Cambridge Additional Mathematics

Quadratics (Chapter 3) 97

5 1800 m of fencing is available to fence six identical pens

as shown in the diagram.

a Explain why 9 x+8y= 1800:

b Show that the area of each pen is given by

A=¡^98 x^2 + 225xm^2.

c If the area enclosed is to be maximised, what are the

dimensions of each pen?

6 500 m of fencing is available

to make 4 rectangular pens

of identical shape. Find the

dimensions that maximise the area

of each pen if the plan is:

ab

7 The graphs of y=x^2 ¡ 3 x and y =2x¡x^2 are

illustrated.

a Show that the graphs meet wherex=0andx=2^12.

b Find the maximum vertical separation between the

curves for 06 x 6212.

8 Infinitely many rectangles may be inscribed within the

right angled triangle shown alongside. One of them is

illustrated.

a Let AB=xcm and BC=ycm.

Use similar triangles to findyin terms ofx.

b Find the dimensions of rectangle ABCD of maximum

area.

Discovery 4 Sum and product of roots

What to do:

1 Suppose ax^2 +bx+c=0 has rootspandq.

Prove that p+q=

¡b

a

and pq=

c

a

.

2 Suppose 2 x^2 ¡ 5 x+1=0 has rootspandq.

Without finding the values ofpandq, find:

a p+q b pq c p^2 +q^2 d

1

p

+

1

q

3 Findallquadratic equations with roots which are:

a one more than the roots of 2 x^2 ¡ 5 x+1=0

b the squares of the roots of 2 x^2 ¡ 5 x+1=0

c the reciprocals of the roots of 2 x^2 ¡ 5 x+1=0.

xm

ym

y

23 x

y=x -3x 2

y=2x-x 2

O

A

D C

B

8 cm

6 cm

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_03\097CamAdd_03.cdr Friday, 3 January 2014 12:04:45 PM GR8GREG