JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Suppose $\sum_{r=0}^{2023} r^2 \ ^{2023}C_r = 2023 \times \alpha \times 2^{2022}$ . Then the value of $\alpha$ is

Choose the correct answer:

Explanation

Step 1: Standard Formula

Hum jante hain ki:

\sum_{r=0}^{n} r^2 \ ^nC_r = n(n+1)2^{n-2}

(Note: Ise $x(1+x)^n$ ko do baar differentiate karke nikala jata hai.)

Step 2: Values put karein

Yahan $n = 2023$ diya gaya hai. Formula mein rakhne par:

\sum_{r=0}^{2023} r^2 \ ^{2023}C_r = 2023(2023 + 1)2^{2023 - 2}

= 2023 \times 2024 \times 2^{2021}

Step 3: Equation ko compare karein

Sawal mein expression diya hai: $2023 \times \alpha \times 2^{2022}$

Hamara result aaya hai: $2023 \times 2024 \times 2^{2021}$

Dono ko barabar rakhein:

2023 \times \alpha \times 2^{2022} = 2023 \times 2024 \times 2^{2021}

$2023$ dono side se cut jayega:

\alpha \times 2^{2022} = 2024 \times 2^{2021}

Ab $2^{2022}$ ko hum $2 \times 2^{2021}$ likh sakte hain:

\alpha \times 2 \times 2^{2021} = 2024 \times 2^{2021}

$2^{2021}$ bhi cut jayega:

2\alpha = 2024

\alpha = \frac{2024}{2} = 1012

Final Answer:

$\alpha$ ki value 1012 hai.