Pre-U Pre-U 9794/2 Specimen — Question 4 6 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
SessionSpecimen
Marks6
TopicComplex Numbers Argand & Loci
TypeComplex conjugate properties
DifficultyStandard +0.3 This is a straightforward algebraic manipulation question involving complex conjugates. Students need to rearrange the equation to isolate p, then separate real and imaginary parts to find p explicitly, before calculating modulus and argument using standard formulas. While it requires careful algebraic handling of conjugates, it's a routine exercise with no novel insight required, making it slightly easier than average.
Spec4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)
4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).
Use cartesian form \(p = a + \mathrm{i}b\) and \(p^* = a - \mathrm{i}b\): M1
Substitution followed by the equating of real and imaginary parts: M1
\(a + \mathrm{i}b + \mathrm{i}(a - \mathrm{i}b) = 2(a + \mathrm{i}b - \mathrm{i}(a - \mathrm{i}b)) - 8\)
\(a - 3b = 8,\ -3a + b = 0\) A1
Attempt at solving linear equations: M1
\((a = -1,\ b = -3)\)
AnswerMarks Guidance
\(p = \sqrt{10}\) A1
\(\arg p = \tan^{-1} 3 - \pi\) A1
Total: 6 marks
Use cartesian form $p = a + \mathrm{i}b$ and $p^* = a - \mathrm{i}b$: M1

Substitution followed by the equating of real and imaginary parts: M1

$a + \mathrm{i}b + \mathrm{i}(a - \mathrm{i}b) = 2(a + \mathrm{i}b - \mathrm{i}(a - \mathrm{i}b)) - 8$

$a - 3b = 8,\ -3a + b = 0$ A1

Attempt at solving linear equations: M1

$(a = -1,\ b = -3)$

$|p| = \sqrt{10}$ A1

$\arg p = \tan^{-1} 3 - \pi$ A1

**Total: 6 marks**
4 The complex number $p$ satisfies the equation

$$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$

Determine the exact values of the modulus and argument of $p$.

\hfill \mbox{\textit{Pre-U Pre-U 9794/2  Q4 [6]}}
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