JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+2}$ , which are in the ratio $1 : 3 : 5$ , is equal to:

Choose the correct answer:

Explanation

Solution

Maan lijiye teen consecutive coefficients $^{n+2}C_{r-1}, ^{n+2}C_{r}$ aur $^{n+2}C_{r+1}$ hain.

Step 1: Ratio ka upyog karke $n$ aur $r$ nikalna

\frac{^{n+2}C_{r-1}}{^{n+2}C_{r}} = \frac{1}{3} \implies \frac{r}{n+2-r+1} = \frac{1}{3} \implies 4r = n+3 \quad \dots(1)

\frac{^{n+2}C_{r}}{^{n+2}C_{r+1}} = \frac{3}{5} \implies \frac{r+1}{n+2-r} = \frac{3}{5} \implies 8r = 3n+1 \quad \dots(2)

Equation $(1)$ aur $(2)$ ko solve karne par:

2(n+3) = 3n+1 \implies 2n+6 = 3n+1 \implies n = 5

$n=5$ ko equation $(1)$ mein rakhne par: $4r = 8 \implies r = 2$ .

Step 2: Coefficients ka sum nikalna

Ab humein $^{n+2}C_{r-1} + ^{n+2}C_{r} + ^{n+2}C_{r+1}$ nikalna hai jahan $n=5, r=2$ :

^7C_1 + ^7C_2 + ^7C_3 = 7 + 21 + 35 = 63

Sahi Option: (1) 63