WMA11 Jan 2023

• WMA11/01 (Edexcel) IAL P1 January 2023, Q8, Quadratics, Inequalities & regions

Figure 2 shows a sketch of the straight line / 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,

• find an equation for​​ l, writing your answer in the form y = mx + c, where m and c are constants to be found.

(2)

Given that C

• has equation y = f(x) where f(x) is a quadratic expression

• has a minimum point at (7, -18)

• cuts the x-axis at 4 and at k, where k is a constant

• deduce the value of k,

(1)

• find f(x).

(3)

The region R is shown shaded in Figure 2.

• Use inequalities to define R.

(2)

SOLUTION

a-​​ Considering two points on the line to find the gradient of the line,​​ l.

The two points are (0, -12) and (4, 0)

m=0--124-0=124=3 

Using the equation of a stright line, where m=3 and c=-12

y=mx+c

y=3x-12 

b-​​ This questions is just of 1 marks, hence, we have to solve by thinking critically.​​ 

Here, we may use the fact that the roots of a quadratic equation are symmetrical about the vertex (turning point or critical point). Now, since the the x-cordinate of the vertex is 7, the distance from the vertex to both roots on left and right side of it must be the same. The root x=4 is 3 units to the left of the vertex. Therefore, the other root must also be 3 units to the right of the vertex.​​ 

Thus, the other root, k must be​​ 

k=7+3 =10

c-​​ 

The quadratic equation can be expressed in the form of completing square form as​​ y=ax-p2+q, where (-p, q) is the vertex of the quardratic function.

In this question, the vertex of a quadratic equation is given as (7, -18).

y=ax-72-18

And, on further substituting the x-intercept coordinate (4, 0), the equation becomes​​ 

0=a4-72 -18

0=a-32-18

9a-18=0

9a=18 

a=2 

Putting back the value of a in the quadratic equation in the completing square form.​​ 

y=2x-72-18

Expanding the brackets.

y=2x2-14x+49-18

y=2x2-28x+98-18

fx=2x2-28x+80 

d-​​ 

Since the region is enclosed between the solid lines, the inequalities must be either​​ ≤​​ or​​ ≥.​​ 

Keeping in view the above flash card.​​ 

Region is​​ 

• above/outside the quadratic curve; therefore,

y≤2x-72-18

• above the straight line; therefore,​​ 

y≥3x-12

• rightside of y-axis (x=0)​​ 

x≥0