If the constant term in the binomial expansion of (x^5/2-4x^)^9 is -84 and the coefficient…

Question

If the constant term in the binomial expansion of $\left(x^{5/2}-\dfrac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha\beta}$, where $\beta<0$ is an odd number, then $|\alpha\ell-\beta|$ is equal to \_\_\_\_\_\_\_\_\_

Answer: 98

Accepted Answer

$T_{r+1}=(-1)^r\binom9r 4^r x^{\frac{45-5r}2-\ell r}$. Setting exponent $=0$: $45-5r-2\ell r=0\Rightarrow r=\dfrac{45}{5+2\ell}\quad\text{...(1)}$

Given $(-1)^r\binom9r4^r=-84$: only natural value is $r=3$ with $\binom93=84$, so $\ell$ from (1) gives $\ell=5$.

For coefficient of $x^{-3\ell}=x^{-15}$: exponent $=\frac{45-5r}2-5r=-15\Rightarrow r=5$.

$$(-1)^5\binom95 4^5=-126\times1024=-63\times2^7$$

So $\alpha=7,\ \beta=-63$. $|\alpha\ell-\beta|=|7(5)-(-63)|=|35+63|=98$

Solutions · 1