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

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1. The diagram shows the line \(y = x + 5\) and the curve \(y = 8 - 2 x - x ^ { 2 }\). The shaded region is the finite region between the line and the curve. The curved part of the boundary is included in the region but the straight part is not included. Write down the inequalities that define the shaded region. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-04_846_716_1379_322} \section*{(b) In this question you must show detailed reasoning.} Solve the inequality \(8 - 2 x - x ^ { 2 } > x + 5\) giving your answer in exact form.

3 Draw a number line to show the values of \(x\) which belong to the set \(\{ x : x \geqslant 2 \} \cap \{ x : x < 7 \}\).

7 The quadratic equation $$( 2 k - 7 ) x ^ { 2 } - ( k - 2 ) x + ( k - 3 ) = 0$$ has real roots.

1. Show that \(7 k ^ { 2 } - 48 k + 80 \leqslant 0\).
2. Find the possible values of \(k\).

8 A curve has equation \(y = x ^ { 2 } + ( 3 k - 4 ) x + 13\) and a line has equation \(y = 2 x + k\), where \(k\) is a constant.

1. Show that the \(x\)-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
2. Given that the line and the curve do not intersect:
   1. show that \(9 k ^ { 2 } - 32 k - 16 < 0\);
   2. find the possible values of \(k\).
      \includegraphics[max width=\textwidth, alt={}]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-18_1657_1714_1050_153}

2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

1. Use algebra to determine the values of \(x\) for which

$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$

1. Use algebra to determine the values of \(x\) for which

$$x ( x - 1 ) > \frac { x - 1 } { x }$$ giving your answer in set notation.

1. Use algebra to find the set of values of \(x\) for which

$$x \geqslant \frac { 2 x + 15 } { 2 x + 3 }$$

1. (a) Use algebra to determine the values of \(x\) for which

$$\frac { 5 x } { x - 2 } \geqslant 12$$ (b) Hence, given that \(x\) is an integer, deduce the value of \(x\).

1. Use algebra to determine the values of \(x\) for which

$$\left| x ^ { 2 } - 2 x \right| \leqslant x$$

8 In this question you must show detailed reasoning.

1. Use differentiation to find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 3 x - 2\).
2. Use the second derivative to determine the nature of the stationary point.
3. Show by shading on a sketch the region defined by the inequality \(y \geqslant 2 x ^ { 2 } - 3 x - 2\), indicating clearly whether the boundary is included or not.
4. Solve the inequality \(2 x ^ { 2 } - 3 x - 2 > 0\) using set notation for your answer.

2 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-4_661_579_831_246} The diagram shows a patio. The perimeter of the patio has to be less than 44 m .
The area of the patio has to be at least \(45 \mathrm {~m} ^ { 2 }\).

1. Write down, in terms of \(x\), an inequality satisfied by
   1. the perimeter of the patio,
   2. the area of the patio.
2. Hence determine the set of possible values of \(x\).

1 State the set of values of \(x\) which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\} \\ & \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\} \\ & \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\} \\ & \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$

1 The graph of \(y = a x ^ { 2 } + b x + c\) has roots \(x = 2\) and \(x = 5\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_905_963_717_625} State the set of values of \(x\) which satisfy $$a x ^ { 2 } + b x + c > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \{ x : x < 2 \} \cup \{ x : x > 5 \} \\ & \{ x : 0 < x < 2 \} \cap \{ x : x > 5 \} \\ & \{ x : 2 < x < 5 \} \\ & \{ x : 2 > x > 5 \} \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_118_115_1950_1087}
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3. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 2 } - 10 x \quad x \in \mathbb { R }$$

1. Solve the equation $$\mathrm { f } ( | x | ) = 48$$
2. Find the set of values of \(x\) for which $$| f ( x ) | \geqslant \frac { 5 } { 2 } x$$

Solve the inequalities

1. \(1 < 4x - 9 < 5\), [3]
2. \(y^2 \geq 4y + 5\). [5]

Solve the inequalities

1. \(3 - 8x > 4\), [2]
2. \((2x - 4)(x - 3) \leq 12\). [5]

Solve the following inequalities.

1. \(5 < 6x + 3 < 14\) [3]
2. \(x(3x - 13) \geqslant 10\) [5]

The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]

Determine the set of values of \(x\) which satisfy the inequality $$3x^2 + 3x > x + 6$$ Give your answer in exact form using set notation. [4 marks]

A curve C has equation \(y = -x^3 + 12x - 20\).

1. Find the coordinates of the stationary points of C and determine their nature. [7]
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation. [3]

\includegraphics{figure_11} The diagram shows a sketch of the curve \(y = 6 + 4x - x^2\) and the line \(y = x + 2\). The point \(P\) has coordinates \((a, b)\). Write down the three inequalities involving \(a\) and \(b\) which are such that the point \(P\) will be strictly contained within the shaded area above, if and only if, all three inequalities are satisfied. [3]