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

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

10 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h] \end{figure}

1. Write down the coordinates of P .
2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).

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

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

5.The function \(f\) is given by $$f ( x ) = \frac { 1 } { \lambda } \left( x ^ { 2 } - 4 \right) \left( x ^ { 2 } - 25 \right)$$ where \(x\) is real and \(\lambda\) is a positive integer.

1. Sketch the graph of \(y = \mathrm { f } ( x )\) showing clearly where the graph crosses the coordinate axes.
2. Find,in terms of \(\lambda\) ,the range of f .
3. Find the sets of positive integers \(k\) and \(\lambda\) such that the equation $$k = | \mathrm { f } ( x ) |$$ has exactly \(k\) distinct real roots.

1. (a) On the same diagram, sketch

$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the \(x\)-coordinates of the points of intersection of these two curves.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.

1. Find the coordinates of \(A\) and \(B\). There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
2. Write down the value of \(k\). The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
3. Find the value of \(\alpha\), leaving your answer in simplified surd form.
4. Find the equation of this normal. The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
5. Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes. The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
6. Find the set of possible values for \(m\).

6. Given that \(a\) and \(b\) are constants and that \(a > b > 0\)

1. on separate diagrams, sketch the graph with equation
   1. \(y = | x - a |\)
   2. \(y = | x - a | - b\) Show on each sketch the coordinates of each point at which the graph crosses or meets the \(x\)-axis and the \(y\)-axis.
2. Hence or otherwise find the complete set of values of \(x\) for which $$| x - a | - b < \frac { 1 } { 2 } x$$ giving your answer in terms of \(a\) and \(b\).

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 $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

7 \includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_419_700_1809_721} The diagram shows the curve \(y = \mathrm { e } ^ { k x } - a\), where \(k\) and \(a\) are constants.

1. Give details of the pair of transformations which transforms the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { k x } - a\).
2. Sketch the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\).
3. Given that the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\) passes through the points \(( 0,13 )\) and \(( \ln 3,13 )\), find the values of \(k\) and \(a\).

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

7 The functions \(\mathrm { f } , \mathrm { g }\) and h are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = | x | , \quad \mathrm { g } ( x ) = 3 x + 5 \quad \text { and } \quad \mathrm { h } ( x ) = \mathrm { gg } ( x ) .$$

1. Solve the equation \(\mathrm { g } ( x + 2 ) = \mathrm { f } ( - 12 )\).
2. Find \(\mathrm { h } ^ { - 1 } ( x )\).
3. Determine the values of \(x\) for which $$x + \mathrm { f } ( x ) = 0 .$$

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

5

1. Give full details of a sequence of two transformations needed to transform the graph of \(y = | x |\) to the graph of \(y = | 2 ( x + 3 ) |\).
2. Solve the inequality \(| x | > | 2 ( x + 3 ) |\), showing all your working.

8 \includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-4_616_1024_296_516} The diagram shows the curve \(y = \frac { 2 x + 4 } { x ^ { 2 } + 5 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the two stationary points.
2. The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$
   1. Sketch the curve \(y = \mathrm { g } ( x )\) and state the range of g .
   2. It is given that the equation \(\mathrm { g } ( x ) = k\), where \(k\) is a constant, has exactly two distinct real roots. Write down the set of possible values of \(k\).

4 It is given that \(| x + 3 a | = 5 a\), where \(a\) is a positive constant. Find, in terms of \(a\), the possible values of $$| x + 7 a | - | x - 7 a |$$

8 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = | 2 x + a | + 3 a \quad \text { and } \quad \mathrm { g } ( x ) = 5 x - 4 a$$ where \(a\) is a positive constant.

1. State the range of f and the range of g .
2. State why f has no inverse, and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Solve for \(x\) the equation \(\operatorname { gf } ( x ) = 31 a\).
4. Show that \(\sin 2 \theta ( \tan \theta + \cot \theta ) \equiv 2\).
5. Hence
   1. find the exact value of \(\tan \frac { 1 } { 12 } \pi + \tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 12 } \pi + \cot \frac { 1 } { 8 } \pi\),
   2. solve the equation \(\sin 4 \theta ( \tan \theta + \cot \theta ) = 1\) for \(0 < \theta < \frac { 1 } { 2 } \pi\),
   3. express \(( 1 - \cos 2 \theta ) ^ { 2 } \left( \tan \frac { 1 } { 2 } \theta + \cot \frac { 1 } { 2 } \theta \right) ^ { 3 }\) in terms of \(\sin \theta\).

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

4 Fig. 4 shows a sketch of the graph of \(y = 2 | x - 1 |\). It meets the \(x\) - and \(y\)-axes at ( \(a , 0\) ) and ( \(0 , b\) ) respectively. \begin{figure}[h] \end{figure} Find the values of \(a\) and \(b\).

2 Solve the inequality \(| 2 x + 1 | > 4\).

1 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.

9
The diagram shows the graph of \(y = | 2 x - 3 |\).

1. State the coordinates of the points of intersection with the axes.
2. Given that the graphs of \(y = | 2 x - 3 |\) and \(y = a x + 2\) have two distinct points of intersection, determine
   1. the set of possible values of \(a\),
   2. the \(x\)-coordinates of the points of intersection of these graphs, giving your answers in terms of \(a\).

5

1. The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
   1. The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\). Find the values of \(a , b\) and \(c\).
   2. Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
2. The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
   1. Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
   2. State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.

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

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 \(k\).