Term independent of x in the expansion of x -1- x 23 n where n is even natural numberA- 3 n -n - - 2 n -C- 3 n -3 n -D- 3 n -n -2

The correct option is A 3n!n!×2n!T_r + 1 = ( 1)^r · ^3nC_r x^3n r×( 1x^2)^r ⇒ T_r + 1 = ( 1)^r · ^3nC_rx^3n 3r For independent term Put 3n 3r = 0 ⇒ r = n ...

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Standard XII

Mathematics

Sum of Product of Binomial Coefficients

Term independ...

Question

Term independent of $x$ in the expansion of ${(x - \frac{1}{x^{2}})}^{3 n}$ (where $n$ is even natural number)

\frac{3 n!}{n! \times 2 n!}

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\frac{3 n!}{n!}

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\frac{3 n!}{2 n!}

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\frac{3 n!}{(n!)^{2}}

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Solution

The correct option is A $\frac{3 n!}{n! \times 2 n!}$
$T_{r + 1} = (- 1)^{r} \cdot^{3 n} C_{r} x^{3 n - r} \times {(\frac{1}{x^{2}})}^{r}$
$\Rightarrow T_{r + 1} = (- 1)^{r} \cdot^{3 n} C_{r} x^{3 n - 3 r}$
For independent term
Put $3 n - 3 r = 0 \Rightarrow r = n .$

So, $T_{n + 1}$ is free from $x .$
And, $T_{n + 1} = (- 1)^{n} \cdot^{3 n} C_{n} =^{3 n} C_{n}$

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Sum of Product of Binomial Coefficients

Standard XII Mathematics

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Term independent of x in the expansion of (x−1x2)3n (where n is even natural number)

The correct option is A 3n!n!×2n!Tr+1=(−1)r⋅ 3nCrx3n−r×(1x2)r ⇒Tr+1=(−1)r⋅ 3nCrx3n−3r For independent term Put 3n−3r=0⇒r=n. So, Tn+1 is free from x. And, Tn+1=(−1)n⋅ 3nCn=3nCn

Term independent of $x$ in the expansion of ${(x - \frac{1}{x^{2}})}^{3 n}$ (where $n$ is even natural number)