Date | November 2018 | Marks available | 1 | Reference code | 18N.3.AHL.TZ0.Hsrg_4 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | Give a reason and State | Question number | Hsrg_4 | Adapted from | N/A |
Question
Consider the functions f, g : R×R→R×R defined by
f((x,y))=(x+y,x−y) and g((x,y))=(xy,x+y).
Find (f∘g)((x,y)).
[3]
a.i.
Find (g∘f)((x,y)).
[2]
a.ii.
State with a reason whether or not f and g commute.
[1]
b.
Find the inverse of f.
[3]
c.
Markscheme
(f∘g)((x,y))=f(g((x,y))) (=f((xy,x+y))) (M1)
=(xy+x+y,xy−x−y) A1A1
[3 marks]
a.i.
(g∘f)((x,y))=g(f((x,y)))
=g((x+y,x−y))
=((x+y)(x−y),x+y+x−y)
=(x2−y2,2x) A1A1
[2 marks]
a.ii.
no because f∘g≠g∘f R1
Note: Accept counter example.
[1 mark]
b.
f((x,y))=(a,b)⇒(x+y,x−y)=(a,b) (M1)
{x=a+b2y=a−b2 (M1)
f−1((x,y))=(x+y2,x−y2) A1
[3 marks]
c.
Examiners report
[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.