WMA11 Jan 2022

4. WMA11/01 Edexcel​​ IAL P1 January 2022,​​ Q4​​ (Quadratics,​​ Equations and​​ Inequalities)

In this question you must show all stages of your working.

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 = 6x – 2x2+ 1​​ 

The line l intersects the curve C at the points P and Q as shown in Figure 1.​​ 

(a) Find, using algebra, the coordinates of P and the coordinates of Q.​​ 

(4)

The region R, shown shaded in Figure 1, is bounded by C, l and the x‑axis.​​ 

(b) Use inequalities to define the region R.​​ 

(3)

SOLUTION

a-​​ Solving​​ both equations simultaneously, to find the points of intersection.​​ 

The equation of line is​​ 

x+y=6

y=6-x

Whereas the equation of a curve is​​ 

y=6x-2x2+1

Now, equating them​​ 

6-x=6x-2x2+1

2x2-7x+5=0

x-12x-5=0

x-1=0       2x-5=0

x=1         x=52

The y-values for x=1.​​ 

y=6-x 

y=5

Hence, the point​​ P(1, 5).​​ 

The y-values for x=5/2.​​ 

 y=6-x ,  y=3.5

Hence, the point​​ Q​​ (5/2,7/2).​​ 

b-​​ 

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≥6x-2x2+1

• below the straight line​​ l; therefore,​​ 

x+y≤6

• right to​​ the point Q (2.5, 0), so​​ 

x≥2.5

• above the x-axis, thus

y≥0