Correct Answer - Option 1 : 13/8
Concept:
The general term in the expansion (a + b)n is given by \(\rm T_{r+1}=^nC_r(a)^{n-r}(b)^r\)
Calculation:
The general term in the expansion \(\rm (1-x^2)^{16}\) is given by,
\(\rm T_{r+1}=(-1)^r 16C_r(1)^{16-r}(x^2)^r\)
\(\rm =(-1)^r16C_r(x^{2r})\) ....(1)
Now, we need the coefficient of \(\rm x^{8}\)
On equating power of x in (1) with 8, we get
2r = 8
⇒ r = 4
∴ \(\rm T_{4+1}=(-1)^416C_4x^8\)
Coefficient of \(\rm x^{8}\) = \(\rm 16C_4=\frac{16× 15×14×13}{4× 3×2×1}\)
= 4 × 5 × 7 × 13 ....(2)
Now, in \(\rm (x+\frac2x)^{8}\), general term is given by,
\(\rm T_{r+1}=8C_r(x)^{8-r}(\frac2x)^r\)
\(\rm =8C_r(x)^{8-r}x^{-r}2^r\)
\(\rm =8C_r(x)^{8-2r}2^r\)
Now, for an independent term power of x should be 0
So, 8 - 2r = 0
⇒ 2r = 8
⇒r = 4
∴ \(\rm T_{4+1}=8C_4(x)^{8-4}(\frac2x)^4\)
Coefficient of independent term = \(\rm 8C_4× 2^4\)
= \(\rm \frac{8× 7×6×5}{4× 3×2×1}× 16\)
= 2 × 7 × 5 × 16 .... (3)
So, Ratio = \(\frac{4 \times 5 \times7 \times13 }{2\times7\times5\times16}\) = 13/8
Hence, option (1) is correct.
