WMA11 Jan 2019

10. WMA11/01 Edexcel IAL P1 January​​ 2019, Q10​​ (Radians)

A sector AOB, of a circle centre O, has radius r cm and angle θ radians.​​ 

Given that the area of the sector is 6cm2​​ and that​​ the perimeter of the sector is 10cm,​​ 

(a) show that​​ 

3θ2  - 13θ + 12 = 0 

(4)​​ 

(b) Hence find possible values of r and θ.​​ 

(3)

SOLUTION​​ 

a-​​ It is given that area of the sector is 6cm2​​ and the perimeter of the sector is 10cm, so finding the respective expressions first.

A=12r2θ

l=rθ

12r2θ=6

For​​ perimeter, consider the sector below.​​ 

P=2r+l

2r+rθ=10

r2+ θ=10

Now, if you pay attention to the expression that is asked to be shown in the question, the expression doesn’t have any value of r. This means, we have to substitute the value of r from one​​ equation to another to​​ get an expression free of any r term. To do so, we will first make r as the subject from the equation derived from perimeter.

r2+ θ=10

r=102+θ

Now, substitute that in the equation derived from the area of a sector.

12r2θ=6

12102+θ2x θ=6

100θ2+4θ+4x θ=12

100θ=12θ2+48θ+48

0=12θ2-52θ+48

3θ2-13θ=12=0

b- Hence, to find the value of θ. solve the above​​ quadratic equation simply. ​​ 

θ-33θ-4=0

θ=3,     θ=43

Now, put the value of​​ θ​​ one by one to get respective values of r in the equation of r.​​ 

r=102+θ

When​​ θ=3,

r=102+3

r=2

When ,

r=102+43

r=10103

r=10÷103

r=10 x310

r=3

Hence, the value r and θ are​​ 

θ=3&43

r=2&3