1.02h Express solutions: using 'and', 'or', set and interval notation

11 The function f is defined by \(\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
2. Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\). The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
3. Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
4. State the smallest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). {www.cie.org.uk} after the live examination series. }

3. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 29 m ,

1. show that \(x > 1.5 \mathrm {~m}\) Given also that the area of the garden is less than \(72 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of possible values of \(x\).
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6. Find the set of values of \(x\) for which

1. \(3 ( 2 x + 1 ) > 5 - 2 x\),
2. \(2 x ^ { 2 } - 7 x + 3 > 0\),
3. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).

1. A rectangular room has a width of \(x \mathrm {~m}\).

The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m ,

1. show that \(x > 2.8\) Given also that the area of the room is less than \(21 \mathrm {~m} ^ { 2 }\),
2. 1. write down an inequality, in terms of \(x\), for the area of the room.
   2. Solve this inequality.
3. Hence find the range of possible values for \(x\).

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) $$| 4 x - 3 | > \frac { 3 } { 2 } - 2 x$$

Find the set of values of \(x\) for which \(\frac { x ^ { 2 } } { x - 2 } > 2 x\).
(Total 6 marks)

2. Using algebra, find the set of values of \(x\) for which $$3 x - 5 < \frac { 2 } { x }$$

2. Use algebra to determine the set of values of \(x\) for which $$\frac { x } { 2 - x } \leqslant \frac { x + 3 } { x }$$ (Solutions relying entirely on graphical methods are not acceptable.)
(8)

8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.

1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
3. By solving the inequalities in parts (i) and (ii), determine the set of possible values of \(x\).

1 Solve the inequality \(x ^ { 2 } - 6 x - 40 \geqslant 0\).

6 List the integers which satisfy both of the following inequalities: $$2 x - 9 < 0 , \quad 8 - x \leq 6$$

1. (i) Solve the inequality

$$x ^ { 2 } + 3 x > 10 .$$ (ii) Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3 \\ & x ^ { 2 } + 3 x > 10 \end{aligned}$$

5. Given that the equation $$x ^ { 2 } + 4 k x - k = 0$$ has no real roots,

1. show that $$4 k ^ { 2 } + k < 0 ,$$
2. find the set of possible values of \(k\).

1. (i) Solve the inequality

$$| x - 0.2 | < 0.03$$ (ii) Hence, find all integers \(n\) such that $$\left| 0.95 ^ { n } - 0.2 \right| < 0.03$$

11 A lawn is to be made in the shape shown below. The units are metres. \includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-4_412_698_486_726}

1. The perimeter of the lawn is \(P \mathrm {~m}\). Find \(P\) in terms of \(x\).
2. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the lawn is given by \(A = 9 x ^ { 2 } + 6 x\). The perimeter of the lawn must be at least 39 m and the area of the lawn must be less than \(99 \mathrm {~m} ^ { 2 }\).
3. By writing down and solving appropriate inequalities, determine the set of possible values of \(x\).

9

1. A rectangular tile has length \(4 x \mathrm {~cm}\) and width \(( x + 3 ) \mathrm { cm }\). The area of the rectangle is less than \(112 \mathrm {~cm} ^ { 2 }\). By writing down and solving an inequality, determine the set of possible values of \(x\).
2. A second rectangular tile of length \(4 y \mathrm {~cm}\) and width \(( y + 3 ) \mathrm { cm }\) has a rectangle of length \(2 y \mathrm {~cm}\) and width \(y \mathrm {~cm}\) removed from one corner as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662} Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of \(y\).

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).

1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.

1. (a) Factorise completely \(9 x - x ^ { 3 }\)

The curve \(C\) has equation $$y = 9 x - x ^ { 3 }$$ (b) Sketch \(C\) showing the coordinates of the points at which the curve cuts the \(x\)-axis. The line \(l\) has equation \(y = k\) where \(k\) is a constant.
Given that \(C\) and \(l\) intersect at 3 distinct points,
(c) find the range of values for \(k\), writing your answer in set notation. Solutions relying on calculator technology are not acceptable.

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

Solutions relying on calculator technology are not acceptable.
Using algebra, solve the inequality $$x ^ { 2 } - x > 20$$ writing your answer in set notation.

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.

6 In this question you must show detailed reasoning.

1. Solve the inequality \(x ^ { 2 } + x - 6 > 0\), giving your answer in set notation.
2. Solve the equation \(x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0\).
3. Find the exact solution of the equation \(\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }\).

3 The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\). This diagram is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}

1. State the values of \(x\) for which \(\mathrm { f } ( x ) < \frac { 1 } { 2 }\), giving your answer in set notation.
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( - x )\).
3. Explain how you can tell that \(\mathrm { f } ( x )\) cannot be expressed as the product of three real linear factors.

2 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-03_835_545_749_244} The diagram shows the line \(y = - 2 x + 4\) and the curve \(y = x ^ { 2 } - 4\). The region \(R\) is the unshaded region together with its boundaries. Write down the inequalities that define \(R\).

1 Write the solution of the inequality \(( x - 2 ) ( x + 3 ) > 0\) using set notation.

7 In this question you must show detailed reasoning.
A curve has equation \(y = 4 x ^ { 3 } - 6 x ^ { 2 } - 9 x + 4\).

1. Sketch the gradient function for this curve, clearly indicating the points where the gradient is zero.
2. Find the set of values of \(x\) for which the gradient function is decreasing. Give your answer using set notation.