F.4 Maths

1(a) f(x)=x^2 - 9x +4,f(a)=f(b)
a^2-9a+4=b^2-9b+4
(a-b)(a+b)=9(a-b)
a+b=9

(b) f(a+b)=f(9)=81-81+4=4

2 h(a-1) - h(a+1)
=[(a-1)^2-3(a-1)-6]-[(a+1)^2-3(a+1)-6]
=[a^2-2a+1-3a+3-6]-[a^2+2a+1-3a-3-6]
=-4a

(b) h(a+1)=4
(a+1)^2-3(a+1)-6=4
Let x=a+1, x^2-3x-10=0
(x-5)(x+2)=0
x=5 or -2
a=4 or -3

1a.
f(a)=a^2-9a+4
f(b)=b^2-9b+4

a^2-9a+4=b^2-9b+4
since f(a)=f(b),but a not equal to b,let b be a negative no.
a^2-9a+4=(-b)^2+9b+4
a^2-b^2=9a+9b
(a-b)(a+b)=9(a+b)
-(a-b)(a+b)=-9(-a-b)
a+b=-9

b.
f(a+b)=(a+b)^2-9(a+b)+4
=(-9)^2-9(-9)+4
=166


2a.
h(a-1)-h(a+1)=(a-1)^2-3(a-1)-6-[(a+1)^2-3(a+1)-6]
=a^2-2a+1-3a+3-6-[a^2+2a+2-3a-3-6]
=a^2-2a+1-3a+3-6-a^2-2a-2+3a+3+6
=-4a+5

b.
h(a+1)=4
(a+1)^2-3(a+1)-6=4
a^2+2a+12-3a-3-6-4=0
a^2-a-11=0
a=4 or -3