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 \(6 - 8i\) 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 \(6 - 8i\) in exact Cartesian form. [5]
Square x + iy and equate real and imaginary parts to 6 and −8 respectively
*M1
Obtain equations x2 – y2 = 6 and 2xy = –8
A1
OE
Eliminate one variable and find an equation in the other (from 2 equations each in 2
Answer
Marks
Guidance
unknowns)
DM1
Condone a slip but not seriously incorrect
algebra, e.g. use of x=−4yis M0.
Answer
Marks
Guidance
Obtain x4 – 6x2 – 16 = 0 or y4 + 6y2 – 16 = 0
A1
Accept 3-term equivalents e.g. x4 =6x2 +16.
Condone missing ‘= 0’ if implied by subsequent
work.
( )
Answer
Marks
Guidance
Obtain answers 2 2− 2i or exact equivalents
A1
Allow if values of x and y stated separately but
the pairing is clear.
Ignore additional correct solutions for x and y
not real, but A0 if any additional incorrect
answers.
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 3:
3 | Square x + iy and equate real and imaginary parts to 6 and −8 respectively | *M1 | Condone +8 in place of -8 and/or i2 =1.
Obtain equations x2 – y2 = 6 and 2xy = –8 | A1 | OE
Eliminate one variable and find an equation in the other (from 2 equations each in 2
unknowns) | DM1 | Condone a slip but not seriously incorrect
algebra, e.g. use of x=−4yis M0.
Obtain x4 – 6x2 – 16 = 0 or y4 + 6y2 – 16 = 0 | A1 | Accept 3-term equivalents e.g. x4 =6x2 +16.
Condone missing ‘= 0’ if implied by subsequent
work.
( )
Obtain answers 2 2− 2i or exact equivalents | A1 | Allow if values of x and y stated separately but
the pairing is clear.
Ignore additional correct solutions for x and y
not real, but A0 if any additional incorrect
answers.
5
Question | Answer | Marks | Guidance
The square roots of $6 - 8i$ 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 $6 - 8i$ in exact Cartesian form. [5]
\hfill \mbox{\textit{CAIE P3 2024 Q3 [5]}}