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

done Multinomial theorem, Terms free from radical sign in the expansion(a1/p+b1/q), Problems regarding to three four consecutive terms or coefficients Total Questions - 17

• question_answer1)
  The coefficient of two consecutive terms in the expansion of \[{{(1+x)}^{n}}\] will be equal, if
  A)
   n is any integer done clear
  B)
  n is an odd integer done clear
  C)
   n is an even integer done clear
  D)
  None of these done clear
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• question_answer2)
  If in the expansion of \[{{(1+x)}^{n}}\], a, b, c are three consecutive coefficients, then n=
  A)
  \[\frac{ac+ab+bc}{{{b}^{2}}+ac}\] done clear
  B)
  \[\frac{2ac+ab+bc}{{{b}^{2}}-ac}\] done clear
  C)
  \[\frac{ab+ac}{{{b}^{2}}-ac}\] done clear
  D)
  None of these done clear
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• question_answer3)
  If n is positive integer and three consecutive coefficients in the expansion of \[{{(1+x)}^{n}}\] are in the ratio 6 : 33 : 110, then n =
  A)
  4 done clear
  B)
  6 done clear
  C)
  12 done clear
  D)
  16 done clear
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• question_answer4)
  The number of terms in the expansion of \[{{(a+b+c)}^{n}}\] will be
  A)
  \[n+1\] done clear
  B)
  \[n+3\] done clear
  C)
  \[\frac{(n+1)(n+2)}{2}\] done clear
  D)
  None of these done clear
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• question_answer5)
  If the three consecutive coefficient in the expansion of \[{{(1+x)}^{n}}\] are 28, 56 and 70, then the value of n is [MP PET 1985]
  A)
  6 done clear
  B)
  4 done clear
  C)
  8 done clear
  D)
  10 done clear
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• question_answer6)
  In the expansion of \[{{({{5}^{1/2}}+{{7}^{1/8}})}^{1024}}\], the number of integral terms is
  A)
  128 done clear
  B)
  129 done clear
  C)
  130 done clear
  D)
  131 done clear
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• question_answer7)
  The number of terms which are free from radical signs in the expansion of \[{{({{y}^{1/5}}+{{x}^{1/10}})}^{55}}\]is
  A)
  5 done clear
  B)
  6 done clear
  C)
  7 done clear
  D)
  None of these done clear
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• question_answer8)
  Let \[R={{(5\sqrt{5}+11)}^{2n+1}}\] and \[f=R-[R]\], where [.] denotes the greatest integer function. The value of R.f is [IIT 1988]
  A)
  \[{{4}^{2n+1}}\] done clear
  B)
  \[{{4}^{2n}}\] done clear
  C)
  \[{{4}^{2n-1}}\] done clear
  D)
  \[{{4}^{-2n}}\] done clear
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• question_answer9)
  The greatest integer less than or equal to \[{{(\sqrt{2}+1)}^{6}}\]is [RPET 2000]
  A)
  196 done clear
  B)
  197 done clear
  C)
  198 done clear
  D)
  199 done clear
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• question_answer10)
  If number of terms in the expansion of \[{{(x-2y+3z)}^{n}}\]are 45, then n=
  A)
  7 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  None of these done clear
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• question_answer11)
  Find the value of\[\frac{({{18}^{3}}+{{7}^{3}}+3.18.7.25)}{{{3}^{6}}+6.243.2+15.81.4+20.27.8+15.9.16+6.3.32+64}\] [IIT 1960]
  A)
  1 done clear
  B)
  5 done clear
  C)
  25 done clear
  D)
  100 done clear
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• question_answer12)
  If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}}\] are the coefficients of any four consecutive terms in the expansion of \[{{(1+x)}^{n}}\], then \[\frac{{{a}_{1}}}{{{a}_{1}}+{{a}_{2}}}+\frac{{{a}_{3}}}{{{a}_{3}}+{{a}_{4}}}\] = [IIT 1975]
  A)
  \[\frac{{{a}_{2}}}{{{a}_{2}}+{{a}_{3}}}\] done clear
  B)
  \[\frac{1}{2}\frac{{{a}_{2}}}{({{a}_{2}}+{{a}_{3}})}\] done clear
  C)
  \[\frac{2{{a}_{2}}}{{{a}_{2}}+{{a}_{3}}}\] done clear
  D)
  \[\frac{2{{a}_{3}}}{{{a}_{2}}+{{a}_{3}}}\] done clear
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• question_answer13)
  The number of integral terms in the expansion of \[{{({{5}^{1/2}}+{{7}^{1/6}})}^{642}}\] is [Kurukshetra CEE 1996]
  A)
  106 done clear
  B)
  108 done clear
  C)
  103 done clear
  D)
  109 done clear
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• question_answer14)
  The expression \[{{(2+\sqrt{2})}^{4}}\] has value, lying between  [AMU 2001]
  A)
  134 and 135 done clear
  B)
  135 and 136 done clear
  C)
  136 and 137 done clear
  D)
  None of these done clear
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• question_answer15)
  The digit in the unit place of the number \[(183\,!)+{{3}^{183}}\] is  [Karnataka CET  2002]
  A)
  7 done clear
  B)
  6 done clear
  C)
  3 done clear
  D)
  0 done clear
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• question_answer16)
  The number of integral terms in the expansion of \[{{\left( \sqrt{3}+\sqrt[8]{5} \right)}^{256}}\] is [AIEEE 2003]
  A)
  32 done clear
  B)
  33 done clear
  C)
  34 done clear
  D)
  35 done clear
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• question_answer17)
   If \[{{a}_{r}}\] is the coefficient of \[{{x}^{r}}\], in the expansion of \[{{(1+x+{{x}^{2}})}^{n}}\], then \[{{a}_{1}}-2{{a}_{2}}+3{{a}_{3}}-....-2n\,{{a}_{2n}}=\] [EAMCET 2003]
  A)
  0 done clear
  B)
  n done clear
  C)
  ? n done clear
  D)
  2n done clear
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