User interface language: English | Español

Date May Example question Marks available 3 Reference code EXM.2.AHL.TZ0.20
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Deduce Question number 20 Adapted from N/A

Question

Let γ = 1 + i 3 2 .

The matrix A is defined by A ( γ 1 0 1 γ ) .

Deduce that

Show that γ 2 = γ 1 .

[2]
a.ii.

Hence find the value of ( 1 γ ) 6 .

[4]
a.iii.

A3 = –I.

[3]
c.i.

A–1 = IA.

[2]
c.ii.

Markscheme

METHOD 1

as  γ is a root of  z 2 z + 1 = 0 then  γ 2 γ + 1 = 0        M1R1

γ 2 = γ 1      AG

Note: Award M1 for the use of  z 2 z + 1 = 0 in any way.

Award R1 for a correct reasoned approach.

 

METHOD 2

γ 2 = 1 + i 3 2        M1

γ 1 = 1 + i 3 2 1 = 1 + i 3 2         A1 

 

[2 marks]

a.ii.

METHOD 1

( 1 γ ) 6 = ( γ 2 ) 6        (M1)

                = ( γ ) 12      A1

                = ( γ 3 ) 4        (M1)

                = ( 1 ) 4

                = 1      A1

 

METHOD 2

( 1 γ ) 6

= 1 6 γ + 15 γ 2 20 γ 3 + 15 γ 4 6 γ 5 + γ 6        M1A1

Note: Award M1 for attempt at binomial expansion.

use of any previous result e.g.  = 1 6 γ + 15 γ 2 + 20 15 γ + 6 γ 2 + 1        M1

= 1      A1

Note: As the question uses the word ‘hence’, other methods that do not use previous results are awarded no marks.

 

[4 marks]

a.iii.

A2 = A I  

A3 = A2 – A      M1A1           

          = AIA         A1           

          = I           AG

Note: Allow other valid methods.

[3 marks]

c.i.

I = A A2

A–1 = A–1AA–1A2        M1A1

⇒ A–1 = I A         AG

Note: Allow other valid methods.

[2 marks]

c.ii.

Examiners report

[N/A]
a.ii.
[N/A]
a.iii.
[N/A]
c.i.
[N/A]
c.ii.

Syllabus sections

Topic 1—Number and algebra » AHL 1.12—Complex numbers introduction
Show 23 related questions
Topic 1—Number and algebra » AHL 1.14—Introduction to matrices
Topic 1—Number and algebra

View options