Representing inequalities graphicallyEdexcel International A Level Maths: Subtopic test

1. 1
   A region $R$ of the $x$-$y$ plane contains all the points $(x,y)$ that satisfy both $y>x+1$ and $y\le5$.

   (a)

   Which of the following points lies in $R$?

   [1 mark]

   • A$(3,2)$
   • B$(1,6)$
   • C$(0,3)$
   • D$(2,3)$

   (b)

   Which of the following correctly describes the boundary lines of $R$ when it is drawn?

   [1 mark]

   • A$y=x+1$ dotted and $y=5$ solid
   • Bboth lines solid
   • C$y=x+1$ solid and $y=5$ dotted
   • Dboth lines dotted

   (c)

   Find the coordinates of the point where the two boundary lines meet, and state, with a reason, whether this point belongs to $R$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The region $S$ contains all the points $(x,y)$ that satisfy both $y\ge x^2-4$ and $y<2x-1$.

   (a)

   Which of the following points lies in $S$?

   [1 mark]

   • A$(0,1)$
   • B$(4,3)$
   • C$(-2,0)$
   • D$(1,0)$

   (b)

   Which of the following gives the $x$-coordinates of the points where the boundaries of $S$ meet?

   [1 mark]

   • A$x=-3$ and $x=1$
   • B$x=-1$ and $x=3$
   • C$x=-2$ and $x=2$
   • D$x=1$ and $x=3$

   (c)

   Show that the point $(3,5)$ lies on the boundary of $S$ but is not in $S$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The region $T$ contains all the points $(x,y)$ that satisfy both $y>x^2-2x-3$ and $y\le0$.

   (a)

   Find the coordinates of the points where the curve $y=x^2-2x-3$ crosses the axes, and the coordinates of its minimum point.

   [3 marks]

   (b)

   Describe how $T$ would be shown on a sketch: state whether each boundary is dotted or solid, which region is shaded, and check your shading using the point $(1,-2)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^2-6x+5$ and the line $L$ has equation $y=x-1$. The region $R$ contains all the points $(x,y)$ that satisfy both $y>x^2-6x+5$ and $y\le x-1$.

   (a)

   (i) Find the coordinates of the points where $C$ and $L$ meet.
   (ii) Describe how $R$ would be drawn on a sketch, stating whether each boundary is dotted or solid and which region is shaded.

   [6 marks]

   (b)

   (i) The point $(4,m)$ lies in $R$. Find the range of possible values of $m$.
   (ii) Find the set of $x$-values of the points of $L$ that lie in $R$, and explain why the points $(1,0)$ and $(6,5)$ are not in $R$.

   [6 marks]

   Total for question 4: 12 marks

End of questions