1.02i Represent inequalities: graphically on coordinate plane

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { x } { | x | - 2 }$$ Use algebra to determine the values of \(x\) for which $$2 x - 5 > \frac { x } { | x | - 2 }$$

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. The equation $$\left| x ^ { 2 } - 8 \right| = m x + c$$ has exactly 3 roots, as shown in Figure 1.

1. Show that $$m ^ { 2 } - 4 c + 32 = 0$$ Given that \(c = 3 m\)
2. determine the value of \(m\) and the value of \(c\)
3. Hence solve $$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$

9 The diagram below shows the graphs of \(y = x ^ { 2 } - 4 x - 12\) and \(y = x + 2\) \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-10_933_912_358_566} 9

1. Write down three inequalities which together describe the shaded region.
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2. Find the coordinates of the points \(A , B\) and \(C\).
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3. Find the exact area of the shaded region.
   Fully justify your answer.
   [0pt] [6 marks]

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | }\) and the line with equation \(y = 5 - 4 x\) Use algebra to determine the values of \(x\) for which $$\frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | } < 5 - 4 x$$

\includegraphics{figure_2} The straight line \(l_1\) has equation \(2y = 3x + 7\) The line \(l_1\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.

1. 1. State the gradient of \(l_1\)
   2. Write down the coordinates of the point \(A\). [2]

Another straight line \(l_2\) intersects \(l_1\) at the point \(B(1, 5)\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle ABC = 90°\),

1. find an equation of \(l_2\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The rectangle \(ABCD\), shown shaded in Figure 2, has vertices at the points \(A\), \(B\), \(C\) and \(D\).

1. Find the exact area of rectangle \(ABCD\). [5]

A circle with centre \(C\) has equation \((x - 2)^2 + (y + 5)^2 = 25\).

1. Show that no part of the circle lies above the \(x\)-axis. [3]
2. The point \(P\) has coordinates \((6, k)\) and lies inside the circle. Find the set of possible values of \(k\). [5]
3. Prove that the line \(2y = x\) does not meet the circle. [4]

Show in a sketch the region of the \(x\)-\(y\) plane within which all three of the following inequalities hold. \(y \geqslant x^2\), \(x + y \leqslant 2\), \(x \geqslant 0\). You should indicate the region for which the inequalities hold by labelling the region \(R\). [3]

Sketch the region defined by the inequalities $$y \leq (1 - 2x)(x + 3) \text{ and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\). [3 marks] \includegraphics{figure_4}