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

done Properties of binomial coefficients Total Questions - 44

• question_answer1)
  \[^{10}{{C}_{1}}{{+}^{10}}{{C}_{3}}{{+}^{10}}{{C}_{5}}{{+}^{10}}{{C}_{7}}{{+}^{10}}{{C}_{9}}=\] [MP PET 1982]
  A)
  \[{{2}^{9}}\] done clear
  B)
  \[{{2}^{10}}\] done clear
  C)
  \[{{2}^{10}}-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  \[{{C}_{0}}{{C}_{r}}+{{C}_{1}}{{C}_{r+1}}+{{C}_{2}}{{C}_{r+2}}+....+{{C}_{n-r}}{{C}_{n}}\]= [BIT Ranchi 1986]
  A)
  \[\frac{(2n)!}{(n-r)\,!\,(n+r)!}\] done clear
  B)
  \[\frac{n!}{(-r)!(n+r)!}\] done clear
  C)
  \[\frac{n!}{(n-r)!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  \[^{n}{{C}_{0}}-\frac{1}{2}{{\,}^{n}}{{C}_{1}}+\frac{1}{3}{{\,}^{n}}{{C}_{2}}-......+{{(-1)} ^{n}}\frac{^{n}{{C}_{n}}}{n+1}=\]
  A)
  n done clear
  B)
  1/n done clear
  C)
  \[\frac{1}{n+1}\] done clear
  D)
  \[\frac{1}{n-1}\] done clear
  View Solution play_arrow

• question_answer4)
  If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+..........+{{C}_{n}}{{x}^{2}},\] then \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+C_{3}^{2}+......+C_{n}^{2}\] = [MP PET 1985; Karnataka CET 1995; MNR 1999]
  A)
  \[\frac{n!}{n!n!}\] done clear
  B)
  \[\frac{(2n)!}{n!n!}\] done clear
  C)
  \[\frac{(2n)!}{n!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+..........+{{C}_{n}}{{x}^{n}}\],  then \[\frac{{{C}_{1}}}{{{C}_{0}}}+\frac{2{{C}_{2}}}{{{C}_{1}}}+\frac{3{{C}_{3}}}{{{C}_{2}}}+....+\frac{n{{C}_{n}}}{{{C}_{n-1}}}=\] [BIT Ranchi 1986; RPET 1996, 97]
  A)
  \[\frac{n(n-1)}{2}\] done clear
  B)
  \[\frac{n(n+2)}{2}\] done clear
  C)
  \[\frac{n(n+1)}{2}\] done clear
  D)
  \[\frac{(n-1)(n-2)}{2}\] done clear
  View Solution play_arrow

• question_answer6)
  \[{{C}_{1}}+2{{C}_{2}}+3{{C}_{3}}+4{{C}_{4}}+....+n{{C}_{n}}=\] [RPET 1995; MP PET 2002; Orissa JEE 2005]
  A)
  \[{{2}^{n}}\] done clear
  B)
  \[n.\,\,{{2}^{n}}\] done clear
  C)
  \[n.\,\,{{2}^{n-1}}\] done clear
  D)
  \[n.\,\,{{2}^{n+1}}\] done clear
  View Solution play_arrow

• question_answer7)
  \[\frac{{{C}_{0}}}{1}+\frac{{{C}_{2}}}{3}+\frac{{{C}_{4}}}{5}+\frac{{{C}_{6}}}{7}+....\]= [RPET 1999]
  A)
  \[\frac{{{2}^{n+1}}}{n+1}\] done clear
  B)
  \[\frac{{{2}^{n+1}}-1}{n+1}\] done clear
  C)
  \[\frac{{{2}^{n}}}{n+1}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  \[\frac{{{C}_{0}}}{1}+\frac{{{C}_{1}}}{2}+\frac{{{C}_{2}}}{3}+....+\frac{{{C}_{n}}}{n+1}=\] [RPET 1996]
  A)
  \[\frac{{{2}^{n}}}{n+1}\] done clear
  B)
  \[\frac{{{2}^{n}}-1}{n+1}\] done clear
  C)
  \[\frac{{{2}^{n+1}}-1}{n+1}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  \[\frac{1}{1!(n-1)\,!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+....=\]  [AMU 2005]
  A)
  \[\frac{{{2}^{n}}}{n!}\]; for all even values of n done clear
  B)
  \[\frac{{{2}^{n-1}}}{n!}\]; for all values of n i.e., all even odd values done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer10)
  The sum to \[(n+1)\] terms of the following series \[\frac{{{C}_{0}}}{2}-\frac{{{C}_{1}}}{3}+\frac{{{C}_{2}}}{4}-\frac{{{C}_{3}}}{5}+\]..... is
  A)
  \[\frac{1}{n+1}\] done clear
  B)
  \[\frac{1}{n+2}\] done clear
  C)
  \[\frac{1}{n(n+1)}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer11)
  If a and d are two complex numbers, then the sum to \[(n+1)\] terms of the following series \[a{{C}_{0}}-(a+d){{C}_{1}}+(a+2d){{C}_{2}}-........\] is
  A)
  \[\frac{a}{{{2}^{n}}}\] done clear
  B)
  \[na\] done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  If  \[{{(1+x)}^{15}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+......+{{C}_{15}}{{x}^{15}},\] then \[{{C}_{2}}+2{{C}_{3}}+3{{C}_{4}}+....+14{{C}_{15}}=\] [IIT 1966]
  A)
  \[{{14.2}^{14}}\] done clear
  B)
  \[{{13.2}^{14}}+1\] done clear
  C)
  \[{{13.2}^{14}}-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  The value of \[\frac{{{C}_{1}}}{2}+\frac{{{C}_{3}}}{4}+\frac{{{C}_{5}}}{6}+.....\]is equal to [Karnataka CET 2000]
  A)
  \[\frac{{{2}^{n}}-1}{n+1}\] done clear
  B)
  \[n{{.2}^{n}}\] done clear
  C)
  \[\frac{{{2}^{n}}}{n}\] done clear
  D)
  \[\frac{{{2}^{n}}+1}{n+1}\] done clear
  View Solution play_arrow

• question_answer14)
  In the expansion of \[{{(1+x)}^{n}}\] the sum of coefficients of odd powers of x is       [MP PET 1986, 93, 2003]
  A)
  \[{{2}^{n}}+1\] done clear
  B)
  \[{{2}^{n}}-1\] done clear
  C)
  \[{{2}^{n}}\] done clear
  D)
  \[{{2}^{n-1}}\] done clear
  View Solution play_arrow

• question_answer15)
  \[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+.....+{{(-1)}^{n}}{{C}_{n}}\] is equal to [MNR 1991; RPET 1995; UPSEAT 2000]
  A)
  \[{{2}^{n}}\] done clear
  B)
  \[{{2}^{n}}-1\] done clear
  C)
  0 done clear
  D)
  \[{{2}^{n-1}}\] done clear
  View Solution play_arrow

• question_answer16)
  The sum of all the coefficients in the binomial expansion of \[{{({{x}^{2}}+x-3)}^{319}}\] is [Bihar CEE 1994]
  A)
  1 done clear
  B)
  2 done clear
  C)
  ? 1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer17)
  If the sum of the coefficients in the expansion of  \[{{(x-2y+3z)}^{n}}\] is 128 then the greatest coefficient in the expansion of \[{{(1+x)}^{n}}\]is
  A)
  35 done clear
  B)
  20 done clear
  C)
  10 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If \[{{(1+x-2{{x}^{2}})}^{6}}=1+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{12}}{{x}^{12}}\], then the expression \[{{a}_{2}}+{{a}_{4}}+{{a}_{6}}+....+{{a}_{12}}\]has the value [RPET 1986, 99; UPSEAT 2003]
  A)
  32 done clear
  B)
  63 done clear
  C)
  64 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  If n is an integer greater than 1, then \[a{{-}^{n}}{{C}_{1}}(a-1){{+}^{n}}{{C}_{2}}(a-2)+....+{{(-1)}^{n}}(a-n)=\]  [IIT 1972]
  A)
  \[a\] done clear
  B)
  0 done clear
  C)
  \[{{a}^{2}}\] done clear
  D)
  \[{{2}^{n}}\] done clear
  View Solution play_arrow

• question_answer20)
  The sum of the coefficients of even power of x in the expansion of \[{{(1+x+{{x}^{2}}+{{x}^{3}})}^{5}}\]is [EAMCET 1988]
  A)
  256 done clear
  B)
  128 done clear
  C)
  512 done clear
  D)
  64 done clear
  View Solution play_arrow

• question_answer21)
  Coefficients of \[{{x}^{r}}[0\le r\le (n-1)]\] in the expansion of  \[{{(x+3)}^{n-1}}+{{(x+3)}^{n-2}}(x+2)\]\[+{{(x+3)}^{n-3}}{{(x+2)}^{2}}+...+{{(x+2)}^{n-1}}\]
  A)
  \[^{n}{{C}_{r}}({{3}^{r}}-{{2}^{n}})\] done clear
  B)
  \[^{n}{{C}_{r}}({{3}^{n-r}}-{{2}^{n-r}})\] done clear
  C)
  \[^{n}{{C}_{r}}({{3}^{r}}+{{2}^{n-r}})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer22)
  If the sum of the coefficients in the expansion of \[{{({{\alpha }^{2}}{{x}^{2}}-2\alpha \text{ }x+1)}^{51}}\]vanishes, then the value of \[\alpha \] is  [IIT 1991; Pb. CET 1988]
  A)
  2 done clear
  B)
  ?1 done clear
  C)
  1 done clear
  D)
  ? 2 done clear
  View Solution play_arrow

• question_answer23)
  If \[x+y=1\], then \[\sum\limits_{r=0}^{n}{{{r}^{2}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}{{y}^{n-r}}}\]equals
  A)
  nxy done clear
  B)
  \[nx(x+yn)\] done clear
  C)
  \[nx(nx+y)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
   The value of \[^{4n}{{C}_{0}}{{+}^{4n}}{{C}_{4}}{{+}^{4n}}{{C}_{8}}+....{{+}^{4n}}{{C}_{4n}}\]is
  A)
  \[{{2}^{4n-2}}+{{(-1)}^{n}}{{2}^{2n-1}}\] done clear
  B)
  \[{{2}^{4n-2}}+{{2}^{2n-1}}\] done clear
  C)
  \[{{2}^{2n-1}}+{{(-1)}^{n}}\,{{2}^{4n-2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  The sum of the last eight coefficients in the expansion of \[{{(1+x)}^{15}}\] is
  A)
  \[{{2}^{16}}\] done clear
  B)
  \[{{2}^{15}}\] done clear
  C)
  \[{{2}^{14}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}}\], then the value of \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+....+(n+1){{C}_{n}}\]will be [MP PET 1996; RPET 1997; DCE 1995; AMU 1995; EAMCET 2001; IIT 1971]
  A)
    \[(n+2){{2}^{n-1}}\] done clear
  B)
    \[(n+1){{2}^{n}}\] done clear
  C)
    \[(n+1){{2}^{n-1}}\] done clear
  D)
    \[(n+2){{2}^{n}}\] done clear
  View Solution play_arrow

• question_answer27)
  The value of \[^{15}C_{0}^{2}{{-}^{15}}C_{1}^{2}{{+}^{15}}C_{2}^{2}-....{{-}^{15}}C_{15}^{2}\]is [MP PET 1996]
  A)
  15 done clear
  B)
  ? 15 done clear
  C)
  0 done clear
  D)
  51 done clear
  View Solution play_arrow

• question_answer28)
  \[2{{C}_{0}}+\frac{{{2}^{2}}}{2}{{C}_{1}}+\frac{{{2}^{3}}}{3}{{C}_{2}}+....+\frac{{{2}^{11}}}{11}{{C}_{10}}\] [MP PET 1999; EAMCET 1992]
  A)
  \[\frac{{{3}^{11}}-1}{11}\] done clear
  B)
  \[\frac{{{2}^{11}}-1}{11}\] done clear
  C)
    \[\frac{{{11}^{3}}-1}{11}\] done clear
  D)
  \[\frac{{{11}^{2}}-1}{11}\] done clear
  View Solution play_arrow

• question_answer29)
  If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}}\], then \[{{C}_{0}}{{C}_{2}}+{{C}_{1}}{{C}_{3}}+{{C}_{2}}{{C}_{4}}+{{C}_{n-2}}{{C}_{n}}\]equals [RPET 1996]
  A)
   \[\frac{(2n)!}{(n+1)!(n+2)!}\] done clear
  B)
   \[\frac{(2n)!}{(n-2)!(n+2)!}\] done clear
  C)
  \[\frac{(2n)!}{(n)!(n+2)!}\] done clear
  D)
   \[\frac{(2n)!}{(n-1)!(n+2)!}\] done clear
  View Solution play_arrow

• question_answer30)
  If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+...+{{C}_{n}}{{x}^{n}}\], then the value of \[{{C}_{0}}+{{C}_{2}}+{{C}_{4}}+{{C}_{6}}+.....\] is [RPET 1997]
  A)
  \[{{2}^{n-1}}\] done clear
  B)
  \[{{2}^{n-1}}\] done clear
  C)
  \[{{2}^{n}}\] done clear
  D)
  \[{{2}^{n-1}}-1\] done clear
  View Solution play_arrow

• question_answer31)
  The number 111......1 (91 times) is [UPSEAT 1999]
  A)
  Not a prime done clear
  B)
  An even number done clear
  C)
  Not an odd number done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  If \[{{C}_{0}},{{C}_{1}},{{C}_{2}},.......,{{C}_{n}}\] are the binomial coefficients, then \[2.{{C}_{1}}+{{2}^{3}}.{{C}_{3}}+{{2}^{5}}.{{C}_{5}}+....\]equals   [AMU 1999]
  A)
  \[\frac{{{3}^{n}}+{{(-1)}^{n}}}{2}\] done clear
  B)
  \[\frac{{{3}^{n}}-{{(-1)}^{n}}}{2}\] done clear
  C)
  \[\frac{{{3}^{n}}+1}{2}\] done clear
  D)
  \[\frac{{{3}^{n}}-1}{2}\] done clear
  View Solution play_arrow

• question_answer33)
  The sum of coefficients in the expansion of \[{{(x+2y+3z)}^{8}}\] is  [RPET 2000]
  A)
  \[{{3}^{8}}\] done clear
  B)
  \[{{5}^{8}}\] done clear
  C)
  \[{{6}^{8}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  In the expansion of \[{{(1+x)}^{50}},\] the sum of the coefficient of odd powers of x is                [UPSEAT 2001; Pb. CET 2004]
  A)
  0 done clear
  B)
  \[{{2}^{49}}\] done clear
  C)
  \[{{2}^{50}}\] done clear
  D)
  \[{{2}^{51}}\] done clear
  View Solution play_arrow

• question_answer35)
  \[{{n}^{n}}{{\left( \frac{n+1}{2} \right)}^{2n}}\] is [AMU 2001]
  A)
  Less than \[{{\left( \frac{n+1}{2} \right)}^{3}}\] done clear
  B)
  Greater than \[{{\left( \frac{n+1}{2} \right)}^{3}}\] done clear
  C)
  Less than \[{{(n!)}^{3}}\] done clear
  D)
  Greater than \[{{(n!)}^{3}}\,\] done clear
  View Solution play_arrow

• question_answer36)
  The sum of coefficients in \[{{(1+x-3{{x}^{2}})}^{2134}}\]is [Kurukshetra CEE 2001]
  A)
  ? 1 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  \[{{2}^{2134}}\] done clear
  View Solution play_arrow

• question_answer37)
  The sum of coefficients in the expansion of \[{{(1+x+{{x}^{2}})}^{n}}\] is  [EAMCET 2002]
  A)
  2 done clear
  B)
  \[{{3}^{n}}\] done clear
  C)
  \[{{4}^{n}}\] done clear
  D)
  \[{{2}^{n}}\] done clear
  View Solution play_arrow

• question_answer38)
  The sum of the coefficients in the expansion of \[{{(1+x-3{{x}^{2}})}^{3148}}\] is [Karnataka CET  2003]
  A)
  7 done clear
  B)
  8 done clear
  C)
  ? 1 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer39)
  If \[{{a}_{k}}=\frac{1}{k(k+1)},\] for \[k=1,\,2,\,3,\,4,.....,\,n\], then \[{{\left( \sum\limits_{k=1}^{n}{{{a}_{k}}} \right)}^{2}}=\] [EAMCET 2000]
  A)
  \[\left( \frac{n}{n+1} \right)\] done clear
  B)
  \[{{\left( \frac{n}{n+1} \right)}^{2}}\] done clear
  C)
  \[{{\left( \frac{n}{n+1} \right)}^{4}}\] done clear
  D)
  \[{{\left( \frac{n}{n+1} \right)}^{6}}\] done clear
  View Solution play_arrow

• question_answer40)
  In the expansion of \[{{(1+x)}^{5}}\], the sum of the coefficient of the terms is           [RPET 1992, 97; Kurukshetra CEE 2000]
  A)
  80 done clear
  B)
  16 done clear
  C)
  32 done clear
  D)
  64 done clear
  View Solution play_arrow

• question_answer41)
  \[\sum\limits_{k=0}^{10}{^{20}{{C}_{k}}=}\] [Orissa JEE 2004]
  A)
  \[{{2}^{19}}+\frac{1}{2}{{\,}^{20}}{{C}_{10}}\] done clear
  B)
  \[{{2}^{19}}\] done clear
  C)
  \[^{20}{{C}_{10}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer42)
  If \[{{S}_{n}}=\sum\limits_{r=0}^{n}{\frac{1}{^{n}{{C}_{r}}}}\] and \[{{t}_{n}}=\sum\limits_{r=0}^{n}{\frac{r}{^{n}{{C}_{r}}}}\], then \[\frac{{{t}_{n}}}{{{S}_{n}}}\] is equal to [AIEEE 2004]
  A)
  \[\frac{2n-1}{2}\] done clear
  B)
  \[\frac{1}{2}n-1\] done clear
  C)
  \[n-1\] done clear
  D)
  \[\frac{1}{2}n\] done clear
  View Solution play_arrow

• question_answer43)
  The value of \[\sum\limits_{n=1}^{\infty }{\frac{^{n}{{C}_{0}}+...{{+}^{n}}{{C}_{n}}}{^{n}{{P}_{n}}}}\] is  [Kerala (Engg.) 2005]
  A)
  \[{{e}^{2}}\] done clear
  B)
  e done clear
  C)
  \[{{e}^{2}}-1\] done clear
  D)
  \[e-1\] done clear
  E)
  \[{{e}^{2}}+1\] done clear
  View Solution play_arrow

• question_answer44)
  What is the sum of the coefficients of \[{{({{x}^{2}}-x-1)}^{99}}\] [Orissa JEE 2005]
  A)
  1 done clear
  B)
  0 done clear
  C)
  ?1 done clear
  D)
  None of these done clear
  View Solution play_arrow