Animated Solution for Mathematics - Complex Numbers: If $\alpha$ and $\beta$ are the roots of the equation $2z^2-3z-2i=0$ where $i=\sqrt{-1}$, then $16\cdot Re(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}})\cdot Im(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}})$ is equal to

Imagine you are standing before a complex quadratic equation: $2z^2-3z-2i=0$. At first glance, it looks like a standard problem, but then you see the target expression:

Calculating ${\alpha }^{19}$ directly sounds like a journey into a computational abyss. However, in the world of JEE Advanced, whenever you see such high powers, there is almost always a hidden symmetry waiting to be uncovered.

The secret lies in the powers themselves: $11,15,19$. Notice the gap? It is exactly $4$.

This is not a coincidence; it is a breadcrumb trail. To follow it, we must transform our quadratic equation into something that relates $z$ to $1/z$.

We start by dividing the entire equation $2z^2-3z-2i=0$ by $2z$:

This is our foundation.

Now, we need to reach the fourth power. We square our equation:

Expanding the left side, we get $z^2+(i/z{)}^2-2(z)(i/z)=9/4$. Since $i^2=-1$, this simplifies to:

One more square will take us to the fourth power. Squaring both sides again:

The left side becomes $z^4+1/z^4-2$, and the right side expands to:

Thus, $z^4+1/z^4-2=17/16+9i$. Adding $2$ to both sides, we finally arrive at the golden key:

Now, look back at our expression $E$. We can group the terms by factoring out ${\alpha }^{15}$ and ${\beta }^{15}$:

Since both $\alpha$ and $\beta$ satisfy our derived relation, we can replace the bracketed terms with our constant $K=\frac{49}{16}+9i$. The expression becomes:

Factoring out $K$, we see the numerator and denominator are identical. They cancel out, leaving us with $E=K=\frac{49}{16}+9i$.

The real part is $\frac{49}{16}$ and the imaginary part is $9$. The final step is a simple multiplication:

It is a moment of pure mathematical zen when the complexity dissolves into a simple integer. You have conquered the beast!