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Date May 2019 Marks available 3 Reference code 19M.2.AHL.TZ2.H_8
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Express Question number H_8 Adapted from N/A

Question

Find the roots of the equation w 3 = 8 i , w C . Give your answers in Cartesian form.

[4]
a.

One of the roots w 1 satisfies the condition Re ( w 1 ) = 0 .

Given that  w 1 = z z i , express z in the form  a + b i , where a , b Q .

[3]
b.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

w 3 = 8 i

writing  8 i = 8 ( cos ( π 2 + 2 π k ) + i sin ( π 2 + 2 π k ) )               (M1)

Note: Award M1 for an attempt to find cube roots of w using modulus-argument form.

cube roots   w = 2 ( cos ( π 2 + 2 π k 3 ) + i sin ( π 2 + 2 π k 3 ) )               (M1)

i.e. w = 3 + i, 3 + i, 2 i          A2

Note: Award A2 for all 3 correct, A1 for 2 correct.

Note: Accept  w = 1.73 + i and  w = 1.73 + i .

 

METHOD 2

w 3 + ( 2 i ) 3 = 0

( w + 2 i ) ( w 2 2 w i 4 ) = 0               M1

w = 2 i ± 12 2               M1

w = 3 + i, 3 + i, 2 i          A2

Note: Award A2 for all 3 correct, A1 for 2 correct.

Note: Accept  w = 1.73 + i and  w = 1.73 + i .

 

[4 marks]

a.

w 1 = 2 i

z z i = 2 i       M1

z = 2 i ( z i )

z ( 1 + 2 i ) = 2

z = 2 1 + 2 i       A1

z = 2 5 + 4 5 i       A1

Note: Accept a = 2 5 , b = 4 5 .

[3 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 1—Number and algebra » AHL 1.13—Polar and Euler form
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Topic 1—Number and algebra » AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers
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