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

done Mathematical induction and Divisibility problems Total Questions - 23

• question_answer1)
  For all positive integral values of n, \[{{3}^{2n}}-2n+1\] is divisible by
  A)
  2 done clear
  B)
  4 done clear
  C)
  8 done clear
  D)
  12 done clear
  View Solution play_arrow

• question_answer2)
  If \[n\in N\], then \[{{x}^{2n-1}}+{{y}^{2n-1}}\] is divisible by
  A)
  \[x+y\] done clear
  B)
  \[x-y\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}\] done clear
  D)
  \[{{x}^{2}}+xy\] done clear
  View Solution play_arrow

• question_answer3)
  If\[n\in N\], then\[{{7}^{2n}}+{{2}^{3n-3}}\]. \[{{3}^{n-1}}\] is always divisible by [IIT 1982]
  A)
  25 done clear
  B)
  35 done clear
  C)
  45 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  If \[n\in N\], then \[{{11}^{n+2}}+{{12}^{2n+1}}\] is divisible by [Roorkee 1982]
  A)
  113 done clear
  B)
  123 done clear
  C)
  133 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  For every natural number n, \[n\,({{n}^{2}}-1)\] is divisible by  [RPET 1991]
  A)
  4 done clear
  B)
  6 done clear
  C)
  10 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer6)
  For every natural number n
  A)
  \[n>{{2}^{n}}\] done clear
  B)
  \[n<{{2}^{n}}\] done clear
  C)
  \[n\ge {{2}^{n}}\] done clear
  D)
  \[n\le {{2}^{n}}\] done clear
  View Solution play_arrow

• question_answer7)
  For each \[n\in N\], the correct statement is
  A)
  \[{{2}^{n}}<n\] done clear
  B)
  \[{{n}^{2}}>2n\] done clear
  C)
  \[{{n}^{4}}<{{10}^{n}}\] done clear
  D)
  \[{{2}^{3n}}>7n+1\] done clear
  View Solution play_arrow

• question_answer8)
  For natural number n, \[{{2}^{n}}\,(n-1)\,!<{{n}^{n}}\], if
  A)
  n < 2 done clear
  B)
  n > 2 done clear
  C)
  n ³ 2 done clear
  D)
  Never done clear
  View Solution play_arrow

• question_answer9)
  If n is a natural number then \[{{\left( \frac{n+1}{2} \right)}^{n}}\ge n\,!\] is true when
  A)
  n > 1 done clear
  B)
  n ³ 1 done clear
  C)
  n > 2 done clear
  D)
  n ³ 2 done clear
  View Solution play_arrow

• question_answer10)
  For positive integer n, \[{{10}^{n-2}}>81n\], if
  A)
  n > 5 done clear
  B)
  n ³ 5 done clear
  C)
  n < 5 done clear
  D)
  n > 6 done clear
  View Solution play_arrow

• question_answer11)
  For every positive integer n, \[{{2}^{n}}<n\,!\] when
  A)
  n < 4 done clear
  B)
  n ³ 4 done clear
  C)
  n < 3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  For every positive integral value of n, \[{{3}^{n}}>{{n}^{3}}\] when
  A)
  n > 2 done clear
  B)
  n ³ 3 done clear
  C)
  n ³ 4 done clear
  D)
  n < 4 done clear
  View Solution play_arrow

• question_answer13)
  For natural number n, \[{{(n\,!)}^{2}}>{{n}^{n}}\], if
  A)
  n > 3 done clear
  B)
  n > 4 done clear
  C)
  n ³ 4 done clear
  D)
  n ³ 3 done clear
  View Solution play_arrow

• question_answer14)
  Let P (n) denote the statement that \[{{n}^{2}}+n\] is odd. It is seen that \[P(n)\Rightarrow P(n+1)\], \[{{P}_{n}}\] is true for all [IIT JEE 1996]
  A)
  n > 1 done clear
  B)
  n done clear
  C)
  n > 2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer15)
  If p is a prime number, then \[{{n}^{p}}-n\] is divisible by p when n is a
  A)
  Natural number greater than 1 done clear
  B)
  Irrational number done clear
  C)
  Complex number done clear
  D)
  Odd number done clear
  View Solution play_arrow

• question_answer16)
  \[x({{x}^{n-1}}-n{{a}^{n-1}})+{{a}^{n}}(n-1)\] is divisible by \[{{(x-a)}^{2}}\] for
  A)
  n > 1 done clear
  B)
  n > 2 done clear
  C)
  All n Î N done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  If P(n) = 2 + 4 + 6 +?.+ 2n, n Î N, then P(k) = k(k + 1) + 2 Þ P(k + 1) = (k + 1)(k + 2) + 2 for all k Î N. So we can conclude that P(n) = n(n + 1) + 2 for
  A)
  All n Î N done clear
  B)
  n > 1 done clear
  C)
  n > 2 done clear
  D)
  Nothing can be said done clear
  View Solution play_arrow

• question_answer18)
  For every natural number n, n(n + 1) is always
  A)
  Even done clear
  B)
  Odd done clear
  C)
  Multiple of 3 done clear
  D)
  Multiple of 4 done clear
  View Solution play_arrow

• question_answer19)
  The statement P(n) ?\[1\times 1\,!\,+\,2\times 2\,!\,+\,3\times 3\,!\,+.....+n\times n\,!=(n+1)\,!\,-1\]? is
  A)
  True for all n > 1 done clear
  B)
  Not true for any n done clear
  C)
  True for all n Î N done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  The remainder when \[{{5}^{99}}\] is divided by 13 is
  A)
  6 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer21)
  When \[{{2}^{301}}\] is divided by 5, the least positive remainder is [Karnataka CET 2005]
  A)
  4 done clear
  B)
  8 done clear
  C)
  2 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer22)
  For a positive integer n, Let\[a\,(n)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+....+\frac{1}{({{2}^{n}})-1}\]. Then [IIT 1999]
  A)
  \[a\,(100)\le 100\] done clear
  B)
  \[a\,(100)>100\] done clear
  C)
  \[a\,(200)\le 100\] done clear
  D)
  \[a\,(200)>100\] done clear
  View Solution play_arrow

• question_answer23)
  \[{{10}^{n}}+3\,({{4}^{n+2}})+5\] is divisible by \[(n\in N)\][Kerala (Engg.) 2005]
  A)
  7 done clear
  B)
  5 done clear
  C)
  9 done clear
  D)
  17 done clear
  E)
  13 done clear
  View Solution play_arrow