WMA11 October 2022

9. WMA11/01 Edexcel IAL P1 October 2022, Q9 (Quadratics: Completing the Square & Finding Vertex)

Figure 3 shows a sketch of the curve C with equation​​ 

y =12x2 – 10x + 22

(a) Write​​ 12x2 – 10x + 22​​ in the form​​ 

ax + b2 + c 

where​​ a, b and c are constants to be found.​​ 

(3)

The point M is the minimum turning point of C, as shown in Figure 3.​​ 

(b) Deduce the coordinates of M​​ 

(2)​​ 

The line l is the normal to C at the point P, as shown in Figure 3. Given that​​ l​​ has equation​​ y = k –18 x, where k​​ is a constant,​​ 

(c)​​ 

• find the coordinates of P​​ 

• find the value of k​​ 

(6)

Figure 4 is a copy of Figure 3. The finite region R, shown shaded in Figure 4, is bounded by l, C and the line through M parallel to the y‑axis.​​ 

(d) Identify the inequalities that​​ define R.​​ 

(3)

SOLUTION

a-​​ 

y=12x2-10x+22

y=12x2-20x+22

y=12x-2022-2022+22

y=12x-102-100+22

On expanding the square brackets.

y=12x-102-50+22

y=12x-102-28

b-​​ 

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.

Thus, the the turning point of C is​​ 

M10,-28

c-​​ 

i-​​ Since the normal line has the slope​​ mN=-18,​​ 

The equation of normal is given as ​​ y = k –18 x,​​ and the coefficient of x is equal to gradient of normal. So the slope/gradient of normal is​​ 

mN=-18

Therefore, the gradient of tangent could be found by the rule that the product of​​ the gradients of two lines perpendicular is always -1.​​ 

mN×mT=-1

-18×mT=-1

mT=8

We may differentiate the curve and find the expression for the gradient of the curve.​​ 

y=12x2-10x+22

dydx=x-10

As found above​​ mT=8, so​​ 

8=x-10

x=18

Putting this x value on the equation of a curve to find the y-coordinate of point P.

y=12182-1018+22

y=4

Hence, the cordinate of point P is​​ P18,4​​ 

c- ii-​​ 

Substituting the coordinate of P in the given equation of a line, passes through P.​​ 

y=k-18x

4=k-1818

4=k-94

k=4+94

k=254

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/inside the quadratic curve; therefore,

y≥12x2-10x+22

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

y≤254-18x

• right side of line​​ x=10, so​​ 

x≥10

For better understanding the graph below has been marked with equation of lines and curve.​​