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Date May 2011 Marks available 8 Reference code 11M.1.hl.TZ0.3
Level HL only Paper 1 Time zone TZ0
Command term List and Prove that Question number 3 Adapted from N/A

Question

Prove that the number 14641 is the fourth power of an integer in any base greater than 6.

[3]
a.

For a,bZ the relation aRb is defined if and only if ab=2k , kZ .

  (i)     Prove that R is an equivalence relation.

  (ii)     List the equivalence classes of R on the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

[8]
b.

Markscheme

14641 (base a>6 ) =a4+4a3+6a2+4a+1 ,     M1A1

=(a+1)4     A1

this is the fourth power of an integer     AG

[3 marks]

a.

(i)     aRa since aa=1=20 , hence R is reflexive     A1

aRbab=2kba=2kbRa

so R is symmetric     A1

aRb and bRcab=2m, mZ and bRcbc=2nnZ     M1

ab×bc=ac=2m+n , m+nZ    A1

aRc so transitive     R1

hence R is an equivalence relation     AG

 

(ii)     equivalence classes are {1, 2, 4, 8} , {3, 6} , {5, 10} , {7} , {9}     A3

Note: Award A2 if one class missing, A1 if two classes missing, A0 if three or more classes missing.

 

[8 marks]

b.

Examiners report

This was not difficult but a surprising number of candidates were unable to do it. Care with notation and logic were lacking.

a.

The question was at first straightforward but some candidates mixed up the properties of an equivalence relation with those of a group. The idea of an equivalence class is still not clearly understood by many candidates so that some were missing.

b.

Syllabus sections

Topic 4 - Sets, relations and groups » 4.2 » Relations: equivalence relations; equivalence classes.

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