Correct Answer - Option 4 : 462/x and -462x
Concept:
The general term in the expansion of (a - b)n is given by, \(\rm T_{r+1}=(-1)^r\times^nC_ra^{n-r}b^r\)
- When n is even, then the middle term = \(\rm (\frac n 2+1)\)th term.
- When n is odd, then the middle term = \(\rm \frac 12(n+1)\)th term and \(\rm \frac 12(n+3)\)th term.
Calculation:
The general term in the expansion of \(\rm (x-\frac1x)^{11}\)is given by,
\(\rm T_{r+1}=(-1)^r\times^{11}C_r\times x^{11-r}\times(\frac1x)^r\)
\(\rm =(-1)^r\times ^{11}C_r\times x^{11-r}\times x^{-r}\)
\(\rm =(-1)^r\times ^{11}C_r\times x^{11-2r}\)
Here, n is 11 (i.e., odd), so there will be two middle terms
\(\rm \frac 12(n+1)\)th term = 6th term and \(\rm \frac 12(n+3)\)th term = 7th term
∴ \(\rm T_{6}=T_{5+1}\) \(\rm =(-1)^5\times^{11}C_5\times x^{11-10}\)
\(\rm =-\frac{11\times 10\times 9\times 8\times 7}{5\times 4\times 3\times 2\times 1}\times x\)
= -462 x
And, \(\rm T_{7}=T_{6+1}\) \(\rm =(-1)^6\times ^{11}C_6\times x^{11-12}\)
\(\rm =\frac{11\times 10\times 9\times 8\times 7}{5\times 4\times 3\times 2\times 1}\times x^{-1}\)
= 462/x
Hence, option (4) is correct.
