WMA11 June 2021

3.​​ WMA11/01​​ Edexcel IAL P1 June 2021 Q3​​ (Trigonometric Ratios:​​ Cosine Rule​​ & Area of Triangles)

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

Solutions relying on calculator technology are not acceptable.

Figure 1 shows the plan view of a flower bed.​​ 

The flowerbed is in the shape of a triangle ABC with​​ 

• AB = p metres​​ 

• AC = q metres​​ 

• BC =​​ 22​​ metres​​ 

• angle BAC = 60°​​ 

(a) Show that​​ 

p2​​ + q2​​ – pq = 8

(2)

Given that side AC is 2 metres longer than side AB, use algebra to find​​ 

(b)

(i) the exact value of p,​​ 

(ii) the exact value of q.​​ 

(5)

Using the answers to part (b),​​ 

(c) calculate the exact area of the flower bed.​​ 

(2)

SOLUTION

a-​​ Using cosine rule.​​ 

a2=b2+c2-2bccos⁡A

222=p2+q2-2pqcos⁡60

42=p2+q2-2pq x12

8=p2+q2-pq

p2+q2-pq=8

b- i-​​ 

It is given that AC is 2m longer than AB where​​ AB=p &​​ AC=q.

So,​​ 

AC=2+AB

q=2+p

Now, substituting the value of q in the equation we have proved in previous part.​​ 

p2+q2-pq=8

p2+p+22-pp+2=8 

p2+p2+4p+4-p2-2p=8

p2+2p-4=0

p2+2p=4

p+12-1=4

p+12=5

p+1=±5

p=-1±5

Since p and q are the lengths of traingle so they have to be positive, we will reject the negative values.​​ 

P=-1+5m

ii-​​ 

Now, putting the value of p in q=2+p.

q=P+2

q=-1+ 5+2

q=1+5m 

Hence,​​ p=-1+5m​​ &​​ q=1+5m.

c-​​ 

Using the formula of area of traingle.​​ 

A=12absin⁡c

In case of our question,​​ 

A=12pqsin⁡A

Area =125-15+1sin⁡60o

Area=12 5-1 x32

Area=2 x32= 3

Area =3 m2