Let the coefficient of $x^r$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}$ be $\alpha_r$. If $\sum_{r=0}^{n}\alpha_r=\beta^n-\gamma^n$, $\beta,\gamma\in\mathbb{N}$, then the value of $\beta^2+\gamma^2$ equals ____________.

JEE Main 2024 · MathsNumerical answerNumerical valueHardMulti-step

JEE Main 31 January 2024, Shift 2, Maths Q23

Question 23 of 90 in this shift, Maths question 23 of 30, Section B.

Let the coefficient of

x^r

in the expansion of

(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}

\alpha_r

. If

\sum_{r=0}^{n}\alpha_r=\beta^n-\gamma^n

\beta,\gamma\in\mathbb{N}

, then the value of

\beta^2+\gamma^2

equals ____________.

Official answer

NTA final key.

Chapter: Binomial Theorem
Topic: Binomial Coefficients
Idea tested: Sum of all Coefficients

Same idea in other shifts

Asked 3× in all

11 Apr 2023, Shift 1 · Q21The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ is ______________.MediumNumerical value
2 Apr 2017 (offline) · Q67The value of $(^{21}C_1-{}^{10}C_1)+(^{21}C_2-{}^{10}C_2)+(^{21}C_3-{}^{10}C_3)+(^{21}C_4-{}^{10}C_4)+\ldots+(^{21}C_{10}-{}^{10}C_{10})$ …MediumSingle correct

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 42 of the 174 JEE Main shifts held from 2017 to 2026.