119351
The greatest value of the term independent of \(x\), as \(\alpha\) varies over \(R\), in the expansion of \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) is
B Given, \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) The general term in the expansion \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}}(\mathrm{x} \cos \alpha)^{10-\mathrm{r}}\left(\frac{\sin \alpha}{\mathrm{x}}\right)^{\mathrm{r}}\) \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-\mathrm{r}}(\cos \alpha)^{10-\mathrm{r}} \cdot(\sin \alpha)^{\mathrm{r}}\left(\mathrm{x}^{-\mathrm{r}}\right)\) \(={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-2 \mathrm{r}}(\cos \alpha)^{10-\mathrm{r}}(\sin \alpha)^{\mathrm{r}}\) For the term independent term power of \(\mathrm{x}\) should be zero \(10-2 \mathrm{r} =0\) \(2 \mathrm{r} =10\) \(\mathrm{r} =5\) For independent term, \(r=5\) The term independent of \(\mathrm{x}\) is \({ }^{10} \mathrm{C}_5(\cos \alpha)^5(\sin \alpha)^5\) \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5(\sin 2 \alpha)^5\) So, the greatest value of the independent term is \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5\)
AMU-2016
Binomial Theorem and its Simple Application
119352
Let \(n\) be a positive integer. If the coefficients of second, third and fourth terms in the expansion of \((1+x)^n\) are in A.P., then \(n=\)
1 2
2 6
3 7
4 None of these
Explanation:
C \(2(\text { Coefficient of third term })=(\text { coefficient of second }\) \(\text { term }+ \text { coefficient of fourth term) }\) \(\because 2^n C_2={ }^n C_1+{ }^n C_3\) \(\frac{2 \times n !}{2 !(n-2) !}=\frac{n !}{1 !(n-1) !}+\frac{n !}{3 !(n-3) !}\) \(\frac{2 n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}\) \(6 n(n-1)=6 n+n(n-1)(n-2)\) \(6 n(n-1)=n\left(6+n^2-3 n+2\right)\) \(\left(n^2-9 n+14\right)=0\) \((n-2)(n-7)=0\) \(n=7\)
AMU-2013
Binomial Theorem and its Simple Application
119353
The total number of terms in the expansion of \((1+x)^{2 n}-(1-x)^{2 n}\) after simplification is
119351
The greatest value of the term independent of \(x\), as \(\alpha\) varies over \(R\), in the expansion of \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) is
B Given, \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) The general term in the expansion \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}}(\mathrm{x} \cos \alpha)^{10-\mathrm{r}}\left(\frac{\sin \alpha}{\mathrm{x}}\right)^{\mathrm{r}}\) \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-\mathrm{r}}(\cos \alpha)^{10-\mathrm{r}} \cdot(\sin \alpha)^{\mathrm{r}}\left(\mathrm{x}^{-\mathrm{r}}\right)\) \(={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-2 \mathrm{r}}(\cos \alpha)^{10-\mathrm{r}}(\sin \alpha)^{\mathrm{r}}\) For the term independent term power of \(\mathrm{x}\) should be zero \(10-2 \mathrm{r} =0\) \(2 \mathrm{r} =10\) \(\mathrm{r} =5\) For independent term, \(r=5\) The term independent of \(\mathrm{x}\) is \({ }^{10} \mathrm{C}_5(\cos \alpha)^5(\sin \alpha)^5\) \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5(\sin 2 \alpha)^5\) So, the greatest value of the independent term is \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5\)
AMU-2016
Binomial Theorem and its Simple Application
119352
Let \(n\) be a positive integer. If the coefficients of second, third and fourth terms in the expansion of \((1+x)^n\) are in A.P., then \(n=\)
1 2
2 6
3 7
4 None of these
Explanation:
C \(2(\text { Coefficient of third term })=(\text { coefficient of second }\) \(\text { term }+ \text { coefficient of fourth term) }\) \(\because 2^n C_2={ }^n C_1+{ }^n C_3\) \(\frac{2 \times n !}{2 !(n-2) !}=\frac{n !}{1 !(n-1) !}+\frac{n !}{3 !(n-3) !}\) \(\frac{2 n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}\) \(6 n(n-1)=6 n+n(n-1)(n-2)\) \(6 n(n-1)=n\left(6+n^2-3 n+2\right)\) \(\left(n^2-9 n+14\right)=0\) \((n-2)(n-7)=0\) \(n=7\)
AMU-2013
Binomial Theorem and its Simple Application
119353
The total number of terms in the expansion of \((1+x)^{2 n}-(1-x)^{2 n}\) after simplification is
119351
The greatest value of the term independent of \(x\), as \(\alpha\) varies over \(R\), in the expansion of \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) is
B Given, \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) The general term in the expansion \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}}(\mathrm{x} \cos \alpha)^{10-\mathrm{r}}\left(\frac{\sin \alpha}{\mathrm{x}}\right)^{\mathrm{r}}\) \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-\mathrm{r}}(\cos \alpha)^{10-\mathrm{r}} \cdot(\sin \alpha)^{\mathrm{r}}\left(\mathrm{x}^{-\mathrm{r}}\right)\) \(={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-2 \mathrm{r}}(\cos \alpha)^{10-\mathrm{r}}(\sin \alpha)^{\mathrm{r}}\) For the term independent term power of \(\mathrm{x}\) should be zero \(10-2 \mathrm{r} =0\) \(2 \mathrm{r} =10\) \(\mathrm{r} =5\) For independent term, \(r=5\) The term independent of \(\mathrm{x}\) is \({ }^{10} \mathrm{C}_5(\cos \alpha)^5(\sin \alpha)^5\) \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5(\sin 2 \alpha)^5\) So, the greatest value of the independent term is \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5\)
AMU-2016
Binomial Theorem and its Simple Application
119352
Let \(n\) be a positive integer. If the coefficients of second, third and fourth terms in the expansion of \((1+x)^n\) are in A.P., then \(n=\)
1 2
2 6
3 7
4 None of these
Explanation:
C \(2(\text { Coefficient of third term })=(\text { coefficient of second }\) \(\text { term }+ \text { coefficient of fourth term) }\) \(\because 2^n C_2={ }^n C_1+{ }^n C_3\) \(\frac{2 \times n !}{2 !(n-2) !}=\frac{n !}{1 !(n-1) !}+\frac{n !}{3 !(n-3) !}\) \(\frac{2 n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}\) \(6 n(n-1)=6 n+n(n-1)(n-2)\) \(6 n(n-1)=n\left(6+n^2-3 n+2\right)\) \(\left(n^2-9 n+14\right)=0\) \((n-2)(n-7)=0\) \(n=7\)
AMU-2013
Binomial Theorem and its Simple Application
119353
The total number of terms in the expansion of \((1+x)^{2 n}-(1-x)^{2 n}\) after simplification is
119351
The greatest value of the term independent of \(x\), as \(\alpha\) varies over \(R\), in the expansion of \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) is
B Given, \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) The general term in the expansion \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}}(\mathrm{x} \cos \alpha)^{10-\mathrm{r}}\left(\frac{\sin \alpha}{\mathrm{x}}\right)^{\mathrm{r}}\) \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-\mathrm{r}}(\cos \alpha)^{10-\mathrm{r}} \cdot(\sin \alpha)^{\mathrm{r}}\left(\mathrm{x}^{-\mathrm{r}}\right)\) \(={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-2 \mathrm{r}}(\cos \alpha)^{10-\mathrm{r}}(\sin \alpha)^{\mathrm{r}}\) For the term independent term power of \(\mathrm{x}\) should be zero \(10-2 \mathrm{r} =0\) \(2 \mathrm{r} =10\) \(\mathrm{r} =5\) For independent term, \(r=5\) The term independent of \(\mathrm{x}\) is \({ }^{10} \mathrm{C}_5(\cos \alpha)^5(\sin \alpha)^5\) \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5(\sin 2 \alpha)^5\) So, the greatest value of the independent term is \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5\)
AMU-2016
Binomial Theorem and its Simple Application
119352
Let \(n\) be a positive integer. If the coefficients of second, third and fourth terms in the expansion of \((1+x)^n\) are in A.P., then \(n=\)
1 2
2 6
3 7
4 None of these
Explanation:
C \(2(\text { Coefficient of third term })=(\text { coefficient of second }\) \(\text { term }+ \text { coefficient of fourth term) }\) \(\because 2^n C_2={ }^n C_1+{ }^n C_3\) \(\frac{2 \times n !}{2 !(n-2) !}=\frac{n !}{1 !(n-1) !}+\frac{n !}{3 !(n-3) !}\) \(\frac{2 n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}\) \(6 n(n-1)=6 n+n(n-1)(n-2)\) \(6 n(n-1)=n\left(6+n^2-3 n+2\right)\) \(\left(n^2-9 n+14\right)=0\) \((n-2)(n-7)=0\) \(n=7\)
AMU-2013
Binomial Theorem and its Simple Application
119353
The total number of terms in the expansion of \((1+x)^{2 n}-(1-x)^{2 n}\) after simplification is
119351
The greatest value of the term independent of \(x\), as \(\alpha\) varies over \(R\), in the expansion of \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) is
B Given, \(\left(x \cos \alpha+\frac{\sin \alpha}{x}\right)^{10}\) The general term in the expansion \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}}(\mathrm{x} \cos \alpha)^{10-\mathrm{r}}\left(\frac{\sin \alpha}{\mathrm{x}}\right)^{\mathrm{r}}\) \(\mathrm{T}_{\mathrm{r}+1} ={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-\mathrm{r}}(\cos \alpha)^{10-\mathrm{r}} \cdot(\sin \alpha)^{\mathrm{r}}\left(\mathrm{x}^{-\mathrm{r}}\right)\) \(={ }^{10} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{10-2 \mathrm{r}}(\cos \alpha)^{10-\mathrm{r}}(\sin \alpha)^{\mathrm{r}}\) For the term independent term power of \(\mathrm{x}\) should be zero \(10-2 \mathrm{r} =0\) \(2 \mathrm{r} =10\) \(\mathrm{r} =5\) For independent term, \(r=5\) The term independent of \(\mathrm{x}\) is \({ }^{10} \mathrm{C}_5(\cos \alpha)^5(\sin \alpha)^5\) \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5(\sin 2 \alpha)^5\) So, the greatest value of the independent term is \(\left(\frac{1}{2}\right)^5{ }^{10} \mathrm{C}_5\)
AMU-2016
Binomial Theorem and its Simple Application
119352
Let \(n\) be a positive integer. If the coefficients of second, third and fourth terms in the expansion of \((1+x)^n\) are in A.P., then \(n=\)
1 2
2 6
3 7
4 None of these
Explanation:
C \(2(\text { Coefficient of third term })=(\text { coefficient of second }\) \(\text { term }+ \text { coefficient of fourth term) }\) \(\because 2^n C_2={ }^n C_1+{ }^n C_3\) \(\frac{2 \times n !}{2 !(n-2) !}=\frac{n !}{1 !(n-1) !}+\frac{n !}{3 !(n-3) !}\) \(\frac{2 n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}\) \(6 n(n-1)=6 n+n(n-1)(n-2)\) \(6 n(n-1)=n\left(6+n^2-3 n+2\right)\) \(\left(n^2-9 n+14\right)=0\) \((n-2)(n-7)=0\) \(n=7\)
AMU-2013
Binomial Theorem and its Simple Application
119353
The total number of terms in the expansion of \((1+x)^{2 n}-(1-x)^{2 n}\) after simplification is