JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The coefficient of $x^{301}$ in $(1 + x)^{500} + x (1 + x)^{499} + x^2 (1 + x)^{498} + \dots + x^{500}$ is:

Choose the correct answer:

Explanation

Solution

1. Series identification (Geometric Progression):

Yahan series ek G.P. hai jahan:

First term ( $a$ ) = $(1+x)^{500}$
Common ratio ( $r$ ) = $\frac{x}{1+x}$
Number of terms ( $n$ ) = $501$

2. Sum of G.P. formula:

S = a \frac{(1 - r^n)}{(1 - r)}

S = (1+x)^{500} \left[ \frac{1 - \left(\frac{x}{1+x}\right)^{501}}{1 - \frac{x}{1+x}} \right]

3. Simplification:

S = (1+x)^{500} \left[ \frac{\frac{(1+x)^{501} - x^{501}}{(1+x)^{501}}}{\frac{1+x-x}{1+x}} \right]

S = (1+x)^{500} \left[ \frac{(1+x)^{501} - x^{501}}{(1+x)^{501}} \times (1+x) \right]

S = (1+x)^{500} \left[ \frac{(1+x)^{501} - x^{501}}{(1+x)^{500}} \right]

S = (1+x)^{501} - x^{501}

4. Coefficient of $x^{301}$ nikalna:

Hume $(1+x)^{501} - x^{501}$ mein $x^{301}$ ka coefficient chahiye.

$x^{501}$ mein $x^{301}$ nahi hota, isliye sirf $(1+x)^{501}$ ko expand karenge:

\text{Coefficient of } x^{301} = \binom{501}{301} = ^{501}C_{301}

5. Property use karna:

Kyuki $^{n}C_{r} = ^{n}C_{n-r}$ :

^{501}C_{301} = ^{501}C_{501-301} = ^{501}C_{200}

Sahi Option: (2)