Suppose x7 occurs in (r + 1)th term of expansion [ax2 + \(\frac{1}{bx^2}\) ] 11 .
Now, Tr + 1 = 11Cr (ax2 ) 11 – r ( \(\frac{1}{bx}
\))
Tr + 1 = 11Cr a11 – r b– r x 22 – 3r (i)
This will contain x7 , if
22 – 3r = 7
⇒ 3r = 15 or r = 5.
Putting r = 5 in (i), we obtain that coefficient of x7 In the expansion of [ax2 + \(\frac{1}{bx}\) ] 11 is 11C5 a6 b – 5
Suppose x-7 occurs in (r + 1)th term of the expansion of [ax - \(\frac{1}{bx^2}\) ] 11
Now, Tr + 1 = 11Cr (ax)11 – r (− 1 2 )
⇒ Tr + 1 = 11Cr a11 – r (– 1)r b – r x11– 3r (ii)
This will contain x-7 , if
11 – 3r = - 7
⇒ 3r = 18 or r = 6
Putting r = 6 in (ii), we obtain that coefficient of x – 7 in the expansion of
[ax - \(\frac{1}{bx^2}\) ] 11
11C6 a 5 b – 6 (– 1)6
If the coefficient of x7 [ax - \(\frac{1}{bx^2}\) ] 11 is equal to the coefficient of x – 7 ( [ax - \(\frac{1}{bx^2}\) ] 11 then
11C5 a 6 b – 5 = 11C6 a5 b – 6 (– 1)6
⇒11C5 ab = 11C6
⇒ ab = 1 [∵ 11C5 = 11C6]