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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
