A-Level Mathematics May/June 2024 Q3: The square roots of 24-7i can be expressed in the Cartesian form x+iy, where x and y ar…

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  Let the square roots of 24-7i be x+iy, where $x, y \in \mathbb{R}$.

  Then $(x+iy)^2 = 24-7i$.

  Expanding the left side: $x^2 + 2xyi + (iy)^2 = 24-7i$ $x^2 + 2xyi - y^2 = 24-7i$ $(x^2 - y^2) + (2xy)i = 24 - 7i$

  Equating the real and imaginary parts: Real part: $x^2 - y^2 = 24$ ---(1) Imaginary part: $2xy = -7$ ---(2)

  From equation (2), we can express y in terms of x: $y = -\frac{7}{2x}$

  Substitute this expression for y into equation (1): $x^2 - \left(-\frac{7}{2x}\right)^2 = 24$ $x^2 - \frac{49}{4x^2} = 24$

  To form a quartic equation, multiply the entire equation by $4x^2$ to eliminate the denominator: $4x^2(x^2) - 4x^2\left(\frac{49}{4x^2}\right) = 4x^2(24)$ $4x^4 - 49 = 96x^2$ $4x^4 - 96x^2 - 49 = 0$

  This is a quadratic equation in terms of $x^2$. Let $u = x^2$. The equation becomes: $4u^2 - 96u - 49 = 0$

  Solve for u using the quadratic formula, $u = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$: $u = \frac{-(-96) \pm \sqrt{(-96)^2 - 4(4)(-49)}}{2(4)}$ $u = \frac{96 \pm \sqrt{9216 + 784}}{8}$ $u = \frac{96 \pm \sqrt{10000}}{8}$ $u = \frac{96 \pm 100}{8}$

  So, $u = \frac{96+100}{8} = \frac{196}{8} = \frac{49}{2}$ or $u = \frac{96-100}{8} = \frac{-4}{8} = -\frac{1}{2}$.

  Since x is a real number, $x^2$ must be non-negative. Therefore, we reject $u = -\frac{1}{2}$. $x^2 = u = \frac{49}{2}$ $x = \pm \sqrt{\frac{49}{2}} = \pm \frac{7}{\sqrt{2}} = \pm \frac{7\sqrt{2}}{2}$

  Now, find the corresponding values of y using $y = -\frac{7}{2x}$:

  Case 1: $x = \frac{7\sqrt{2}}{2}$ $y = -\frac{7}{2(\frac{7\sqrt{2}}{2})} = -\frac{7}{7\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$

  Case 2: $x = -\frac{7\sqrt{2}}{2}$ $y = -\frac{7}{2(-\frac{7\sqrt{2}}{2})} = -\frac{7}{-7\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

  The two square roots are: $ \frac{7$\sqrt{2}$}{2} - \frac{$\sqrt{2}$}{2}i \quad $\text{and}$ \quad -\frac{7$\sqrt{2}$}{2} + \frac{$\sqrt{2}$}{2}i $

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  How the marks are awarded

  • M1 — Setting $(x+iy)^2 = 24-7i$, expanding to get three terms, and equating real and imaginary parts.
  • A1 — Correctly obtaining the two simultaneous equations: $x^2 - y^2 = 24$ and $2xy = -7$.
  • M1 — Correctly eliminating one variable to form a single equation in the other. For example, substituting $y = -7/(2x)$ into the equation for the real parts and clearing the denominator.
  • A1 — Obtaining the correct quartic equation with integer coefficients, $4x^4 - 96x^2 - 49 = 0$ (or the equivalent in y).
  • A1 — Solving the quartic to find the exact values for x and y, and stating the two final square roots correctly paired and simplified. For example, $\frac{7\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$ and $-\frac{7\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$.

  Common mistakes

  • A sign error when equating imaginary parts, leading to $2xy = 7$ instead of $2xy = -7$. This results in the wrong pairing of x and y values at the end.
  • Making an algebraic mistake when eliminating a variable, such as incorrectly squaring $y = -7/(2x)$ to get $y^2 = 49/(2x^2)$, which leads to an incorrect quartic equation.
  • After finding the two possible values for x and the two for y, failing to pair them correctly using the equation $2xy = -7$. This often leads to stating four incorrect roots, such as $\pm \frac{7\sqrt{2}}{2} \pm \frac{\sqrt{2}}{2}i$.
  • Failing to provide exact answers, either by using a calculator to get decimal approximations or by not fully simplifying radicals, for example leaving answers with $\sqrt{49/2}$ or $1/\sqrt{2}$.

  Examiner tip: This question rewards the ability to translate a complex number problem into a system of real simultaneous equations and solve it methodically using algebraic substitution and manipulation.

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