.(a)On separate diagrams,sketch the curves with the following equations.On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes.

1. \(y = 8 + 2 x - x ^ { 2 }\)
2. \(y = 8 + 2 | x | - x ^ { 2 }\)
3. \(y = 8 + x + | x | - x ^ { 2 }\) (b)Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

1 Solve the inequality \(\left| 2 ^ { x } - 8 \right| < 5\).

2 Given that \(x\) satisfies the equation \(| 2 x + 3 | = | 2 x - 1 |\), find the value of $$| 4 x - 3 | - | 6 x |$$

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 \(| 2 x + 1 | = | 3 x - 2 |\).

1 Solve the equation \(\left| x ^ { 3 } - 14 \right| = 13\), showing all your working.

5. Sketch the graph of \(y = \ln | x |\), stating the coordinates of any points of intersection with the axes.

3 It is given that \(a\) is a positive constant.

1. 1. Sketch on a single diagram the graphs of \(y = | 2 x - 3 a |\) and \(y = | 2 x + 4 a |\).
   2. State the coordinates of each of the points where each graph meets an axis.
2. Solve the inequality \(| 2 x - 3 a | < | 2 x + 4 a |\).

2

1. Sketch the graph of \(y = | 2 x - 3 |\).
2. Hence, or otherwise, solve the inequality \(| 2 x - 3 | < 5\). Illustrate your answer on your graph.

3 Fig. 1 shows the graphs of \(y = | x |\) and \(y = a | x + b |\), where \(a\) and \(b\) are constants. The intercepts of \(y = a | x + b |\) with the \(x\)-and \(y\)-axes are \(( - 1,0 )\) and \(\left( 0 , \frac { 1 } { 2 } \right)\) respectively. \begin{figure}[h] \end{figure}

1. Find \(a\) and \(b\).
2. Find the coordinates of the two points of intersection of the graphs.

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

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = | 2 - 4 \ln ( x + 1 ) | \quad x > k$$ where \(k\) is a constant.
Given that the curve

• has an asymptote at \(x = k\)
• cuts the \(y\)-axis at point \(A\)
• meets the \(x\)-axis at point \(B\) as shown in Figure 2,
  1. state the value of \(k\)
  2. 1. find the \(y\) coordinate of \(A\)
     2. find the exact \(x\) coordinate of \(B\)
  3. Using algebra and showing your working, find the set of values of \(x\) such that

$$| 2 - 4 \ln ( x + 1 ) | > 3$$

2 Express \(1 < x < 3\) im th \(\quad | x - a | < b\), where \(a\) and \(b\) are to be determined.

1. Sketch, on the same axes, the graph of \(y = |(x - 2)(x - 4)|\), and the line with equation \(y = 6 - 2x\). [4]
2. Find the exact values of \(x\) for which \(|(x - 2)(x - 4)| = 6 - 2x\). [5]
3. Hence solve the inequality \(|(x - 2)(x - 4)| < 6 - 2x\). [2]

2 Solve the equation \(2 \left| 3 ^ { x } - 1 \right| = 3 ^ { x }\), giving your answers correct to 3 significant figures.

1. Sketch the graph of \(y = |3x - 7|\), stating the coordinates of the points where the graph meets the axes. [2]
2. Hence find the set of values of the constant \(k\) for which the equation \(|3x - 7| = k(x - 4)\) has exactly two real roots. [2]