Key Concepts and Formulas

• Complex Number Properties: A complex number $Z = x + iy$ is real if and only if its imaginary part, $y$, is zero.
• Geometric Progression (GP) Sum: The sum of a GP with first term $A$, common ratio $R$, and $N$ terms is $S_N = \frac{A(R^N - 1)}{R - 1}$, for $R \neq 1$.
• De Moivre's Theorem: For a complex number $z = r(\cos\theta + i\sin\theta)$ and an integer $n$, $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$.

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Step-by-Step Solution

Step 1: Determine the value of 'a' and the complex number 'z'

We are given $z = 1 + ai$ with $a > 0$, and $z^3$ is a real number. We need to find $a$. Expand $z^3$: $z^3 = (1 + ai)^3$ Using the binomial expansion $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$: $z^3 = 1^3 + 3(1^2)(ai) + 3(1)(ai)^2 + (ai)^3$ $z^3 = 1 + 3ai + 3a^2i^2 + a^3i^3$ Substitute $i^2 = -1$ and $i^3 = -i$: $z^3 = 1 + 3ai + 3a^2(-1) + a^3(-i)$ $z^3 = 1 + 3ai - 3a^2 - a^3i$ Group the real and imaginary parts: $z^3 = (1 - 3a^2) + i(3a - a^3)$ Since $z^3$ is a real number, its imaginary part must be zero: $\text{Im}(z^3) = 3a - a^3 = 0$ Factor out $a$: $a(3 - a^2) = 0$ The possible values for $a$ are $0$ or $a^2 = 3$. Since we are given $a > 0$, we have $a = \sqrt{3}$. Therefore, the complex number is $z = 1 + \sqrt{3}i$.

Step 2: Convert 'z' to Polar Form

To make calculations with powers of $z$ easier, we convert $z = 1 + \sqrt{3}i$ to its polar form $z = r(\cos\theta + i\sin\theta)$. Calculate the modulus $r$: $r = |z| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$ Calculate the argument $\theta$. Since $z$ is in the first quadrant: $\theta = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$ So, the polar form of $z$ is: $z = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$

Step 3: Calculate $z^{12}$ using De Moivre's Theorem

We need $z^{12}$ for the sum of the geometric progression. Using De Moivre's Theorem: $z^{12} = r^{12}(\cos(12\theta) + i\sin(12\theta))$ Substitute $r=2$ and $\theta=\frac{\pi}{3}$: $z^{12} = 2^{12}\left(\cos\left(12 \cdot \frac{\pi}{3}\right) + i\sin\left(12 \cdot \frac{\pi}{3}\right)\right)$ $z^{12} = 2^{12}(\cos(4\pi) + i\sin(4\pi))$ Since $\cos(4\pi) = 1$ and $\sin(4\pi) = 0$: $z^{12} = 2^{12}(1 + 0i) = 2^{12}$ Calculate $2^{12}$: $2^{12} = 4096$ So, $z^{12} = 4096$.

Step 4: Calculate the Sum of the Geometric Progression

We need to find the sum $S = 1 + z + z^2 + \dots + z^{11}$. This is a geometric progression with:

• First term $A = 1$.
• Common ratio $R = z$.
• Number of terms $N = 12$ (from $z^0$ to $z^{11}$).

Using the GP sum formula $S_N = \frac{A(R^N - 1)}{R - 1}$: $S = \frac{1 \cdot (z^{12} - 1)}{z - 1} = \frac{z^{12} - 1}{z - 1}$ Substitute $z^{12} = 4096$ and $z = 1 + \sqrt{3}i$: $S = \frac{4096 - 1}{(1 + \sqrt{3}i) - 1}$ $S = \frac{4095}{\sqrt{3}i}$ To simplify, multiply the numerator and denominator by $i$: $S = \frac{4095i}{\sqrt{3}i^2}$ Since $i^2 = -1$: $S = \frac{4095i}{-\sqrt{3}} = -\frac{4095}{\sqrt{3}}i$ Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$: $S = -\frac{4095\sqrt{3}}{3}i$ Divide 4095 by 3: $4095 \div 3 = 1365$ So, the sum is: $S = -1365\sqrt{3}i$

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Common Mistakes & Tips

• Sign Errors with $i^2$ and $i^3$: Be careful when substituting the values of $i^2 = -1$ and $i^3 = -i$ during the expansion of $z^3$.
• Rationalizing the Denominator: When a complex number is in the denominator, always rationalize it by multiplying by its conjugate or by $i$ (if the denominator is purely imaginary) to express the result in the standard $x+iy$ form.
• De Moivre's Theorem Application: Ensure the angle in De Moivre's theorem is correctly calculated and simplified, especially when dealing with multiples of $\pi$.

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Summary

The problem required us to first determine the complex number $z$ by using the condition that $z^3$ is real. This led to $z = 1 + \sqrt{3}i$. We then converted $z$ to its polar form to efficiently calculate $z^{12}$ using De Moivre's Theorem. Finally, we applied the formula for the sum of a geometric progression to find the required sum $1 + z + z^2 + \dots + z^{11}$. The calculation involved careful handling of complex number arithmetic and simplification.

The final answer is $\boxed{-1365\,\sqrt 3 \,i}$. This corresponds to option (A).