A-LevelMathematicsComplex numbers9709/32 · Oct/Nov 2024 · Q35 Marks

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.

(a)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]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[*M1] Square x + iy and equate real and imaginary parts to 6 and -8 respectively
[A1] Obtain equations x² – y² = 6 and 2xy = -8
[DM1] Eliminate one variable and find an equation in the other (from 2 equations each in 2 unknowns)
[A1] Obtain x⁴ – 6x² – 16 = 0 or y⁴ + 6y² – 16 = 0
[A1] Obtain answers ±(2√2 - √2i) or exact equivalents

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Topic: Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source: A-Level Mathematics, Oct/Nov 2024 Paper 3 Variant 2, question 3.

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