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Date November 2008 Marks available 13 Reference code 08N.1.hl.TZ0.13
Level HL only Paper 1 Time zone TZ0
Command term Show that Question number 13 Adapted from N/A

Question

(a)     Use de Moivre’s theorem to find the roots of the equation z4=1i .

(b)     Draw these roots on an Argand diagram.

(c)     If z1 is the root in the first quadrant and z2 is the root in the second quadrant, find z2z1 in the form a + ib .

[12]
Part A.

(a)     Expand and simplify (x1)(x4+x3+x2+x+1) .

(b)     Given that b is a root of the equation z51=0 which does not lie on the real axis in the Argand diagram, show that 1+b+b2+b3+b4=0 .

(c)     If u=b+b4 and v=b2+b3 show that

(i)     u + v = uv = −1;

(ii)     uv=5 , given that uv>0 .

[13]
Part B.

Markscheme

(a)     z=(1i)14

Let 1i=r(cosθ+isinθ)

r=2     A1

θ=π4     A1

z=(2(cos(π4)+isin(π4)))14     M1

=(2(cos(π4+2nπ)+isin(π4+2nπ)))14

=218(cos(π16+nπ2)+isin(π16+nπ2))     M1

=218(cos(π16)+isin(π16))

Note: Award M1 above for this line if the candidate has forgotten to add 2π and no other solution given.

 

=218(cos(7π16)+isin(7π16))

=218(cos(15π16)+isin(15π16))

=218(cos(9π16)+isin(9π16))     A2

Note: Award A1 for 2 correct answers. Accept any equivalent form.

 

[6 marks]

 

(b)

    A2

 

Note: Award A1 for roots being shown equidistant from the origin and one in each quadrant.

A1 for correct angular positions. It is not necessary to see written evidence of angle, but must agree with the diagram.

 

[2 marks]

 

(c)     z2z1=218((cos15π16)+isin(15π16))218((cos7π16)+isin(7π16))     M1A1

=cosπ2+isinπ2     (A1)

=i     A1     N2

(a=0, b=1)

[4 marks] 

Part A.

(a)     (x1)(x4+x3+x2+x+1)

=x5+x4+x3+x2+xx4x3x2x1     (M1)

=x51     A1

[2 marks]

 

(b)     b is a root

f(b)=0

b5=1     M1

b51=0     A1

(b1)(b4+b3+b2+b+1)=0

b1     R1

1+b+b2+b3+b4=0 as shown.     AG

[3 marks]

 

(c)     (i)     u+v=b4+b3+b2+b=1     A1

uv=(b+b4)(b2+b3)=b3+b4+b6+b7     A1

Now b5=1     (A1)

Hence uv=b3+b4+b+b2=1     A1

Hence u+v=uv=1     AG

 

(ii)     (uv)2=(u2+v2)2uv     (M1)

=((u+v)22uv)2uv(=(u+v)24uv)     (M1)A1

Given uv>0 

uv=(u+v)24uv

=(1)24(1)

=1+4     A1

=5     AG

Note: Award A0 unless an indicator is given that uv=5 is invalid.

 

[8 marks]

Total [13 marks]

Part B.

Examiners report

The response to Part A was disappointing. Many candidates did not know that they had to apply de Moivre’s theorem and did not appreciate that they needed to find four roots.

Part A.

 

Part B started well for most candidates, but in part (b) many candidates did not appreciate the significance of b not lying on the real axis. A majority of candidates started (c) (i) and many fully correct answers were seen. Part (c) (ii) proved unsuccessful for all but the very best candidates.

 

Part B.

Syllabus sections

Topic 1 - Core: Algebra » 1.5 » Sums, products and quotients of complex numbers.

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