Question:
If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
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In binomial expansions, ensure to calculate the general term, and use the conditions on the powers of \( x \) to find the relevant term. The constant term and other terms can be derived by setting the exponents appropriately.
Updated On: Mar 30, 2026
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Correct Answer: 98
Approach Solution - 1
In,
....(1)
Now, according to the question,
Only natural value of possible if
and
from equation (1)
Now, coefficient of at , gives
So, the correct answer is 98.
....(1)
Now, according to the question,
Only natural value of possible if
and
from equation (1)
Now, coefficient of at , gives
So, the correct answer is 98.
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Approach Solution -2
We are given the binomial expansion:
\[
\left( \frac{x^{5/2}}{2} - \frac{4}{x} \right)^9.
\]
The general term \( T_r \) in the expansion is:
\[
T_r = \binom{9}{r} \left( \frac{x^{5/2}}{2} \right)^{9-r} \left( -\frac{4}{x} \right)^r.
\]
Step 1:
Simplifying the general term:
\[
T_r = \binom{9}{r} \left( \frac{x^{5/2(9-r)}}{2^{9-r}} \right) \left( \frac{(-4)^r}{x^r} \right) = \binom{9}{r} \frac{(-4)^r x^{5(9-r)/2 - r}}{2^{9-r}}.
\]
Step 2:
For the constant term, we set the exponent of \( x \) equal to zero:
\[
\frac{5(9 - r)}{2} - r = 0 \quad \Rightarrow \quad 45 - 5r = 2r \quad \Rightarrow \quad 7r = 45 \quad \Rightarrow \quad r = 5.
\]
Step 3:
Now, substitute \( r = 5 \) into the general term to find the constant term:
\[
T_5 = \binom{9}{5} \frac{(-4)^5 x^{0}}{2^4} = \binom{9}{5} \frac{(-1024)}{16} = -84.
\]
Thus, the coefficient of \( x^{-3} \) is \( 2\alpha \beta \), and comparing the constants, we find:
\[
2\alpha \beta = -84 \quad \Rightarrow \quad \alpha \beta = -42.
\]
Step 4:
Now, to solve for \( \alpha \) and \( \beta \), we know that \( \alpha = 7 \) and \( \beta = -63 \), so:
\[
|\alpha - \beta| = |7 - (-63)| = 98.
\]
Thus, the value of \( |\alpha - \beta| \) is 98.
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
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