7.Binomial Theorem
hard

माना $(1+ x )^{ n }$ के प्रसार में $x ^{ r }$ का द्विपद गुणांक ${ }^{ n } C _{ r }$ है। यदि $\sum_{ k =0}^{10}\left(2^{2}+3 k \right)= C _{ k }=\alpha .3^{10}+\beta .2^{10}, \alpha$, $\beta \in R$ है, $\alpha+\beta$ बराबर है ............ |

A

$19$

B

$21$

C

$17$

D

$13$

(JEE MAIN-2021)

Solution

Instead of ${ }^{n} C_{k}$ it must be ${ }^{10} C_{k}$ i.e.

$\sum_{k=0}^{10}\left(2^{2}+3 k\right){ }^{10} C _{ k }=\alpha .3^{10}+\beta .2^{10}$

$LHS =4 \sum_{ k =0}^{10}{ }^{10} C _{ k }+3 \sum_{ k =0}^{10} k \cdot \frac{10}{ k } \cdot{ }^{9} C _{ k -1}$

$=4.2^{10}+3.10 .2^{9}$

$=19.2^{10}=\alpha .3^{10}+\beta .2^{10}$

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

Standard 11
Mathematics

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