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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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