A-LevelMathematicsComplex numbers9709/33 · May/June 2025 · Q66 Marks
Find the complex numbers z for which (z+4)/(z+4i) is real and |z|= √10. Give your answers in the form z = x+iy, where x and y are real.
- Find the complex numbers z for which (z+4)/(z+4i) is real and |z|= √10. Give your answers in the form z = x+iy, where x and y are real.[6]
Mark scheme
Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.
- [M1*] Multiply numerator (and denominator) by the conjugate of the denominator.
- [DM1] Equate imaginary part of numerator to zero. Numerator = x(x+4)+y(y+4)+i[xy-(y+4)(x+4)].
- [A1] Obtain x + y = −4. OE
- [DM1] Correct use of modulus and solve for x or y. x²+(-x-4)² =10 => x²+4x+3=0 or (−y−4)² + y² = 10 => y²+4y+3=0.
- [A1] One correct solution A1. SC A1 A0 for both pairs of x and y correct but not in given form.
- [A1] Two solutions only.
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