Processing math: 45%

User interface language: English | Español

Date November 2019 Marks available 2 Reference code 19N.1.AHL.TZ0.H_5
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Find Question number H_5 Adapted from N/A

Question

Consider the equation z4=4, where zC.

Solve the equation, giving the solutions in the form a+ib, where abR.

[5]
a.

The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.

[2]
b.

Markscheme

METHOD 1

|z|=44(=2)       (A1)

arg(z1)=π4       (A1)

first solution is 1+i       A1

valid attempt to find all roots (De Moivre or +/− their components)        (M1)

other solutions are 1+i1i1i       A1

 

METHOD 2

z4=4

(a+ib)4=4

attempt to expand and equate both reals and imaginaries.        (M1)

a4+4a3bi6a2b24ab3i+b4=4

(a46a4+a4=4)a=±1 and (4a3b4ab3=0)a=±b       (A1)

first solution is 1+i       A1

valid attempt to find all roots (De Moivre or +/− their components)        (M1)

other solutions are 1+i1i1i       A1

 

[5 marks]

a.

complete method to find area of ‘rectangle'        (M1)

=4      A1

[2 marks]

b.

Examiners report

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

Syllabus sections

Topic 1—Number and algebra » AHL 1.12—Complex numbers – Cartesian form and Argand diag
Show 43 related questions
Topic 1—Number and algebra

View options