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question_answer1)
The inequality \[n!>{{2}^{n-1}}\] is true for
A)
\[n>2\] done
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B)
\[n\in N\] done
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C)
\[n>3\] done
clear
D)
None of these done
clear
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question_answer2)
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
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C)
True for all \[n\in N\] done
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D)
None of these done
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question_answer3)
If \[n\in N\], then \[{{11}^{n+2}}+{{12}^{2n+1}}\] is divisible by
A)
113 done
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B)
123 done
clear
C)
133 done
clear
D)
None of these done
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question_answer4)
For a positive integer n, Let \[a(n)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{({{2}^{n}})-1}.\] Then
A)
\[a(100)\le 100\] done
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B)
\[a(100)>100\] done
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C)
\[a(200)\le 100\] done
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D)
\[a(200)<100\] done
clear
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question_answer5)
If m, n are any two odd positive integers with \[n<m\], then the largest positive integer which divides all the numbers of the type \[{{m}^{2}}-{{n}^{2}}\] is
A)
4 done
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B)
6 done
clear
C)
8 done
clear
D)
9 done
clear
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question_answer6)
Which one of the following is true?
A)
\[{{\left( 1+\frac{1}{n} \right)}^{n}}<{{n}^{2}},n\] is a positive integer done
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B)
\[{{\left( 1+\frac{1}{n} \right)}^{n}}<2,n\] is a positive integer done
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C)
\[{{\left( 1+\frac{1}{n} \right)}^{n}}<{{n}^{3}},n\] is a positive integer done
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D)
\[{{\left( 1+\frac{1}{n} \right)}^{n}}>2,n\] is a positive integer done
clear
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question_answer7)
By the principle of induction \[\forall n\in N,{{3}^{2n}}\] when divided by 8, leaves remainder
A)
2 done
clear
B)
3 done
clear
C)
7 done
clear
D)
1 done
clear
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question_answer8)
Let \[P(n):''{{2}^{n}}<(1\times 2\times 3\times ...\times n)''.\] Then the smallest positive integer for which P(n) is true is
A)
1 done
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B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer9)
If \[\frac{(n+2)!}{6(n-1)!}\] divisible by n, \[n\in N\]and \[1\le n\le 9,\] Then n is
A)
4 done
clear
B)
2 done
clear
C)
6 done
clear
D)
1 done
clear
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question_answer10)
For every positive integer n, \[{{7}^{n}}-{{3}^{n}}\] is divisible by
A)
7 done
clear
B)
3 done
clear
C)
4 done
clear
D)
5 done
clear
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question_answer11)
If \[\frac{{{4}^{n}}}{n+1}<\frac{(2n)!}{{{(n!)}^{2}}},\] then P(n) is true for
A)
\[n\ge 1\] done
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B)
\[n>0\] done
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C)
\[n<0\] done
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D)
\[n\ge 2\] done
clear
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question_answer12)
A student was asked to prove a statement P (n) by induction. He proved that \[P(k+1)\]is true whenever \[P(k)\] is true for all \[k>5\in N\] and also that P (5) is true. On the basis of this he could conclude that P(n) is true
A)
For all \[n\in N\] done
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B)
For all \[n>5\] done
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C)
For all \[n\ge 5\] done
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D)
For all \[n<5\] done
clear
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question_answer13)
If \[n\in N\] and \[n>1,\] then
A)
\[n!>{{\left( \frac{n+1}{2} \right)}^{n}}\] done
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B)
\[n!\ge {{\left( \frac{n+1}{2} \right)}^{n}}\] done
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C)
\[n!<{{\left( \frac{n+1}{2} \right)}^{n}}\] done
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D)
None of these done
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question_answer14)
For all \[n\in N,\,\,\,{{41}^{n}}-{{14}^{n}}\] is a multiple of
A)
26 done
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B)
27 done
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C)
25 done
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D)
None of these done
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question_answer15)
For all \[n\in N,\] \[1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+n}\] is equal to
A)
\[\frac{3n}{n+1}\] done
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B)
\[\frac{n}{n+1}\] done
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C)
\[\frac{2n}{n-1}\] done
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D)
\[\frac{2n}{n+1}\] done
clear
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question_answer16)
\[{{10}^{n}}+3({{4}^{n+2}})+5\] is divisible by \[(n\in N)\]
A)
7 done
clear
B)
5 done
clear
C)
9 done
clear
D)
17 done
clear
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question_answer17)
The greatest positive integer, which divides \[n(n+1)(n+2)(n+3)\] for all \[n\in N,\] is
A)
2 done
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B)
6 done
clear
C)
24 done
clear
D)
120 done
clear
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question_answer18)
Which of the following result is valid?
A)
\[{{(1+x)}^{n}}>(1+nx),\] For all natural numbers n done
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B)
\[{{(1+x)}^{n}}\ge (1+nx),\] For all natural numbers n, Where \[x>-1\] done
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C)
\[{{(1+x)}^{n}}\le (1+nx),\] For all natural numbers n done
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D)
\[{{(1+x)}^{n}}<(1+nx),\] For all natural numbers n done
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question_answer19)
When \[{{2}^{301}}\] is divided by 5, the least positive remainder is
A)
4 done
clear
B)
8 done
clear
C)
2 done
clear
D)
6 done
clear
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question_answer20)
\[{{10}^{n}}+3({{4}^{n+2}})+5\] is divisible by \[(n\in N)\]
A)
7 done
clear
B)
5 done
clear
C)
9 done
clear
D)
17 done
clear
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question_answer21)
Let \[S(k)=1+3+5...+(2k-1)=3+{{k}^{2}}.\] Then which of the following is true?
A)
Principle of mathematical induction can be used to prove the formula done
clear
B)
\[S(k)\Rightarrow S(k+1)\] done
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C)
\[S(k)\not{\Rightarrow }S(k+1)\] done
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D)
\[S(1)\] is correct done
clear
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question_answer22)
For every positive integral value of n, \[{{3}^{n}}>{{n}^{3}}\] when
A)
\[n>2\] done
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B)
\[n\ge 3\] done
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C)
\[n\ge 4\] done
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D)
\[n<4\] done
clear
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question_answer23)
If n is a natural number, then
A)
\[{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}<\frac{{{n}^{3}}}{3}\] done
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B)
\[{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}=\frac{{{n}^{3}}}{3}\] done
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C)
\[{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}>{{n}^{3}}\] done
clear
D)
\[{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}>\frac{{{n}^{3}}}{3}\] done
clear
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question_answer24)
If \[P(n):{{3}^{n}}<n!,n\in N,\] then P(n) is true
A)
For \[n\ge 6\] done
clear
B)
For \[n\ge 7\] done
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C)
For \[n\ge 3\] done
clear
D)
For all n. done
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question_answer25)
If \[P=n({{n}^{2}}-{{1}^{2}})({{n}^{2}}-{{2}^{2}})({{n}^{2}}-{{3}^{2}})...({{n}^{2}}-{{r}^{2}}),\] \[n>r,n\in N\] then P is necessarily divisible by
A)
\[(2r+2)!\] done
clear
B)
\[(2r+4)!\] done
clear
C)
\[(2r+1)!\] done
clear
D)
None of these done
clear
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question_answer26)
If \[P(n):''{{46}^{n}}+{{19}^{n}}+k\] is divisible by 64 for\[n\in N\]? is true, then the least negative integer value of k is.
A)
-1 done
clear
B)
1 done
clear
C)
2 done
clear
D)
-2 done
clear
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question_answer27)
If \[\frac{1}{2\times 4}+\frac{1}{4\times 6}+\frac{1}{6\times 8}+...n\] terms \[=\frac{kn}{n+1},\] then k is equal to
A)
\[\frac{1}{4}\] done
clear
B)
\[\frac{1}{2}\] done
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C)
1 done
clear
D)
\[\frac{1}{8}\] done
clear
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question_answer28)
For given series: \[{{1}^{2}}+2\times {{2}^{2}}+{{3}^{2}}+2\times {{4}^{2}}+{{5}^{2}}+2\times {{6}^{2}}+...,\] If \[{{S}_{n}}\] is the sum of n terms, then
A)
\[{{S}_{n}}=\frac{n{{(n+1)}^{2}}}{2},\] If n is even done
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B)
\[{{S}_{n}}=\frac{{{n}^{2}}(n+1)}{2},\] If n is odd done
clear
C)
Both and are true done
clear
D)
Both and are false done
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question_answer29)
The remainder when \[{{5}^{4n}}\] is divided by 13, is
A)
1 done
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B)
8 done
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C)
9 done
clear
D)
10 done
clear
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question_answer30)
Using mathematical induction, the numbers \[{{a}_{n}}'s\]are defined by \[{{a}_{0}}=1,\,\,{{a}_{n+1}}=3{{n}^{2}}+n+{{a}_{n'}}\] \[(n\ge 0).\]Then, \[{{a}_{n}}\] is equal to
A)
\[{{n}^{3}}+{{n}^{2}}+1\] done
clear
B)
\[{{n}^{3}}-{{n}^{2}}+1\] done
clear
C)
\[{{n}^{3}}-{{n}^{2}}\] done
clear
D)
\[{{n}^{3}}+{{n}^{2}}\] done
clear
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