Question

Question #32b20

Answer 1

`m=-7/3.`

Explanation:

Recall that, the `"General "(r+1)^(th)" Term "T_(r+1)` in the

Binomial Expansion of `(a+b)^n` is given by,

`T_(r+1)=""_nC_ra^(n-r)b^r, r=0,1,...,n.`

In our Problem, we have, `a=x^(1/2), b=m^(1/2)*x^-2, n=10.`

`:. T_(r+1)=""_10C_r(x^(1/2))^(10-r)*(-m^(1/2)*x^-2)^r,`

`=""_10C_r*(-m^(1/2))^r*x^{1/2(10-r)+(-2r)},`

`=""_10C_r*(-m^(1/2))^r*x^(5-r/2--2r),`

`rArr T_(r+1) =""_10C_r*(-m^(1/2))^r*x^(5-5/2*r)......(star).`

For the term free from `x,` its index must be `0.`

`:. 5-5/2*r=0 rArr r=2.`

`:. T_(2+1)=T_3" is free from x."`

Sub.ing `r=2` in `(star), T_3=""_10C_2*(-m^(1/2))^2*x^(5-5/2*2), i.e.,`

`T_3=""_10C_2*(-m),=105......................................."[Given]."`

`:. m=-105/(""_10C_2)=-105/{(10xx9)/(1xx2)}=-7/3.`