Class 11: Binomial Theorem – Very Short Answer Questions

$\displaystyle \textbf{Question 1. }\text{Write the number of terms in the expansion of } \\ (2+\sqrt{3}x)^{10}+(2-\sqrt{3}x)^{10}.$
$\displaystyle \text{Answer:}$
$\displaystyle (2+\sqrt{3}x)^{10}+(2-\sqrt{3}x)^{10}$
$\displaystyle \text{In the sum, all odd powers of }x\text{ cancel out.}$
$\displaystyle \text{Only even powers }x^0,x^2,x^4,x^6,x^8,x^{10}\text{ remain.}$
$\displaystyle \therefore \text{Number of terms}=6$
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$\displaystyle \textbf{Question 2. }\text{Write the sum of the coefficients in the expansion of } \\ (1-3x+x^2)^{111}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Sum of coefficients is obtained by putting }x=1.$
$\displaystyle (1-3x+x^2)^{111}\big|_{x=1}=(1-3+1)^{111}$
$\displaystyle =(-1)^{111}=-1$
$\\$

$\displaystyle \textbf{Question 3. }\text{Write the number of terms in the expansion of } \\ (1-3x+3x^2-x^3)^8.$
$\displaystyle \text{Answer:}$
$\displaystyle 1-3x+3x^2-x^3=(1-x)^3$
$\displaystyle \therefore (1-3x+3x^2-x^3)^8=(1-x)^{24}$
$\displaystyle \text{Number of terms in }(1-x)^{24}\text{ is }24+1=25$
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$\displaystyle \textbf{Question 4. }\text{Write the middle term in the expansion of }\left(\frac{2x^2}{3}+\frac{3}{2x^2}\right)^{10}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since }n=10,\text{ the middle term is the }6\text{th term.}$
$\displaystyle T_6={} ^{10}C_5\left(\frac{2x^2}{3}\right)^5\left(\frac{3}{2x^2}\right)^5$
$\displaystyle ={}^{10}C_5$
$\displaystyle =252$
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$\displaystyle \textbf{Question 5. }\text{Which term is independent of }x,\text{ in the expansion of }\left(x-\frac{1}{3x^2}\right)^9?$
$\displaystyle \text{Answer:}$
$\displaystyle T_{r+1}={}^9C_r(x)^{9-r}\left(-\frac{1}{3x^2}\right)^r$
$\displaystyle ={}^9C_r\left(-\frac{1}{3}\right)^r x^{9-r-2r}$
$\displaystyle ={}^9C_r\left(-\frac{1}{3}\right)^r x^{9-3r}$
$\displaystyle \text{For the term independent of }x,\ 9-3r=0$
$\displaystyle r=3$
$\displaystyle \therefore \text{Required term is }T_{r+1}=T_4$
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$\displaystyle \textbf{Question 6. }\text{If }a\text{ and }b\text{ denote respectively the coefficients of }x^m\text{ and }x^n\text{ in}$
$\displaystyle \text{the expansion of }(1+x)^{m+n},\text{ then write the relation between }a\text{ and }b.$
$\displaystyle \text{Answer:}$
$\displaystyle a={} ^{m+n}C_m$
$\displaystyle b={} ^{m+n}C_n$
$\displaystyle \text{Since }{}^{m+n}C_m={} ^{m+n}C_n$
$\displaystyle \therefore a=b$
$\\$

$\displaystyle \textbf{Question 7. }\text{If }a\text{ and }b\text{ are coefficients of }x^n\text{ in the expansions of }(1+x)^{2n}$
$\displaystyle \text{and }(1+x)^{2n-1}\text{ respectively, then write the relation between }a\text{ and }b.$
$\displaystyle \text{Answer:}$
$\displaystyle a={} ^{2n}C_n$
$\displaystyle b={} ^{2n-1}C_n$
$\displaystyle \frac{a}{b}=\frac{{}^{2n}C_n}{{}^{2n-1}C_n}=\frac{2n}{n}=2$
$\displaystyle \therefore a=2b$
$\\$

$\displaystyle \textbf{Question 8. }\text{Write the middle term in the expansion of }\left(x+\frac{1}{x}\right)^{10}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since }n=10,\text{ the middle term is the }6\text{th term.}$
$\displaystyle T_6={} ^{10}C_5x^5\left(\frac{1}{x}\right)^5$
$\displaystyle ={}^{10}C_5=252$
$\\$

$\displaystyle \textbf{Question 9. }\text{If }a\text{ and }b\text{ denote the sum of the coefficients in the expansions of}$
$\displaystyle (1-3x+10x^2)^n\text{ and }(1+x^2)^n\text{ respectively, then write the relation between }a\text{ and }b.$
$\displaystyle \text{Answer:}$
$\displaystyle a=(1-3+10)^n=8^n$
$\displaystyle b=(1+1)^n=2^n$
$\displaystyle \therefore a=(2^n)^3=b^3$
$\\$

$\displaystyle \textbf{Question 10. }\text{Write the coefficient of the middle term in the expansion of } \\ (1+x)^{2n}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since the power is }2n,\text{ the middle term is the }(n+1)\text{th term.}$
$\displaystyle \therefore \text{Coefficient of middle term}={} ^{2n}C_n$
$\\$

$\displaystyle \textbf{Question 11. }\text{Write the number of terms in the expansion of }\{(2x+y^3)^4\}^7.$
$\displaystyle \text{Answer:}$
$\displaystyle \{(2x+y^3)^4\}^7=(2x+y^3)^{28}$
$\displaystyle \therefore \text{Number of terms}=28+1=29$
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$\displaystyle \textbf{Question 12. }\text{Find the sum of the coefficients of two middle terms in the} \\ \text{binomial expansion of }(1+x)^{2n-1}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since the power is }2n-1,\text{ there are two middle terms.}$
$\displaystyle \text{Their coefficients are }{}^{2n-1}C_{n-1}\text{ and }{}^{2n-1}C_n.$
$\displaystyle \therefore \text{Required sum}={} ^{2n-1}C_{n-1}+{}^{2n-1}C_n$
$\displaystyle ={}^{2n}C_n$
$\\$

$\displaystyle \textbf{Question 13. }\text{Find the ratio of the coefficients of }x^p\text{ and }x^q\text{ in the expansion of } \\ (1+x)^{p+q}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Coefficient of }x^p={} ^{p+q}C_p$
$\displaystyle \text{Coefficient of }x^q={} ^{p+q}C_q$
$\displaystyle \text{But }{}^{p+q}C_p={} ^{p+q}C_q$
$\displaystyle \therefore \text{Required ratio}=1:1$
$\\$

$\displaystyle \textbf{Question 14. }\text{Write last two digits of the number }3^{400}.$
$\displaystyle \text{Answer:}$
$\displaystyle 3^{20}\equiv1\pmod{100}$
$\displaystyle \therefore 3^{400}=(3^{20})^{20}\equiv1^{20}\pmod{100}$
$\displaystyle \therefore \text{Last two digits are }01$
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$\displaystyle \textbf{Question 15. }\text{Find the number of terms in the expansion of }(a+b+c)^n.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{General term in the expansion is }a^pb^qc^r$
$\displaystyle \text{where }p+q+r=n$
$\displaystyle \text{The number of non-negative integral solutions of }p+q+r=n$
$\displaystyle \text{is }{}^{n+2}C_2$
$\displaystyle \therefore \text{Number of terms}={}^{n+2}C_2=\frac{(n+2)(n+1)}{2}$
$\\$

$\displaystyle \textbf{Question 16. }\text{If }a\text{ and }b\text{ are the coefficients of }x^n\text{ in the expansions of }$
$\displaystyle (1+x)^{2n} \text{and }(1+x)^{2n-1}\text{ respectively, find }\frac{a}{b}.$
$\displaystyle \text{Answer:}$
$\displaystyle a={} ^{2n}C_n$
$\displaystyle b={} ^{2n-1}C_n$
$\displaystyle \therefore \frac{a}{b}=\frac{{}^{2n}C_n}{{}^{2n-1}C_n}$
$\displaystyle =\frac{2n}{n}=2$
$\\$

$\displaystyle \textbf{Question 17. }\text{Write the total number of terms in the expansion of } \\ (x+a)^{100}+(x-a)^{100}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{In the sum, terms containing odd powers of }a\text{ cancel out.}$
$\displaystyle \text{Only even powers of }a\text{ remain.}$
$\displaystyle \therefore \text{Number of terms}=51$
$\\$

$\displaystyle \textbf{Question 18. }\text{If }(1-x+x^2)^n=a_0+a_1x+a_2x^2+\cdots+a_{2n}x^{2n},\text{ find the value of}$
$\displaystyle a_0+a_2+a_4+\cdots+a_{2n}.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Sum of even coefficients}=\frac{f(1)+f(-1)}{2}$
$\displaystyle f(x)=(1-x+x^2)^n$
$\displaystyle f(1)=(1-1+1)^n=1$
$\displaystyle f(-1)=(1+1+1)^n=3^n$
$\displaystyle \therefore a_0+a_2+a_4+\cdots+a_{2n}=\frac{1+3^n}{2}$
$\\$