Standard +0.3 This is a standard complex number square root problem with a prescribed method (forming a quartic equation). While it requires algebraic manipulation through multiple steps, the technique is routine for P3 level and follows a well-established procedure. The 5 marks reflect computational work rather than conceptual difficulty, making it slightly easier than average.
The square roots of \(24 - 7i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact.
By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7i\) in exact Cartesian form. [5]
Square x + iy obtaining three terms when simplified and equate real
and imaginary parts to 24 and −7 respectively
M1
Having used i2 = −1.
Obtain equations x2 – y2 = 24 and 2xy = –7
A1
Allow 2xyi = –7i.
Eliminate one variable by correct method and find a horizontal
Answer
Marks
Guidance
equation in the other
M1
All powers of x or y are positive and are in the numerator.
Obtain 4x4 – 96x2 – 49 = 0 or 4y4 + 96y2 – 49 = 0
Answer
Marks
or 3-term equivalents
A1
7 2 2 7 2 2
Obtain answers i and i or exact equivalents
2 2 2 2
Answer
Marks
Guidance
and no others
A1
7 2 2 7 2 2
E.g. i, but not i or
2 2 2 2
7 2 2
i.
2 2
Allow coordinates or x =…, y =… paired correctly.
ISW converting to different form.
Must simplify 49.
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 3:
3 | Square x + iy obtaining three terms when simplified and equate real
and imaginary parts to 24 and −7 respectively | M1 | Having used i2 = −1.
Obtain equations x2 – y2 = 24 and 2xy = –7 | A1 | Allow 2xyi = –7i.
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 4x4 – 96x2 – 49 = 0 or 4y4 + 96y2 – 49 = 0
or 3-term equivalents | A1
7 2 2 7 2 2
Obtain answers i and i or exact equivalents
2 2 2 2
and no others | A1 | 7 2 2 7 2 2
E.g. i, but not i or
2 2 2 2
7 2 2
i.
2 2
Allow coordinates or x =…, y =… paired correctly.
ISW converting to different form.
Must simplify 49.
5
Question | Answer | Marks | Guidance
The square roots of $24 - 7i$ can be expressed in the Cartesian form $x + iy$, where $x$ and $y$ are real and exact.
By first forming a quartic equation in $x$ or $y$, find the square roots of $24 - 7i$ in exact Cartesian form. [5]
\hfill \mbox{\textit{CAIE P3 2024 Q3 [5]}}