1.02i Represent inequalities: graphically on coordinate plane

4. \begin{figure}[h] \end{figure} Figure 1 shows a line \(l _ { 1 }\) with equation \(2 y = x\) and a curve \(C\) with equation \(y = 2 x - \frac { 1 } { 8 } x ^ { 2 }\) The region \(R\), shown unshaded in Figure 1, is bounded by the line \(l _ { 1 }\), the curve \(C\) and a line \(l _ { 2 }\) Given that \(l _ { 2 }\) is parallel to the \(y\)-axis and passes through the intercept of \(C\) with the positive \(x\)-axis, identify the inequalities that define \(R\).

4. \begin{figure}[h] \end{figure} The points \(P\) and \(Q\), as shown in Figure 2, have coordinates ( \(- 2,13\) ) and ( \(4 , - 5\) ) respectively. The straight line \(l\) passes through \(P\) and \(Q\).

1. Find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found. The quadratic curve \(C\) passes through \(P\) and has a minimum point at \(Q\).
2. Find an equation for \(C\). The region \(R\), shown shaded in Figure 2, lies in the second quadrant and is bounded by \(C\) and \(l\) only.
3. Use inequalities to define region \(R\). \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-11_2255_50_314_34}
   

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4. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} Figure 1 shows a line \(l\) with equation \(x + y = 6\) and a curve \(C\) with equation \(y = 6 x - 2 x ^ { 2 } + 1\) The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 1, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).
   

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8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\) and the curve \(C\).
Given that \(l\) cuts the \(y\)-axis at - 12 and cuts the \(x\)-axis at 4 , as shown in Figure 2,

1. find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. Given that \(C\)
   The region \(R\) is shown shaded in Figure 2.
2. Use inequalities to define \(R\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\)

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• the maximum turning point on \(C\) has coordinates \(( - 2,12 )\)
• \(C\) cuts the negative \(x\)-axis at - 5
  1. find \(\mathrm { f } ( x )\)

The line \(l _ { 1 }\) has equation \(y = \frac { 4 } { 5 } x\) Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(( - 5,0 )\)

find an equation for \(l _ { 2 }\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
(3) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve \(C\) and the lines \(l _ { 1 }\) and \(l _ { 2 }\)

Define the region \(R\), shown shaded in Figure 2, using inequalities.

7. \begin{figure}[h] \end{figure} The region \(R _ { 1 }\), shown shaded in Figure 2, is defined by the inequalities $$0 \leqslant y \leqslant 2 \quad y \leqslant 10 - 2 x \quad y \leqslant k x$$ where \(k\) is a constant.
The line \(x = a\), where \(a\) is a constant, passes through the intersection of the lines \(y = 2\) and \(y = k x\) Given that the area of \(R _ { 1 }\) is \(\frac { 27 } { 4 }\) square units,

1. find
   1. the value of \(a\)
   2. the value of \(k\)
2. Define the region \(R _ { 2 }\), also shown shaded in Figure 2, using inequalities.

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = x ^ { 2 } - 5 x + 13\) The point \(M\) is the minimum point of \(C\). The straight line \(l\) passes through the origin \(O\) and intersects \(C\) at the points \(M\) and \(N\) as shown. Find, showing your working,

1. the coordinates of \(M\),
2. the coordinates of \(N\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_531_561_1793_680} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the curve \(C\) and the line \(l\). The finite region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
3. Use inequalities to define the region \(R\).

3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only. Given that

• \(\quad l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(\quad C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only.
Given that

• \(l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).
  

8. \section*{Diagram NOT to scale} \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).

• \(B C = 8 \mathrm {~m}\)
• \(O A = O B = 3 \mathrm {~m}\)
• angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
• angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
  1. Calculate (i) the exact area of the sector \(A O B X\),
     (ii) the exact perimeter of the sector \(A O B X\).
  2. Calculate the exact area of the triangle \(A O B\).
  3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
  4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.

11. \begin{figure}[h] \end{figure} Figure 5 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$

1. Write \(2 x ^ { 2 } - 12 x + 14\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. Given that \(C\) has a minimum at the point \(P\)
2. state the coordinates of \(P\) The line \(l\) intersects \(C\) at \(( - 1,28 )\) and at \(P\) as shown in Figure 5.
3. Find the equation of \(l\) giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. The finite region \(R\), shown shaded in Figure 5, is bounded by the \(x\)-axis, \(l\), the \(y\)-axis, and \(C\).
4. Use inequalities to define the region \(R\).

1. (i) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\).

Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$

7 Fig. 7 shows part of the curve with equation \(y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the coordinates of the two points where the curve crosses the axes.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Determine how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
4. On the copy of Fig. 7, sketch the rest of the curve.
5. Solve the inequality \(\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0\).

7 A curve has equation \(y = \frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 }\).

1. Find the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or from below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.
5. Solve the inequality \(\frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 } < 2\).

7 Fig. 7 shows a sketch of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes. Hence write down the solution of the inequality \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } > 0\).
2. The equation \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } = k\) has no real solutions. Show that \(- 1 < k < - \frac { 1 } { 9 }\). Relate this result to the graph of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\).

4 Solve the inequality \(\frac { 3 } { x - 4 } > 1\).

7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.

1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
3. Sketch the curve.
4. Find the set of values of \(x\) for which \(y > 0\).

3

1. In this question you must show detailed reasoning.
   Find the coordinates of the points of intersection of the curves with equations \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\).
2. The diagram shows the curves \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\). This diagram is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-5_377_542_603_322} On the diagram in the Printed Answer Booklet, draw the line \(y = 2 x - 2\).
3. Show on your diagram in the Printed Answer Booklet the region of the \(x - y\) plane within which all three of the following inequalities are satisfied. \(y \geqslant x ^ { 2 } - 2 x + 1 \quad y \leqslant - x ^ { 2 } + 6 x - 5 \quad y \leqslant 2 x - 2\) You should indicate the region for which all the inequalities hold by labelling the region \(R\).[1]

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\).
Given that

• \(C\) has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression in \(x\)
• \(C\) cuts the \(x\)-axis at 0 and 6
• \(l\) cuts the \(y\)-axis at 60 and intersects \(C\) at the point \(( 10,80 )\) use inequalities to define the region \(R\) shown shaded in Figure 3.

12. Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\).
2. [diagram]
   The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\). The curve \(C\) meets \(l\) at the points \(( 2,4 )\) and \(( 6,0 )\) as shown. The shaded region \(R\), shown shaded in Figure 1, is bounded by \(\mathrm { C } , l\) and the \(y\)-axis. Given that \(\mathrm { f } ( x )\) is a quadratic function in \(x\), use inequalities to define region \(R\).

3 The diagram in the Printed Answer Booklet shows part of the graph of \(y = x ^ { 2 } - 4 x + 3\).

1. It is required to solve the equation \(x ^ { 2 } - 3 x + 1 = 0\) graphically by drawing a straight line with equation \(y = m x + c\) on the diagram, where \(m\) and \(c\) are constants. Find the values of \(m\) and \(c\).
2. Use the graph to find approximate values of the roots of the equation \(x ^ { 2 } - 3 x + 1 = 0\).
3. By shading, or otherwise, indicate clearly the regions where all of the following inequalities are satisfied. You should use the values of \(m\) and \(c\) found in part (a). \(x \geqslant 0\) \(x \leqslant 4\) \(y \leqslant x ^ { 2 } - 4 x + 3\) \(y \geqslant m x + c\)

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 \(R\).

7 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).

1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the inequalities that define the shaded region, including its boundaries.

3

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.

7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$