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Chapter 7: Problem 494

Middle term in expansion of \([(2 / x)-3 x y]^{12}\) is (a) \(14370048 \mathrm{y}^{6}\) (b) \(14370024 \mathrm{y}^{6}\) (c) \(43110144 \mathrm{y}^{6}\) (d) \(43110124 \mathrm{y}^{6}\)

Short Answer

Expert verified
The middle term in the expansion of \([(2/x)-3xy]^{12}\) is (a) \(14370048 \mathrm{y}^{6}\).

Step by step solution

01

Find the middle term's index

Since n is even (12 is even), the middle term will be at position \(\frac{n}{2} + 1 = \frac{12}{2} + 1 = 7\). So, we are looking for the 7th term in the expansion: \(T_7\).
02

Apply the Binomial theorem to find the middle term

Using the Binomial theorem, we can write the 7th term as \[T_7 = ^{12}C_6(2/x)^{12-6}(-3xy)^6\]
03

Calculate the binomial coefficient

Now, let's calculate the binomial coefficient: \[^{12}C_6 = \frac{12!}{6!(12-6)!} = \frac{12!}{6!6!} = 924\]
04

Simplify the term

Now we plug in the binomial coefficient and simplify the term: \[T_7 = 924\left(\frac{2}{x}\right)^6(-3xy)^6\] \[T_7 = 924\left(\frac{2^6}{x^6}\right)(-3^6x^6y^6)\] \[T_7 = 924(64)(-729x^6y^6)\] \[T_7 = 14370048y^6\] So, the middle term in the expansion of \([(2/x)-3xy]^{12}\) is \(14370048y^6\). The answer is (a) \(14370048 \mathrm{y}^{6}\).

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Most popular questions from this chapter

If rth term in the expansion of \(\left[a x^{p}+\left(b / x^{q}\right)\right]^{n}\) is constant, then \(\mathrm{r}=\) (a) \([(q n) \bar{m}+q)]+1\) (b) \([(\mathrm{pn}) /(\mathrm{p}-\mathrm{q})]+1\) (c) \([(\mathrm{pn}) /(\mathrm{p}+\mathrm{q})]+1\) (d) \([(\mathrm{pn}) /(\mathrm{p}-\mathrm{q})]-1\)

Co efficient of middle term in expansion of \(\left[x-\left(x^{3} / 5\right)\right]^{8}=\) (a) \([(14) /(625)]\) (b) \([(70) /(62 \overline{5)}]\) (c) \([(14) /(125)]\) (d) \([(70) /(125)]\)

\(\mathrm{s}(\mathrm{k}): 1+3+5+\ldots+(2 \mathrm{k}-1)=3+\mathrm{k}^{2}\) then which statement is true ? (a) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (b) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (c) s (1) is true (d) Result is proved by Principle of Mathematical induction

Co-efficient of \(\mathrm{x}^{4}\) in expansion of \(\left(1+\mathrm{x}+\mathrm{x}^{2}+\mathrm{x}^{3}\right)^{11}\) is (a) 330 (b) 990 (c) 1040 (d) 900

Co-efficient of \(\mathrm{x}^{53}\) in \(\sqrt{ }^{100} \sum_{\mathrm{r}=0}{ }^{100} \mathrm{C}_{\mathrm{r}}(\mathrm{x}-5)^{100-\mathrm{r}} 4^{\mathrm{r}}\) is (a) \({ }^{100} \mathrm{C}_{53}\) (b) \({ }^{100} \mathrm{C}_{48}\) (c) \(-{ }^{100} \mathrm{C}_{53}\) (d) \({ }^{100} \mathrm{C}_{51}\)
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