CAIE A-Level Mathematics 3.9.6 Complex numbers
Practise finding exact square roots of complex numbers in Cartesian or polar form, showing the equations that determine both roots.
- Syllabus
- 2028–2030
- Course
- Mathematics 9709
- Level
- A2
Exam points
- square x+iy and equate real and imaginary parts to form two equations
- eliminate x or y to find exact Cartesian square roots in conjugate-sign pairs
- halve the argument and root the modulus when square roots are requested in polar form
3.9.6—Complex numbers question 1
[Maximum number: 5]
The square roots of 24-7i can be expressed in the Cartesian form x+i y, where x and y are real and exact.
By first forming a quartic equation in x or y, find the square roots of 24-7 i in exact Cartesian form.
Square x+i y obtaining three terms when simplified and equate real
and imaginary parts to 24 and -7 respectively
M1
Having used i^2=-1.
Obtain equations x^2-y^2=24 and 2 x y=-7
A1
Marking guidance:
Allow 2 x y i=-7 i.
Eliminate one variable by correct method and find a horizontal equation in the other
M1
All powers of x or y are positive and are in the numerator.
Obtain 4 x^4-96 x^2-49=0 or 4 y^4+96 y^2-49=0
or 3-term equivalents
A1
Obtain answers 7 22-22 i and -7 22+22 i or exact equivalents
and no others
A1
E.g. +/-(7 22-22 i), but not +/-(7 22 22 i) or
( +/- 7 22 22 i).
Allow coordinates or x=, y= paired correctly. ISW converting to different form.
Must simplify 49.
All question bank results loaded