If the constant term in the binomial expansion of x 5 2 2 - 4 x l 9 is - 84 and the coefficient of x - 3 l is 2 α β where β < 0 is an odd number, then α l - β is equal to _____ .

Question

If the constant term in the binomial expansion of x 5 2 2 - 4 x l 9 is - 84 and the coefficient of x - 3 l is 2 &#945; &#946; where &#946; &#60; 0 is an odd number, then &#945; l - &#946; is equal to _____ .

Aicharya · Accepted Answer

Given,

The constant term in the binomial expansion of ${(\frac{x^{\frac{5}{2}}}{2} - \frac{4}{x^{l}})}^{9}$ is $- 84$ and the coefficient of $x^{- 3}$ is $2^{α} β$

Now General Term of the binomial is given by, $T_{r + 1} = {}^{9}C_{r} {(\frac{x^{\frac{5}{2}}}{2})}^{9 - r} {(\frac{- 4}{x^{l}})}^{r} = {(- 1)}^{r} C_{r} x^{\frac{45}{2} - \frac{5 r}{2} - l r} 2^{3 - 9}$

Now for constant term $45 - 5 r - 2 l r = 0 \Rightarrow r = \frac{45}{5 + 2 l}$

So, coefficient of constant term will be,

${(- 1)}^{9} \times^{9} C_{r} 2^{3 r - 9} = - 84 \Rightarrow r = 3$

Now putting the value of $r$ in $r = \frac{45}{5 + 2 l}$ we get, $l = 5$

Now coefficient of $x^{- 3 l} = x^{- 15} \Rightarrow \frac{45}{2} - \frac{5 r}{2} - l r = - 15 \Rightarrow r = 5$ when $l = 5$
Now coefficient of $x^{- 15} = {(- 1)}^{59} C_{5} 2^{6} = - 63 (2^{7}) = β (2^{α})$

Hence, on comparing we get, $α = 7, β = - 63 & l = 5$

So, the value of $|α l - β| = |7 \times 5 - (- 63)| = 98$

If the constant term in the binomial expansion of x 5 2 2 - 4 x l 9 is - 84 and the coefficient of x - 3 l…

Solution