Animated Solution for Mathematics - Binomial Theorem: If the constant term in the binomial expansion of $(\frac{x^{5/2}}{2}-\frac{4}{x^l}{)}^9$ is $-84$ and the Coefficient of $x^{-3l}$ is $2^{\alpha }\beta$, where $\beta <0$ is an odd number, Then $|\alpha l-\beta |$ is equal to ______.

Unlocking the Constant Term

Welcome, fellow traveler on the path to JEE mastery! Today, we are going to dissect a beautiful problem that tests not just your knowledge of the Binomial Theorem, but your ability to act as a mathematical detective.

We are given the expansion of $(\frac{x^{5/2}}{2}-\frac{4}{x^l}{)}^9$ and two clues: the constant term is $-84$, and the coefficient of $x^{-3l}$ takes the form $2^{\alpha }\beta$. Our goal is to find $|\alpha l-\beta |$.

Let us begin.

The General Term as a Swiss Army Knife

Whenever you face a binomial expansion problem, the general term formula is your most reliable companion. It is defined as $T_{r+1}=\left(\genfrac{}{}{0ex}{}{n}{r}\right)a^{n-r}{b}^r$.

In our case, $n=9$, $a=\frac{x^{5/2}}{2}$, and $b=-\frac{4}{x^l}$. Substituting these into the formula, we get:

This formula is like a telescope; it allows us to zoom in on any specific term in the expansion without having to write out all ten terms. The key is to keep the negative sign with the $4$ inside the parenthesis. If you lose that sign, the entire calculation will collapse later on.

The Quest for the Constant Term

Now, let us separate the constants from the variables. We pull out the numerical parts: $\left(\genfrac{}{}{0ex}{}{9}{r}\right)(\frac{1}{2}{)}^{9-r}(-4{)}^r$.

What remains are the powers of $x$:

A constant term is, by definition, independent of $x$. This means the exponent of $x$ must be zero. Setting $\frac{45-5r-2lr}{2}=0$ leads us to the elegant relation $r(5+2l)=45$.

Since $r$ must be an integer between $0$ and $9$, $r$ must be a divisor of $45$. Testing $r=3$, we find $3(5+2l)=45$, which simplifies to $5+2l=15$, giving us $l=5$. A quick verification confirms that $r=3$ and $l=5$ indeed yields the constant term $-84$.

We have our first breakthrough!

The Second Challenge

With $l=5$ firmly in our grasp, the second part of the problem becomes clear. We need the coefficient of $x^{-3l}$, which is $x^{-15}$.

We return to our exponent formula:

This simplifies to $\frac{45-15r}{2}=-15$, or $45-15r=-30$. Solving this gives $15r=75$, so $r=5$.

Now, we calculate the coefficient for $r=5$:

The Final Synthesis

We are almost at the finish line. We need to express $-8064$ in the form $2^{\alpha }\beta$, where $\beta$ is an odd negative integer.

Factorizing $-8064$, we get:

Comparing this to $2^{\alpha }\beta$, we identify $\alpha =7$ and $\beta =-63$.

Finally, we compute:

The elegance of this result is a testament to the power of systematic algebraic manipulation. You have successfully navigated the traps of the Binomial Theorem. Keep this confidence, and carry it into your next challenge!