A-level · Algebra & functions

Inequalities on graphs and regionsExam-style questions

Reading an inequality as a region of the plane. The boundary as the matching equation; which side of a curve or line to shade, and testing a point; dashed boundaries for strict inequalities and solid ones for inclusive; a region as the overlap of several inequalities; finding its corners by solving the boundaries together; and writing down the inequalities for a shaded region.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The curve with equation $y = 5 + 4x - x^2$ and the line with equation $x + y = 3$ meet at the points $A$ and $B$ , as shown in the diagram below.(a) On the diagram above, shade the region where

$0 \le y \le 5 + 4x - x^2$

and

$x + y \ge 3$

Label this region $R$

(2 marks)
(b) Find the exact coordinates of $A$

(3 marks)
Level 4 · 5 marksTry it
2
The diagram below shows the graphs of $y = 6 - x - x^2$ and $y - 2x = 1$

The points $A$ and $B$ lie on both graphs.(a) The region $R$ satisfies the inequalities

$0 \le y \le 6 - x - x^2 \quad \text{and} \quad y - 2x \ge 1$

Shade and label $R$ on the diagram above.

(2 marks)
(b) Find the exact coordinates of $A$

(3 marks)
Level 4 · 5 marksTry it
3
(a) In this question you must show detailed reasoning.

Determine the coordinates of the two points where the curve $y = x^2 - x - 4$ meets the curve $y = 2 + 3x - x^2$ .

(4 marks)
(b) The two curves from part (a) are drawn in the diagram, together with a straight line $L$ . The line $L$ crosses the $x$ -axis at the point $(1, 0)$ and the $y$ -axis at the point $(0, -2)$ .Find the equation of $L$ , giving your answer in the form $y = mx + c$ .

(2 marks)
(c) Write down three inequalities which together define the shaded region $R$ .

(1 mark)
Level 4 · 7 marksTry it
4
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

The curve $C$ has equation $y = 2x^2 - 7x + 8$ and the straight line $l$ has equation $2x + y = 20$

Figure 1 shows a sketch of $C$ and $l$ , which meet at the points $A$ and $B$ .
(a) Use algebra to find the coordinates of $A$ and of $B$ .

(4 marks)

The boundary of the finite region $R$ , shaded in Figure 1, consists of parts of $C$ , $l$ and the $y$ -axis.
(b) Use inequalities to define the region $R$ .

(3 marks)
Level 4 · 7 marksTry it
5
Figure 1

The straight line $l$ $l$ passes through the points $A(-5,\ 0)$ $A (- 5, 0)$ and $B(1,\ 12)$ $B (1, 12)$ , as shown in Figure 1.
(a) Find the equation of $l$ $l$ in the form $y = mx + c$ $y = m x + c$ .

(2 marks)

The curve $C$ $C$ , also shown in Figure 1, has equation $y = \text{f}(x)$ $y = f (x)$ , where $\text{f}(x)$ $f (x)$ is a quadratic expression.

Given that
- $C$ passes through $A$
- the maximum point of $C$ is $(1,\ 36)$
- $C$ crosses the $x$ -axis again at the point $(k,\ 0)$
(b) deduce the value of $k$ $k$ ,

(1 mark)
(c) find $\text{f}(x)$ $f (x)$ .

(3 marks)

The shaded region $R$ $R$ in Figure 1 has boundaries $l$ $l$ , $C$ $C$ and the $x$ $x$ -axis.
(d) Use inequalities to define $R$ $R$ .

(2 marks)
Level 4 · 8 marksTry it