\displaystyle \textbf{Question 1. }\text{If in the expansion of }(1+x)^{20},\text{ the coefficients of }r\text{th and } \\ (r+4)\text{th terms}   \text{are equal, then }r\text{ is equal to}
\displaystyle \text{(a) }7\qquad \text{(b) }8\qquad \text{(c) }9\qquad \text{(d) }10
\displaystyle \text{Answer:}
\displaystyle \text{Coefficient of }r\text{th term}={} ^{20}C_{r-1}
\displaystyle \text{Coefficient of }(r+4)\text{th term}={} ^{20}C_{r+3}
\displaystyle {}^{20}C_{r-1}={} ^{20}C_{r+3}
\displaystyle \therefore (r-1)+(r+3)=20
\displaystyle 2r+2=20
\displaystyle r=9
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 2. }\text{The term without }x\text{ in the expansion of }\left(2x-\frac{1}{2x^2}\right)^{12}\text{ is}
\displaystyle \text{(a) }495\qquad \text{(b) }-495\qquad \text{(c) }-7920\qquad \text{(d) }7920
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{12}C_r(2x)^{12-r}\left(-\frac{1}{2x^2}\right)^r
\displaystyle ={}^{12}C_r2^{12-2r}(-1)^rx^{12-3r}
\displaystyle \text{For the term independent of }x,\ 12-3r=0
\displaystyle r=4
\displaystyle \therefore \text{Required term}=T_5={} ^{12}C_42^4
\displaystyle =495\times16=7920
\displaystyle \therefore \text{Correct option is (d).}
\\

\displaystyle \textbf{Question 3. }\text{If }r\text{th term in the expansion of }\left(2x^2-\frac{1}{x}\right)^{12}\text{ is without }x,\text{ then }r\text{ is equal to}
\displaystyle \text{(a) }8\qquad \text{(b) }7\qquad \text{(c) }9\qquad \text{(d) }10
\displaystyle \text{Answer:}
\displaystyle T_r={} ^{12}C_{r-1}(2x^2)^{13-r}\left(-\frac{1}{x}\right)^{r-1}
\displaystyle \text{Power of }x=2(13-r)-(r-1)
\displaystyle =26-2r-r+1=27-3r
\displaystyle \text{For term independent of }x,\ 27-3r=0
\displaystyle r=9
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 4. }\text{If in the expansion of }(a+b)^n\text{ and }(a+b)^{n+3},\text{ the ratio of the} \\ \text{coefficients of second and third terms, and third and fourth terms respectively} \\ \text{are equal, then }n\text{ is}
\displaystyle \text{(a) }3\qquad \text{(b) }4\qquad \text{(c) }5\qquad \text{(d) }6
\displaystyle \text{Answer:}
\displaystyle \text{In }(a+b)^n,
\displaystyle \text{Second term coefficient}={}^{n}C_1=n
\displaystyle \text{Third term coefficient}={}^{n}C_2=\frac{n(n-1)}{2}
\displaystyle \therefore \frac{\text{Second coefficient}}{\text{Third coefficient}}=\frac{n}{\frac{n(n-1)}{2}}=\frac{2}{n-1}
\displaystyle \text{In }(a+b)^{n+3},
\displaystyle \text{Third term coefficient}={}^{n+3}C_2=\frac{(n+3)(n+2)}{2}
\displaystyle \text{Fourth term coefficient}={}^{n+3}C_3=\frac{(n+3)(n+2)(n+1)}{6}
\displaystyle \therefore \frac{\text{Third coefficient}}{\text{Fourth coefficient}}=\frac{\frac{(n+3)(n+2)}{2}}{\frac{(n+3)(n+2)(n+1)}{6}}=\frac{3}{n+1}
\displaystyle \text{Given ratios are equal,}
\displaystyle \frac{2}{n-1}=\frac{3}{n+1}
\displaystyle 2(n+1)=3(n-1)
\displaystyle 2n+2=3n-3
\displaystyle n=5
\displaystyle \therefore\text{Correct option is (c).}
\\

\displaystyle \textbf{Question 5. }\text{If }A\text{ and }B\text{ are the sums of odd and even terms respectively in the expansion}
\displaystyle \text{of }(x+a)^n,\text{ then }(x+a)^{2n}-(x-a)^{2n}\text{ is equal to}
\displaystyle \text{(a) }4(A+B)\qquad \text{(b) }4(A-B)\qquad \text{(c) }AB\qquad \text{(d) }4AB
\displaystyle \text{Answer:}
\displaystyle A+B=(x+a)^n
\displaystyle B-A=(x-a)^n
\displaystyle \therefore (x+a)^n=A+B,\quad (x-a)^n=B-A
\displaystyle (x+a)^{2n}-(x-a)^{2n}=(A+B)^2-(B-A)^2
\displaystyle =4AB
\displaystyle \therefore \text{Correct option is (d).}
\\

\displaystyle \textbf{Question 6. }\text{The number of irrational terms in the expansion of }\left(4^{1/5}+7^{1/10}\right)^{45}\text{ is}
\displaystyle \text{(a) }40\qquad \text{(b) }5\qquad \text{(c) }41\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{45}C_r\left(4^{1/5}\right)^{45-r}\left(7^{1/10}\right)^r
\displaystyle ={}^{45}C_r4^{\frac{45-r}{5}}7^{\frac{r}{10}}
\displaystyle \text{For rational terms, }\frac{r}{10}\text{ must be an integer.}
\displaystyle \therefore r=0,10,20,30,40
\displaystyle \text{Hence there are }5\text{ rational terms.}
\displaystyle \text{Total number of terms}=46
\displaystyle \therefore \text{Number of irrational terms}=46-5=41
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 7. }\text{The coefficient of }x^{-17}\text{ in the expansion of }\left(x^4-\frac{1}{x^3}\right)^{15}\text{ is}
\displaystyle \text{(a) }1365\qquad \text{(b) }-1365\qquad \text{(c) }3003\qquad \text{(d) }-3003
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{15}C_r(x^4)^{15-r}\left(-\frac{1}{x^3}\right)^r
\displaystyle ={}^{15}C_r(-1)^rx^{60-7r}
\displaystyle \text{For the coefficient of }x^{-17},\ 60-7r=-17
\displaystyle 7r=77
\displaystyle r=11
\displaystyle \therefore \text{Required coefficient}={} ^{15}C_{11}(-1)^{11}
\displaystyle =-{}^{15}C_4=-1365
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 8. }\text{In the expansion of }\left(x^2-\frac{1}{3x}\right)^9,\text{ the term without }x\text{ is equal to}
\displaystyle \text{(a) }\frac{28}{81}\qquad \text{(b) }-\frac{28}{243}\qquad \text{(c) }\frac{28}{243}\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={}^9C_r(x^2)^{9-r}\left(-\frac{1}{3x}\right)^r
\displaystyle ={}^9C_r\left(-\frac{1}{3}\right)^r x^{18-3r}
\displaystyle \text{For the term independent of }x,\ 18-3r=0
\displaystyle r=6
\displaystyle \therefore \text{Required term}={}^9C_6\left(-\frac{1}{3}\right)^6
\displaystyle =84\times\frac{1}{729}=\frac{28}{243}
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 9. }\text{If in the expansion of }(1+x)^{15},\text{ the coefficients of }(2r+3)\text{th}
\displaystyle \text{and }(r-1)\text{th terms are equal, then the value of }r\text{ is}
\displaystyle \text{(a) }5\qquad \text{(b) }6\qquad \text{(c) }4\qquad \text{(d) }3
\displaystyle \text{Answer:}
\displaystyle \text{Coefficient of }(2r+3)\text{th term}={} ^{15}C_{2r+2}
\displaystyle \text{Coefficient of }(r-1)\text{th term}={} ^{15}C_{r-2}
\displaystyle {}^{15}C_{2r+2}={} ^{15}C_{r-2}
\displaystyle \therefore (2r+2)+(r-2)=15
\displaystyle 3r=15
\displaystyle r=5
\displaystyle \therefore \text{Correct option is (a).}
\\

\displaystyle \textbf{Question 10. }\text{The middle term in the expansion of }\left(\frac{2x^2}{3}+\frac{3}{2x^2}\right)^{10}\text{ is}
\displaystyle \text{(a) }251\qquad \text{(b) }252\qquad \text{(c) }250\qquad \text{(d) none of these}
\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=252
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 11. }\text{If in the expansion of }\left(x^4-\frac{1}{x^3}\right)^{15},\ x^{-17}\text{ occurs in }r\text{th term, then}
\displaystyle \text{(a) }r=10\qquad \text{(b) }r=11\qquad \text{(c) }r=12\qquad \text{(d) }r=13
\displaystyle \text{Answer:}
\displaystyle T_r={} ^{15}C_{r-1}(x^4)^{16-r}\left(-\frac{1}{x^3}\right)^{r-1}
\displaystyle \text{Power of }x=4(16-r)-3(r-1)
\displaystyle =64-4r-3r+3=67-7r
\displaystyle \text{Given }67-7r=-17
\displaystyle 7r=84
\displaystyle r=12
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 12. }\text{In the expansion of }\left(x-\frac{1}{3x^2}\right)^9,\text{ the term independent of }x\text{ is}
\displaystyle \text{(a) }T_3\qquad \text{(b) }T_4\qquad \text{(c) }T_5\qquad \text{(d) none of these}
\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^{9-3r}
\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
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 13. }\text{If in the expansion of }(1+y)^n,\text{ the coefficients of }5\text{th, }6\text{th and }7\text{th terms}
\displaystyle \text{are in A.P., then }n\text{ is equal to}
\displaystyle \text{(a) }7,11\qquad \text{(b) }7,14\qquad \text{(c) }8,16\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{Coefficients of }5\text{th, }6\text{th and }7\text{th terms are }{}^nC_4,\ {}^nC_5,\ {}^nC_6.
\displaystyle \text{Since they are in A.P., }2{}^nC_5={}^nC_4+{}^nC_6
\displaystyle \frac{{}^nC_5}{{}^nC_4}=\frac{n-4}{5},\qquad \frac{{}^nC_6}{{}^nC_5}=\frac{n-5}{6}
\displaystyle 2=\frac{5}{n-4}+\frac{n-5}{6}
\displaystyle 12(n-4)=30+(n-5)(n-4)
\displaystyle n^2-21n+98=0
\displaystyle (n-7)(n-14)=0
\displaystyle \therefore n=7,14
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 14. }\text{In the expansion of }\left(\frac{1}{2}x^{1/3}+x^{-1/5}\right)^8,\text{ the term independent of }x\text{ is}
\displaystyle \text{(a) }T_5\qquad \text{(b) }T_6\qquad \text{(c) }T_7\qquad \text{(d) }T_8
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={}^8C_r\left(\frac{1}{2}x^{1/3}\right)^{8-r}\left(x^{-1/5}\right)^r
\displaystyle \text{Power of }x=\frac{8-r}{3}-\frac{r}{5}
\displaystyle \text{For the term independent of }x,\ \frac{8-r}{3}-\frac{r}{5}=0
\displaystyle 5(8-r)=3r
\displaystyle 40=8r
\displaystyle r=5
\displaystyle \therefore \text{Required term is }T_{r+1}=T_6
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 15. }\text{If the sum of odd numbered terms and the sum of even numbered terms in}
\displaystyle \text{the expansion of }(x+a)^n\text{ are }A\text{ and }B\text{ respectively, then the value of } \\ (x^2-a^2)^n\text{ is}
\displaystyle \text{(a) }A^2-B^2\qquad \text{(b) }A^2+B^2\qquad \text{(c) }4AB\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle A+B=(x+a)^n
\displaystyle A-B=(x-a)^n
\displaystyle \therefore (x^2-a^2)^n=(x+a)^n(x-a)^n
\displaystyle =(A+B)(A-B)=A^2-B^2
\displaystyle \therefore \text{Correct option is (a).}
\\

\displaystyle \textbf{Question 16. }\text{If the coefficient of }x\text{ in }\left(x^2+\frac{\lambda}{x}\right)^5\text{ is }270,\text{ then }\lambda=
\displaystyle \text{(a) }3\qquad \text{(b) }4\qquad \text{(c) }5\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={}^5C_r(x^2)^{5-r}\left(\frac{\lambda}{x}\right)^r
\displaystyle ={}^5C_r\lambda^r x^{10-3r}
\displaystyle \text{For coefficient of }x,\ 10-3r=1
\displaystyle r=3
\displaystyle \therefore {}^5C_3\lambda^3=270
\displaystyle 10\lambda^3=270
\displaystyle \lambda^3=27
\displaystyle \lambda=3
\displaystyle \therefore \text{Correct option is (a).}
\\

\displaystyle \textbf{Question 17. }\text{The coefficient of }x^4\text{ in }\left(\frac{x}{2}-\frac{3}{x^2}\right)^{10}\text{ is}
\displaystyle \text{(a) }\frac{405}{256}\qquad \text{(b) }\frac{504}{259}\qquad \text{(c) }\frac{450}{263}\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{10}C_r\left(\frac{x}{2}\right)^{10-r}\left(-\frac{3}{x^2}\right)^r
\displaystyle \text{Power of }x=10-r-2r=10-3r
\displaystyle \text{For coefficient of }x^4,\ 10-3r=4
\displaystyle r=2
\displaystyle \therefore \text{Required coefficient}={} ^{10}C_2\left(\frac{1}{2}\right)^8(-3)^2
\displaystyle =45\times\frac{1}{256}\times9=\frac{405}{256}
\displaystyle \therefore \text{Correct option is (a).}
\\

\displaystyle \textbf{Question 18. }\text{The total number of terms in the expansion of } \\ (x+a)^{100}+(x-a)^{100}\text{ after simplification is}
\displaystyle \text{(a) }202\qquad \text{(b) }51\qquad \text{(c) }50\qquad \text{(d) none of these}
\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 \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 19. }\text{If }T_2/T_3\text{ in the expansion of }(a+b)^n\text{ and }T_3/T_4\text{ in the expansion of} \\ (a+b)^{n+3}\text{ are equal, then }n=
\displaystyle \text{(a) }3\qquad \text{(b) }4\qquad \text{(c) }5\qquad \text{(d) }6
\displaystyle \text{Answer:}
\displaystyle \text{In }(a+b)^n,
\displaystyle T_2={}^{n}C_1a^{n-1}b,\qquad T_3={}^{n}C_2a^{n-2}b^2
\displaystyle \therefore \frac{T_2}{T_3}=\frac{{}^{n}C_1}{{}^{n}C_2}\cdot\frac{a}{b}=\frac{2}{n-1}\cdot\frac{a}{b}
\displaystyle \text{In }(a+b)^{n+3},
\displaystyle T_3={}^{n+3}C_2a^{n+1}b^2,\qquad T_4={}^{n+3}C_3a^nb^3
\displaystyle \therefore \frac{T_3}{T_4}=\frac{{}^{n+3}C_2}{{}^{n+3}C_3}\cdot\frac{a}{b}=\frac{3}{n+1}\cdot\frac{a}{b}
\displaystyle \text{Given, }\frac{T_2}{T_3}=\frac{T_3}{T_4}
\displaystyle \therefore \frac{2}{n-1}\cdot\frac{a}{b}=\frac{3}{n+1}\cdot\frac{a}{b}
\displaystyle \frac{2}{n-1}=\frac{3}{n+1}
\displaystyle 2(n+1)=3(n-1)
\displaystyle 2n+2=3n-3
\displaystyle n=5
\displaystyle \therefore\text{Correct option is (c).}
\\

\displaystyle \textbf{Question 20. }\text{The coefficient of }\frac{1}{x}\text{ in the expansion of }(1+x)^n\left(1+\frac{1}{x}\right)^n\text{ is}
\displaystyle \text{(a) }\frac{n!}{(n-1)!(n+1)!}\qquad \text{(b) }\frac{(2n)!}{(n-1)!(n+1)!}\qquad \text{(c) }\frac{(2n)!}{(2n-1)!(2n+1)!}\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle (1+x)^n\left(1+\frac{1}{x}\right)^n=(1+x)^n\left(\frac{1+x}{x}\right)^n
\displaystyle =\frac{(1+x)^{2n}}{x^n}
\displaystyle \text{Coefficient of }\frac{1}{x}\text{ is coefficient of }x^{n-1}\text{ in }(1+x)^{2n}
\displaystyle \therefore \text{Required coefficient}={} ^{2n}C_{n-1}
\displaystyle =\frac{(2n)!}{(n-1)!(n+1)!}
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 21. }\text{If the sum of the binomial coefficients of the expansion }\left(2x+\frac{1}{x}\right)^n
\displaystyle \text{ is equal to }256, \text{ then the term independent of }x\text{ is}
\displaystyle \text{(a) }1120\qquad \text{(b) }1020\qquad \text{(c) }512\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{Sum of binomial coefficients}=2^n=256
\displaystyle \therefore n=8
\displaystyle T_{r+1}={}^8C_r(2x)^{8-r}\left(\frac{1}{x}\right)^r
\displaystyle ={}^8C_r2^{8-r}x^{8-2r}
\displaystyle \text{For the term independent of }x,\ 8-2r=0
\displaystyle r=4
\displaystyle \therefore \text{Required term}={}^8C_4 2^4=70\times16=1120
\displaystyle \therefore \text{Correct option is (a).}
\\

\displaystyle \textbf{Question 22. }\text{If the fifth term of the expansion }\left(a^{2/3}+a^{-1}\right)^n\text{ does not contain }a,
\displaystyle \text{then }n\text{ is equal to}
\displaystyle \text{(a) }2\qquad \text{(b) }5\qquad \text{(c) }10\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle T_5={}^nC_4\left(a^{2/3}\right)^{n-4}\left(a^{-1}\right)^4
\displaystyle \text{Power of }a=\frac{2}{3}(n-4)-4
\displaystyle \text{Since }T_5\text{ does not contain }a,\ \frac{2}{3}(n-4)-4=0
\displaystyle 2(n-4)=12
\displaystyle n=10
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 23. }\text{The coefficient of }x^{-3}\text{ in the expansion of }\left(x-\frac{m}{x}\right)^{11}\text{ is}
\displaystyle \text{(a) }-924m^7\qquad \text{(b) }-792m^5\qquad \text{(c) }-792m^6\qquad \text{(d) }-330m^7
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{11}C_r x^{11-r}\left(-\frac{m}{x}\right)^r
\displaystyle ={}^{11}C_r(-m)^r x^{11-2r}
\displaystyle \text{For coefficient of }x^{-3},\ 11-2r=-3
\displaystyle r=7
\displaystyle \therefore \text{Required coefficient}={} ^{11}C_7(-m)^7
\displaystyle =-330m^7
\displaystyle \therefore \text{Correct option is (d).}
\\

\displaystyle \textbf{Question 24. }\text{The coefficient of the term independent of }x\text{ in the expansion of }\left(ax+\frac{b}{x}\right)^{14}\text{ is}
\displaystyle \text{(a) }14!a^7b^7\qquad \text{(b) }\frac{14!}{7!}a^7b^7\qquad \text{(c) }\frac{14!}{(7!)^2}a^7b^7\qquad \text{(d) }\frac{14!}{(7!)^3}a^7b^7
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{14}C_r(ax)^{14-r}\left(\frac{b}{x}\right)^r
\displaystyle ={}^{14}C_r a^{14-r}b^r x^{14-2r}
\displaystyle \text{For term independent of }x,\ 14-2r=0
\displaystyle r=7
\displaystyle \therefore \text{Required term}={} ^{14}C_7a^7b^7
\displaystyle =\frac{14!}{(7!)^2}a^7b^7
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 25. }\text{The coefficient of }x^5\text{ in the expansion of }(1+x)^{21}+(1+x)^{22}+\cdots+(1+x)^{30}\text{ is}
\displaystyle \text{(a) }{}^{51}C_5\qquad \text{(b) }{}^{9}C_5\qquad \text{(c) }{}^{31}C_6-{}^{21}C_6\qquad \text{(d) }{}^{30}C_5+{}^{20}C_5
\displaystyle \text{Answer:}
\displaystyle \text{Required coefficient}={}^{21}C_5+{}^{22}C_5+\cdots+{}^{30}C_5
\displaystyle \text{Using }\sum_{r=m}^{n}{}^{r}C_k={}^{n+1}C_{k+1}-{}^{m}C_{k+1},
\displaystyle {}^{21}C_5+{}^{22}C_5+\cdots+{}^{30}C_5={}^{31}C_6-{}^{21}C_6
\displaystyle \therefore\text{Correct option is (c).}
\\

\displaystyle \textbf{Question 26. }\text{The coefficient of }x^8y^{10}\text{ in the expansion of }(x+y)^{18}\text{ is}
\displaystyle \text{(a) }{}^{18}C_8\qquad \text{(b) }{}^{18}P_{10}\qquad \text{(c) }2^{18}\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{18}C_r x^{18-r}y^r
\displaystyle \text{For }x^8y^{10},\ r=10
\displaystyle \therefore \text{Required coefficient}={} ^{18}C_{10}={} ^{18}C_8
\displaystyle \therefore \text{Correct option is (a).}
\\

\displaystyle \textbf{Question 27. }\text{If the coefficients of the }(n+1)\text{th term and the }(n+3)\text{th term in the expansion}
\displaystyle \text{of }(1+x)^{20}\text{ are equal, then the value of }n\text{ is}
\displaystyle \text{(a) }10\qquad \text{(b) }8\qquad \text{(c) }9\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{Coefficient of }(n+1)\text{th term}={} ^{20}C_n
\displaystyle \text{Coefficient of }(n+3)\text{th term}={} ^{20}C_{n+2}
\displaystyle {}^{20}C_n={} ^{20}C_{n+2}
\displaystyle \therefore n+(n+2)=20
\displaystyle 2n=18
\displaystyle n=9
\displaystyle \therefore \text{Correct option is (c).}
\\

\displaystyle \textbf{Question 28. }\text{If the coefficients of 2nd, 3rd and 4th terms in the expansion of } \\ (1+x)^n,\ n\in N
\displaystyle \text{are in A.P., then }n=
\displaystyle \text{(a) }7\qquad \text{(b) }14\qquad \text{(c) }2\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{Coefficients of 2nd, 3rd and 4th terms are }{}^nC_1,\ {}^nC_2,\ {}^nC_3.
\displaystyle \text{Since they are in A.P., }2{}^nC_2={}^nC_1+{}^nC_3
\displaystyle 2\cdot\frac{n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6}
\displaystyle 6(n-1)=6+(n-1)(n-2)
\displaystyle n^2-9n+14=0
\displaystyle (n-7)(n-2)=0
\displaystyle \therefore n=7\text{ or }2
\displaystyle \therefore \text{Correct options are (a) and (c).}
\\

\displaystyle \textbf{Question 29. }\text{The middle term in the expansion of }\left(\frac{2x}{3}-\frac{3}{2x^2}\right)^{2n}\text{ is}
\displaystyle \text{(a) }{}^{2n}C_n\qquad \text{(b) }(-1)^n{}^{2n}C_nx^{-n}\qquad \text{(c) }{}^{2n}C_nx^{-n}\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{Since the power is }2n,\text{ the middle term is }T_{n+1}.
\displaystyle T_{n+1}={} ^{2n}C_n\left(\frac{2x}{3}\right)^n\left(-\frac{3}{2x^2}\right)^n
\displaystyle =(-1)^n{}^{2n}C_nx^{-n}
\displaystyle \therefore \text{Correct option is (b).}
\\

\displaystyle \textbf{Question 30. }\text{If }r\text{th term is the middle term in the expansion of }\left(x^2-\frac{1}{2x}\right)^{20},
\displaystyle \text{then }(r+3)\text{th term is}
\displaystyle \text{(a) }{}^{20}C_{14}\left(\frac{x}{2^{14}}\right)\qquad \text{(b) }{}^{20}C_{12}x^2\,2^{-12}\qquad \text{(c) }-{}^{20}C_7x\,2^{-13}\qquad \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{Since the power is }20,\text{ the middle term is the }11\text{th term.}
\displaystyle \therefore r=11
\displaystyle \therefore (r+3)\text{th term}=14\text{th term}
\displaystyle T_{14}={} ^{20}C_{13}(x^2)^7\left(-\frac{1}{2x}\right)^{13}
\displaystyle =-{}^{20}C_{13}\frac{x}{2^{13}}
\displaystyle =-{}^{20}C_7x\,2^{-13}
\displaystyle \therefore \text{Correct option is (c).}
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\displaystyle \textbf{Question 31. }\text{The number of terms with integral coefficients in the expansion of } \\ \left(17^{1/3}+35^{1/2}x\right)^{600}  \text{ is}
\displaystyle \text{(a) }100\qquad \text{(b) }50\qquad \text{(c) }150\qquad \text{(d) }101
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{600}C_r\left(17^{1/3}\right)^{600-r}\left(35^{1/2}x\right)^r
\displaystyle ={}^{600}C_r17^{\frac{600-r}{3}}35^{\frac{r}{2}}x^r
\displaystyle \text{For integral coefficient, }600-r\text{ must be divisible by }3\text{ and }r\text{ by }2.
\displaystyle \therefore r\text{ must be a multiple of }6.
\displaystyle r=0,6,12,\ldots,600
\displaystyle \therefore \text{Number of terms}=101
\displaystyle \therefore \text{Correct option is (d).}
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\displaystyle \textbf{Question 32. }\text{Constant term in the expansion of }\left(x-\frac{1}{x}\right)^{10}\text{ is}
\displaystyle \text{(a) }152\qquad \text{(b) }-152\qquad \text{(c) }-252\qquad \text{(d) }252
\displaystyle \text{Answer:}
\displaystyle T_{r+1}={} ^{10}C_r x^{10-r}\left(-\frac{1}{x}\right)^r
\displaystyle ={}^{10}C_r(-1)^rx^{10-2r}
\displaystyle \text{For constant term, }10-2r=0
\displaystyle r=5
\displaystyle \therefore \text{Constant term}={} ^{10}C_5(-1)^5=-252
\displaystyle \therefore \text{Correct option is (c).}
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\displaystyle \textbf{Question 33. }\text{If the coefficients of }x^2\text{ and }x^3\text{ in the expansion of }(3+ax)^9\text{ are the same,}
\displaystyle \text{then the value of }a\text{ is}
\displaystyle \text{(a) }-\frac{7}{9}\qquad \text{(b) }-\frac{9}{7}\qquad \text{(c) }\frac{7}{9}\qquad \text{(d) }\frac{9}{7}
\displaystyle \text{Answer:}
\displaystyle \text{Coefficient of }x^2={} ^9C_2\,3^7a^2
\displaystyle \text{Coefficient of }x^3={} ^9C_3\,3^6a^3
\displaystyle {}^9C_2\,3^7a^2={}^9C_3\,3^6a^3
\displaystyle 36\cdot3a^2=84a^3
\displaystyle 108a^2=84a^3
\displaystyle a=\frac{108}{84}=\frac{9}{7}
\displaystyle \therefore \text{Correct option is (d).}
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