Let Cr n denote the binomial coefficient of x ^ r in the expansion of ( 1 + x ) ^ n . If ∑ _ k = 0 ^ 10 ( 2 ^ 2 + 3k) Ck n = α . 3 ^ 10 + β · 2 ^ 10 , α, β ∈ R , then α + β is equal to ___ .

Correct answer: 0 Instead of Ck n it must be Ck 10 i.e. ∑ _ k = 0 ^ 10 ( 2 ^ 2 + 3k) Ck 10 = α · 3 ^ 10 + β · 2 ^ 10 LHS = 4 ∑ _ k = 0 ^ 10 Ck 10 + 3 ∑ _ k = 0 ^ 10 k · (10)/(k) · Ck - 1 9 = 4 . 2 ^ 10 + 3 . 10 . 2 ^ 9 = 19 . 2 ^ 10 = α . 3 ^ 10 + β . 2 ^ 10 ⇒ α = 0, β = 19 ⇒ α + β = 19

JEE Main 202118 MarchEvening Shift

Exam: JEE Main
Year: 2021
Date: 18 March
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2021 · 18 March · Evening Shift · Mathematics · Binomial Theorem

Let Cr n denote the binomial coefficient of x ^ r in the expansion...

Let $Cr n$ denote the binomial coefficient of ${x}^{r}$ in the expansion of $( 1 + x{)}^{n}$ . If $\mathop{∑}_{k = 0}^{10}\left({2}^{2} + 3k\right) Ck n = \alpha . {3}^{10} + \beta \cdot {2}^{10}, \alpha, \beta \in R$ , then $\alpha + \beta$ is equal to ___ .

Correct answer

Step-by-step solution

Instead of $Ck n$ it must be $Ck 10$ i.e.

$\mathop{∑}_{k = 0}^{10}\left({2}^{2} + 3k\right) Ck 10 = \alpha \cdot {3}^{10} + \beta \cdot {2}^{10}$

$LHS = 4\mathop{∑}_{k = 0}^{10} Ck 10 + 3\mathop{∑}_{k = 0}^{10}k \cdot \dfrac{10}{k} \cdot Ck - 1 9$

$= 4 . {2}^{10} + 3 . 10 . {2}^{9}$

$= 19 . {2}^{10} = \alpha . {3}^{10} + \beta . {2}^{10}$

$⇒ \alpha = 0, \beta = 19 ⇒ \alpha + \beta = 19$

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