JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank

done Mock Test - Binomial Theorem Total Questions - 25

• question_answer1)
  If \[{{x}^{m}}\]occurs in the expansion of \[{{(x+1/{{x}^{2}})}^{2n}}\], then the coefficient of \[{{x}^{m}}\]is
  A)
  \[\frac{(2n)!}{(m)!(2n-m)!}\]       done clear
  B)
  \[\frac{(2n)!3!3!}{(2n-m)!}\] done clear
  C)
  \[\frac{(2n)!}{\left( \frac{2n-m}{3} \right)!\left( \frac{4n+m}{3} \right)!}\] done clear
  D)
  none of these done clear
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• question_answer2)
  In the expansion of \[{{(1+3x+2{{x}^{2}})}^{6}}\], the coefficient of \[{{x}^{11}}\]is
  A)
  144                   done clear
  B)
  288 done clear
  C)
  216                  done clear
  D)
  576 done clear
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• question_answer3)
  The coefficient of the middle term in the binomial expansion in powers of x of \[{{(1+ax)}^{4}}\]and of \[{{(1-ax)}^{6}}\]is the same, if \[\alpha \]equals
  A)
  \[-\frac{5}{3}\]             done clear
  B)
  \[\frac{10}{3}\] done clear
  C)
  \[-\frac{3}{10}\]            done clear
  D)
  \[\frac{3}{5}\] done clear
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• question_answer4)
  The coefficient of \[{{x}^{5}}\]in the expansion of \[{{({{x}^{2}}-x-2)}^{5}}\]is
  A)
  -83                    done clear
  B)
  -82 done clear
  C)
  -86                   done clear
  D)
  -81 done clear
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• question_answer5)
  The coefficient of \[{{x}^{10}}\]in the expansion of \[{{(1+{{x}^{2}}-{{x}^{3}})}^{8}}\]is
  A)
  476       done clear
  B)
  496 done clear
  C)
  506                  done clear
  D)
  528 done clear
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• question_answer6)
  If coefficient of \[{{a}^{2}}{{b}^{3}}{{c}^{4}}\]in \[{{(a+b+c)}^{m}}\] (where m\[\in \]N) is L (L\[\ne \]0), then in same expansion coefficient of \[{{a}^{4}}{{b}^{4}}{{c}^{1}}\]will be
  A)
  L                      done clear
  B)
  \[\frac{L}{3}\] done clear
  C)
  \[\frac{mL}{4}\]                       done clear
  D)
  \[\frac{L}{2}\] done clear
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• question_answer7)
  If the \[{{6}^{th}}\]term in the expansion of \[{{\left( \frac{1}{{{x}^{8/3}}}+{{x}^{2}}{{\log }_{10}}x \right)}^{8}}\]is 5600, then x equals
  A)
  1                      done clear
  B)
  \[{{\log }_{e}}10\] done clear
  C)
  10                    done clear
  D)
  x does not exist done clear
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• question_answer8)
  if\[I{{f}^{n+1}}{{C}_{r+1}}{{:}^{n}}{{C}_{r}}{{:}^{n-1}}{{C}_{r-1}}=11:6:3\], then nr=
  A)
  20                     done clear
  B)
  30 done clear
  C)
  40                    done clear
  D)
  50 done clear
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• question_answer9)
  The number of distinct terms in the expansion of \[{{\left( x+\frac{1}{x}+{{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{15}}\]is/are (with respect to different power of x)
  A)
  255                   done clear
  B)
  61 done clear
  C)
  127                  done clear
  D)
  none of these done clear
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• question_answer10)
  If the sum of the coefficients in the expansion of \[{{(a+b)}^{n}}\]is 4096, then the greatest coefficient in the expansion is
  A)
  924       done clear
  B)
  792 done clear
  C)
  1594                done clear
  D)
  none of these done clear
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• question_answer11)
  The fractional part of \[{{2}^{4n}}/15\]is \[(n\in N)\]
  A)
  \[\frac{1}{15}\]             done clear
  B)
  \[\frac{2}{15}\] done clear
  C)
  \[\frac{4}{15}\]             done clear
  D)
  none of these done clear
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• question_answer12)
  The value of \[\frac{^{n}{{C}_{0}}}{n}+\frac{^{n}{{C}_{1}}}{n+1}+\frac{^{n}{{C}_{2}}}{n+2}+...+\frac{^{n}{{C}_{n}}}{2n}\]is equal to
  A)
  \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1-x)}^{n}}dx}\] done clear
  B)
  \[\int\limits_{0}^{1}{{{x}^{n}}{{(x-1)}^{n-1}}dx}\] done clear
  C)
  \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1+x)}^{n}}dx}\] done clear
  D)
  \[\int\limits_{0}^{1}{{{(1-x)}^{n}}{{x}^{n-1}}dx}\] done clear
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• question_answer13)
  If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}},\]then \[{{C}_{0}}{{C}_{2}}+{{C}_{1}}{{C}_{3}}+{{C}_{2}}{{C}_{4}}+...+{{C}_{n-2}}{{C}_{n}}=\]
  A)
  \[\frac{(2n)!}{{{(n!)}^{2}}}\]   done clear
  B)
  \[\frac{(2n)!}{(n-1)!(n+1)!}\] done clear
  C)
  \[\frac{(2n)!}{(n-2)!(n+2)!}\] done clear
  D)
  none of these done clear
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• question_answer14)
  If \[{{(3+{{x}^{2008}}+{{x}^{2009}})}^{2010}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{n}}{{x}^{n}}\], then the value of \[{{a}_{0}}-\frac{1}{2}{{a}_{1}}-\frac{1}{2}{{a}_{2}}+{{a}_{3}}-\frac{1}{2}{{a}_{4}}-\frac{1}{2}{{a}_{5}}+{{a}_{6}}-...\]is
  A)
  \[{{3}^{2010}}\]         done clear
  B)
  1 done clear
  C)
  \[{{2}^{2010}}\]         done clear
  D)
  none of these done clear
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• question_answer15)
  The value of \[\sum\limits_{r=0}^{50}{{{(-1)}^{r}}}\]\[\frac{^{50}{{C}_{r}}}{r+2}\]is equal to
  A)
  \[\frac{1}{50\times 51}\]            done clear
  B)
  \[\frac{1}{52\times 50}\] done clear
  C)
  \[\frac{1}{52\times 51}\]            done clear
  D)
  none of these done clear
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• question_answer16)
  \[1+\frac{1}{3}x+\frac{1\times 4}{3\times 6}{{x}^{2}}+\frac{1\times 4\times 7}{3\times 6\times 9}{{x}^{3}}+...\]is equal to
  A)
  x                      done clear
  B)
  \[{{(1+x)}^{1/3}}\] done clear
  C)
  \[{{(1-x)}^{1/3}}\]      done clear
  D)
  \[{{(1-x)}^{-1/3}}\] done clear
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• question_answer17)
  The coefficient of \[{{x}^{5}}\]in \[{{(1+2x+3{{x}^{2}}+...)}^{-3/2}}\]is \[\left( \left| x \right|<1 \right)\]
  A)
  21                     done clear
  B)
  25 done clear
  C)
  26                    done clear
  D)
  none of these done clear
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• question_answer18)
  If \[f(x)=1-x+{{x}^{2}}-{{x}^{3}}+...-{{x}^{15}}+{{x}^{16}}-{{x}^{17}}\]then the coefficient of \[{{x}^{2}}\]in f(x-1) is
  A)
  826                   done clear
  B)
  816 done clear
  C)
  822                  done clear
  D)
  none of these done clear
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• question_answer19)
  Let \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+...+{{a}_{n}}{{x}^{n}}+...\]and \[\frac{f(x)}{1-x}={{b}_{0}}+{{b}_{1}}x+{{b}_{2}}{{x}^{2}}+...+{{b}_{n}}{{x}^{n}}+....,\]then
  A)
  \[{{b}_{n}}={{b}_{n-1}}={{a}_{n}}\] done clear
  B)
  \[{{b}_{n}}-{{b}_{n-1}}={{a}_{n}}\] done clear
  C)
  \[{{b}_{n}}/{{b}_{n-1}}={{a}_{n}}\] done clear
  D)
  none of these done clear
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• question_answer20)
  p is a prime number and n<p<2n. if N=\[^{2n}{{C}_{n}}\], then
  A)
  p divides N         done clear
  B)
  \[{{p}^{2}}\]divides N done clear
  C)
  p cannot divide N done clear
  D)
  none of these done clear
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• question_answer21)
  The sum of the coefficients of the first 10 terms in the expansion of (\[{{(1-x)}^{-3}}\] is ___________.
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• question_answer22)
  The number of distinct terms in the expansion of \[{{(x+{{y}^{2}})}^{13}}+{{({{x}^{2}}+y)}^{14}}\]is___.
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• question_answer23)
  If \[{{6}^{th}}\]term in the expansion of \[{{\left( \frac{3}{2}+\frac{x}{3} \right)}^{n}}\]is numerically greatest, when x=3, then the sum of possible integral values of 'n' is _____________.
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• question_answer24)
  The positive integer just greater than \[{{(1+0.0001)}^{10000}}\] is ____________.
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• question_answer25)
  If the sum of the coefficients in the expansion of (a+b) is 4096, then the greatest coefficient in the expansion is __________.
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