1.02t Solve modulus equations: graphically with modulus function

2

1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 9 |\) and \(y = 5 x - 3\).
2. Solve the equation \(| 2 x - 9 | = 5 x - 3\).

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 $$\left| 2 ^ { t } + 2 k \right| \geqslant \left| 2 ^ { t + 1 } - 3 k \right| .$$

1

1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = x + 2\).
2. Solve the equation \(| 3 x - 5 | = x + 2\).

3

1. Sketch, on a single diagram, the graphs of \(y = \left| \frac { 1 } { 2 } x - a \right|\) and \(y = \frac { 3 } { 2 } x - \frac { 1 } { 2 } a\), where \(a\) is a positive constant.
2. Find the coordinates of the point of intersection of the two graphs.
3. Deduce the solution of the inequality \(\left| \frac { 1 } { 2 } x - a \right| > \frac { 3 } { 2 } x - \frac { 1 } { 2 } a\).

2

1. Sketch, on the same diagram, the graphs of \(y = x + 3\) and \(y = | 2 x - 1 |\).
2. Solve the equation \(x + 3 = | 2 x - 1 |\).
3. Find the value of \(y\) such that \(5 ^ { \frac { 1 } { 2 } y } + 3 = \left| 2 \times 5 ^ { \frac { 1 } { 2 } y } - 1 \right|\). Give your 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.

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

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
4. 1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
   2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
   3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).

6. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 2 x - 5 | + 3 \quad x \geqslant 0$$ The vertex of the graph is at point \(P\) as shown.

1. State the coordinates of \(P\).
2. Solve the equation \(\mathrm { f } ( x ) = 3 x - 2\) Given that the equation $$f ( x ) = k x + 2$$ where \(k\) is a constant, has exactly two roots,
3. find the range of values of \(k\).
   

6. Given that \(k\) is a positive constant,

1. on separate diagrams, sketch the graph with equation
   1. \(y = k - 2 | x |\)
   2. \(y = \left| 2 x - \frac { k } { 3 } \right|\) Show on each sketch the coordinates, in terms of \(k\), of each point where the graph meets or cuts the axes.
2. Hence find, in terms of \(k\), the values of \(x\) for which $$\left| 2 x - \frac { k } { 3 } \right| = k - 2 | x |$$ giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-23_2647_1840_118_111}

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | k x - 9 | - 2 \quad x \in \mathbb { R }$$ and \(k\) is a positive constant. The graph intersects the \(y\)-axis at the point \(A\) and has a minimum point at \(B\) as shown.

1. 1. Find the \(y\) coordinate of \(A\)
   2. Find, in terms of \(k\), the \(x\) coordinate of \(B\)
2. Find, in terms of \(k\), the range of values of \(x\) that satisfy the inequality $$| k x - 9 | - 2 < 0$$ Given that the line \(y = 3 - 2 x\) intersects the graph \(y = \mathrm { f } ( x )\) at two distinct points,
3. find the range of possible values of \(k\)

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 21 - 2 | 2 - x | \quad x \geqslant 0$$

1. Find ff(6)
2. Solve the equation \(\mathrm { f } ( x ) = 5 x\) Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
3. state the set of possible values of \(k\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x - b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x - b )\) is (6, 3). Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\). \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-11_2255_50_314_34}

3. \begin{figure}[h] \end{figure} Figure 1 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. 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,
2. state the set of possible values for \(k\).
   

   VIIIV SIHI NI JIIIM ION OC

   VIIV SIHI NI JAHAM ION OO

   VI4V SIHIL NI JIIIM ION OC

3. The function \(g\) is defined by $$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$

1. Sketch the graph with equation \(y = \mathrm { g } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
2. Solve the equation $$| 8 - 2 x | = x + 5$$ The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
3. Find fg(5).
4. Find the range of f. You must make your method clear.

12. Given that \(k\) is a positive constant,

1. sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis.
2. Find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$

1. 1. The functions \(f\) and \(g\) are defined by

$$\begin{array} { l l } \mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } - 5 , & x \in \mathbb { R } \\ \mathrm {~g} : x \rightarrow \ln ( 3 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 3 } \end{array}$$

1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
2. Find \(\mathrm { fg } ( 3 )\), giving your answer in its simplest form.
   (ii) (a) Sketch the graph with equation $$y = | 4 x - a |$$ where \(a\) is a positive constant. State the coordinates of each point where the graph cuts or meets the coordinate axes. Given that $$| 4 x - a | = 9 a$$ where \(a\) is a positive constant,
3. find the possible values of $$| x - 6 a | + 3 | x |$$ giving your answers, in terms of \(a\), in their simplest form.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 4 x + 10 a |\), where \(a\) is a positive constant. The graph cuts the \(y\)-axis at the point \(P\) and meets the \(x\)-axis at the point \(Q\) as shown.

1. 1. State the coordinates of \(P\).
   2. State the coordinates of \(Q\).
2. A copy of Figure 1 is shown on page 15. On this copy, sketch the graph with equation $$y = | x | - a$$ Show on the sketch the coordinates of each point where your graph cuts or meets the coordinate axes.
3. Hence, or otherwise, solve the equation $$| 4 x + 10 a | = | x | - a$$ giving your answers in terms of \(a\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-15_860_1128_447_392} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \(\_\_\_\_\) 7

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$ The graph cuts the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 2 .

1. State the range of g .
2. State the coordinates of
   1. point \(A\)
   2. point \(B\)
3. Find \(\operatorname { gg } ( x )\) in its simplest form.
4. Sketch the graph with equation \(y = | \mathrm { g } ( x ) |\) On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
5. Find the exact solution of the equation \(| \mathrm { g } ( x ) | = 8\)

5. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \ln ( 2 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 2 } \\ \mathrm {~g} : x \mapsto \frac { 2 } { x - 3 } , & x \in \mathbb { R } , x \neq 3 \end{array}$$

1. Find the exact value of fg(4).
2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.
3. Sketch the graph of \(y = | \mathrm { g } ( x ) |\). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the \(y\)-axis.
4. Find the exact values of \(x\) for which \(\left| \frac { 2 } { x - 3 } \right| = 3\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation \(y = 2 | x | - 5\).
The graph intersects the positive \(x\)-axis at the point \(P\) and the negative \(y\)-axis at the point \(Q\).

1. State the coordinates of \(P\) and the coordinates of \(Q\).
2. Solve the equation $$2 | x | - 5 = 3 - x$$

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 5 - x | + 3 , \quad x \geqslant 0$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly one root,

1. state the set of possible values of \(k\).
2. Solve the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x + 10\) The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\). The vertex on the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\) has coordinates \(( p , q )\).
3. State the value of \(p\) and the value of \(q\).