JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The mean of the coefficients of $x, x^2, \dots, x^7$ in the binomial expansion of $(2 + x)^9$ is ________.

Choose the correct answer:

Explanation

General Term: Binomial expansion $(a+x)^n$ mein general term $T_{r+1} = \binom{n}{r} a^{n-r} x^r$ hota hai.

Yahan expansion $(2+x)^9$ hai, toh $x^r$ ka coefficient $C_r = \binom{9}{r} 2^{9-r}$ hoga.
Sum of coefficients ( $S$ ): Hame $x^1$ se $x^7$ tak ke coefficients ka sum chahiye.

$S = \sum_{r=1}^{7} \binom{9}{r} 2^{9-r}$

Puri series ka sum: $\sum_{r=0}^{9} \binom{9}{r} 2^{9-r} = (2+1)^9 = 3^9$ .

Isliye, $S = 3^9 - \text{term}(r=0) - \text{term}(r=8) - \text{term}(r=9)$ .
- $r=0: \binom{9}{0} 2^9 = 512$
- $r=8: \binom{9}{8} 2^1 = 9 \times 2 = 18$
- $r=9: \binom{9}{9} 2^0 = 1$
  
  $S = 19683 - 512 - 18 - 1 = 19152$
Mean: Yahan total $7$ terms hain ( $x^1, x^2, \dots, x^7$ ).

$\text{Mean} = \frac{S}{7} = \frac{19152}{7} = 2736$

Answer: 2736