WMA11 Jan 2021

8. WMA11/01 Edexcel IAL P1 January 2021 IAL Q8 (Graphs and Transformation: Cubic Graphs)

Figure 4 shows a sketch of part of the curve C with equation y = f(x), where​​ 

fx= 3x – 22x – 4

• Deduce the​​ values of x for which f(x) > 0​​ 

(1)

• Expand f(x) to the form​​ 

ax3+ bx2+ cx + d 

where a, b, c and d are integers to be found.

(3)

The line l, also shown in Figure 4, passes through the y intercept of C and is parallel to the x-axis.​​ 

The line l cuts C again at points P and​​ Q, also shown in Figure 4.​​ 

• Using algebra and showing your working, find the length of line PQ. Write your answer in the form​​ k3, where k is a constant to be found.​​ 

(Solutions relying entirely on calculator technology are not acceptable.)

(5)

SOLUTION​​ 

a-​​ 

3x-22x-4=0 

3x-22=0 & x-4=0 

x=23 & x=4

Since​​ fx>0, we are going to consider only that region which is above the x-axis.​​ 

Hence,​​ x>4​​ for which​​ fx>0.

b-​​ 

fx=3x-22x-4

fx=x-49x2-12x+4

fx=9x3-12x2+4x-36x2+48x-16

fx=9x3-48x2+52 x-16

Hence,​​ a=9,   b=-48m  c=52   d=-16.

c-​​ Since the line passes through the y-intercept of C and is also parallel to x-axis; therefore, the equation of a line is​​ 

y=-16

If you are​​ confused how toidentify the y-intercept of a curve, so always remember that it is a constant term in any equation. As shown below

Since curve and line intersects each other at point P and Q, so lets equate the equations of both.

9x3-48x2 +52x-16=-16

9x3-48x2+52x=0

x9x2-48x+52=0

9x2-48x+52=0

Solving quadratic equation using quadratic formula, where​​ a=9,​​ b=-98, and​​ c=52.

x=-b±b2-4ac2a

x=48±-482-495229

x=8+233 or 8-233

x=83±233

So the length of PQ will be found using the formula of length/distance between two points.​​ 

PQ=x2-x12+y2-y12

Since line is horizontal, no change in y-cordinates, so its 0.​​ 

PQ=x2-x12+02

PQ=x2-x12

PQ=x2-x1

PQ=8+233-8-233

PQ=8+23-8+233

PQ=433

PQ=433