A-Level Mathematics October/November 2024 Q4: Find the complex number z satisfying the equation (z-3i)/(z+3i) = (2-9i)/5. Give your a…

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  Let the complex number $z = x + iy$, where $x$ and $y$ are real.

  Substitute $z = x + iy$ into the given equation: $\frac{(x+iy)-3i}{(x+iy)+3i} = \frac{2-9i}{5}$

  Group the real and imaginary parts in the numerator and denominator: $\frac{x + i(y-3)}{x + i(y+3)} = \frac{2-9i}{5}$

  To eliminate the fraction, cross-multiply: $5(x + i(y-3)) = (2-9i)(x + i(y+3))$

  Expand both sides of the equation:

  LHS: $5x + i(5y - 15)$

  RHS: $2(x + i(y+3)) - 9i(x + i(y+3))$ $= 2x + i(2y+6) - 9ix - 9i^2(y+3)$

  Using the property $i^2 = -1$: $= 2x + i(2y+6) - 9ix + 9(y+3)$ $= 2x + 9y + 27 + i(2y + 6 - 9x)$

  Now, equate the expanded LHS and RHS: $5x + i(5y - 15) = (2x + 9y + 27) + i(2y - 9x + 6)$

  Equate the real parts and the imaginary parts to form two simultaneous linear equations:

  Real parts: $5x = 2x + 9y + 27$ $3x - 9y = 27$ $x - 3y = 9 \quad \cdots (1)$

  Imaginary parts: $5y - 15 = 2y - 9x + 6$ $9x + 3y = 21$ $3x + y = 7 \quad \cdots (2)$

  Now, solve the simultaneous equations. From equation (2), we can express $y$ in terms of $x$: $y = 7 - 3x$

  Substitute this expression for $y$ into equation (1): $x - 3(7 - 3x) = 9$ $x - 21 + 9x = 9$ $10x = 30$ $x = 3$

  Substitute $x=3$ back into the expression for $y$: $y = 7 - 3(3) = 7 - 9 = -2$

  Therefore, the complex number $z$ is $x + iy$.

  $z = 3 - 2i$

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  How the marks are awarded

  • M1 — Substituting z = x + iy into the equation and cross-multiplying to get $5(x + i(y-3)) = (2-9i)(x + i(y+3))$.
  • M1 — Using $i^2 = -1$ when expanding the right-hand side, specifically turning $-9i^2(y+3)$ into $+9(y+3)$.
  • A1 — Correctly expanding and collecting terms to obtain the equation $5x + i(5y - 15) = (2x + 9y + 27) + i(2y - 9x + 6)$.
  • DM1 — Equating the real and imaginary parts to form a pair of simultaneous linear equations: $x - 3y = 9$ and $3x + y = 7$.
  • A1 — Correctly solving the simultaneous equations to find $x=3$ and $y=-2$, and stating the final answer as $z = 3 - 2i$.

  Common mistakes

  • Sign errors when expanding brackets, particularly with the $i^2$ term, for example writing $(2-9i)(i(y+3)) = 2i(y+3) - 9(y+3)$ instead of $+9(y+3)$.
  • Incorrectly forming the simultaneous equations by mixing up real and imaginary parts, for example setting $5x = 2y - 9x + 6$.
  • Making algebraic errors when rearranging or solving the simultaneous equations, such as incorrectly substituting one equation into another.
  • Solving for $x$ and $y$ correctly but failing to write the final answer in the required form $z = x+iy$.

  Examiner tip: When an equation involves a complex variable z, a reliable strategy is to substitute z = x + iy and equate the real and imaginary parts to form solvable simultaneous equations.

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