Inequalities represented graphicallyAQA A-Level Maths: Subtopic test

1. 1
   A region $R$ of the $xy$-plane is defined by the inequalities $y>x+1$ and $y\le5-x$.

   (a)

   Which of these points lies in $R$?

   [1 mark]

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

   (b)

   When $R$ is drawn, which statement about its boundary lines is correct?

   [1 mark]

   • Aboth lines are solid
   • B$y=x+1$ is solid and $y=5-x$ is dashed
   • Cboth lines are dashed
   • D$y=x+1$ is dashed and $y=5-x$ is solid

   (c)

   Find the coordinates of the vertex of $R$, and state whether this vertex belongs to $R$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x^2-4x+3$ and the line $L$ has equation $y=x-1$.

   (a)

   Which pair of inequalities describes the region above $C$ and below $L$?

   [1 mark]

   • A$y<x^2-4x+3$ and $y>x-1$
   • B$y>x^2-4x+3$ and $y<x-1$
   • C$y>x^2-4x+3$ and $y>x-1$
   • D$y<x^2-4x+3$ and $y<x-1$

   (b)

   Which statement is true about the point $(2,0)$?

   [1 mark]

   • Ait is below $C$ and above $L$
   • Bit is above both $C$ and $L$
   • Cit is above $C$ and below $L$
   • Dit is below both $C$ and $L$

   (c)

   Find the set of values of $x$ for which $C$ lies on or above $L$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A baker plans a day's work using $x$ trays of rolls and $y$ trays of cakes, where $x$ and $y$ are whole numbers. The constraints are $x\ge1$, $y\ge2$ and $2x+y\le10$.

   (a)

   The inequalities define a triangular region when drawn. Find the coordinates of its three vertices.

   [3 marks]

   (b)

   Find the number of different pairs $(x,y)$ the baker can choose, counting points on the boundaries.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^2-2x-8$ and the line $L$ has equation $y=2x-3$.

   (a)

   (i) Find the coordinates of the points where $C$ crosses the axes, and of the minimum point of $C$.
   (ii) Use your answers to explain how the shape of $C$ shows that $x^2-2x-8<0$ for $-2<x<4$.

   [6 marks]

   (b)

   (i) Show that $C$ and $L$ meet where $x^2-4x-5=0$, and find the coordinates of the points of intersection.
   (ii) State the values of $x$ for which $C$ is below $L$, and write down the inequalities which define the region between $C$ and $L$, not including the boundaries.

   [6 marks]

   Total for question 4: 12 marks

End of questions