- Home
- Standard 11
- Mathematics
7.Binomial Theorem
hard
જો ગુણાકાર $\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ માં $x$ ની બધીજ યુગ્મ ઘાતાંકનો સરવાળો $61,$ હોય તો $\mathrm{n}$ મેળવો.
A
$30$
B
$26$
C
$22$
D
$20$
(JEE MAIN-2020)
Solution
Let $\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$
$=a_{0}+a_{1} x_{+} a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+\ldots+a_{4 n} x^{4 n}$
$\mathrm{So}$
$a_{0}+a_{1}+a_{2}+\ldots+a_{4 n}=2 n+1$
$a_{0}-a_{1}+a_{2}-a_{3} \ldots+a_{4 n}=2 n+1$
$\Rightarrow a_{0}+a_{2}+a_{4}+\ldots+a_{4 n}=2 n+1$
$\Rightarrow 2 \mathrm{n}+1=61 \quad \Rightarrow \mathrm{n}=30$
Standard 11
Mathematics
