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question_answer1)
The sum of 1+3+5+7..upto n terms is
A)
2n+1 done
clear
B)
\[{{n}^{2}}\] done
clear
C)
\[2{{n}^{2}}-1\] done
clear
D)
\[2{{n}^{2}}\] done
clear
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question_answer2)
If \[p(n)={{n}^{2}}+n,\]\[\forall \,n\in N\], then p(n) is
A)
even positive integer done
clear
B)
prime number done
clear
C)
odd integer done
clear
D)
multiple of 4 done
clear
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question_answer3)
Sum of \[\frac{{{1}^{3}}}{1}+\frac{{{1}^{3}}+{{2}^{3}}}{1+2}+...\]to n terms is
A)
\[\frac{(n+1)(n+2)}{3}\] done
clear
B)
\[n(n+1)(n+2)\] done
clear
C)
\[\frac{n(n+1)(n+2)}{6}\] done
clear
D)
\[\frac{n(n+1)(n+2)}{2}\] done
clear
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question_answer4)
\[p(n)={{n}^{3}}+{{(n+1)}^{3}}{{(n+2)}^{3}}\]where \[n\in N\]is divisible by
A)
9 done
clear
B)
7 done
clear
C)
13 done
clear
D)
15 done
clear
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question_answer5)
\[p(n)={{3}^{4n+2}}+{{5}^{2n+1}}\]is divisible by
A)
12 done
clear
B)
14 done
clear
C)
16 done
clear
D)
61 done
clear
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question_answer6)
The sum of the series \[\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+.....\]to n terms is
A)
\[n-\frac{1}{2}(1-{{3}^{-n}})\] done
clear
B)
\[n-\frac{1}{2}({{3}^{-n}}-1)\] done
clear
C)
\[n-\frac{1}{2}({{3}^{n}}-1)\] done
clear
D)
\[n+\frac{1}{2}({{3}^{n}}-1)\] done
clear
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question_answer7)
If \[n\in N\], \[p(n)={{2}^{n-1}}\] then p(n)\[\le \]
A)
n done
clear
B)
\[{{n}^{2}}\] done
clear
C)
(n-1)! done
clear
D)
n ! done
clear
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question_answer8)
The sum of the series 1\[\times \]3+3\[\times \]5+5\[\times \]7+..to n terms is
A)
\[\frac{n(4{{n}^{2}}+6n)-1}{3}\] done
clear
B)
\[\frac{n(n+1)(n+2)}{2}\] done
clear
C)
\[\frac{(2n+1)(n+1)}{2}\] done
clear
D)
\[\frac{{{n}^{2}}+1}{4}\] done
clear
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question_answer9)
\[\frac{(2n)!}{{{2}^{2n}}{{(n!)}^{2}}}\,\,is\,\,\le \]
A)
\[\frac{1}{3n+1}\] done
clear
B)
\[\frac{1}{{{(3n+1)}^{1/2}}}\] done
clear
C)
\[\frac{1}{{{(3n+1)}^{2}}}\] done
clear
D)
\[\frac{1}{{{(3n+1)}^{1/2}}}\] done
clear
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question_answer10)
If\[n\in N\],then \[\sin \theta +\sin 2\theta +\sin 3\theta +....+\sin n\theta =\]
A)
\[\sin \left[ \frac{n(n+1)\theta }{2} \right]\] done
clear
B)
\[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]\sin \theta }{\sin \left( \frac{\theta }{2} \right)}\] done
clear
C)
\[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]\sin \left( \frac{n\theta }{2} \right)}{\sin \left( \frac{\theta }{2} \right)}\] done
clear
D)
\[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]{{\sin }^{n}}\left( \frac{\theta }{2} \right)}{\sin \left( \frac{\theta }{2} \right)}\] done
clear
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question_answer11)
\[\frac{(n+2)!}{(n-1)!}\]is divisible by
A)
4 done
clear
B)
4 done
clear
C)
6 done
clear
D)
7 done
clear
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question_answer12)
The sum \[\underbrace{{{(666..6)}^{2}}}_{n\,digits}+\underbrace{(888..8)}_{n\,digits}\]is equal to
A)
\[\frac{4}{9}{{({{10}^{n}}-1)}^{2}}\] done
clear
B)
\[\frac{4}{9}({{10}^{n}}-1)\] done
clear
C)
\[\frac{4}{9}({{10}^{2n}}-1)\] done
clear
D)
\[4({{10}^{n}}+1)\] done
clear
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question_answer13)
Sum of 1+2+\[{{2}^{2}}\]+\[{{2}^{n-1}}\]is
A)
\[{{2}^{n}}\] done
clear
B)
\[{{2}^{n}}+1\] done
clear
C)
\[{{2}^{n}}-1\] done
clear
D)
\[{{2}^{n-1}}\] done
clear
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question_answer14)
The number \[({{49}^{2}}-4)({{49}^{3}}-49)\]is divisible by__
A)
6! done
clear
B)
5! done
clear
C)
7! done
clear
D)
9! done
clear
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question_answer15)
The sum of n terms of \[1+\frac{1+2}{2}+\frac{1+2+3}{3}+...\]is
A)
\[\frac{m(n+3)}{4}\] done
clear
B)
\[\frac{(n+3)}{3}\] done
clear
C)
\[\frac{n(n-3)}{3}\] done
clear
D)
\[\frac{n(n-3)}{4}\] done
clear
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question_answer16)
If \[x\in \{1,2,3....9\}\]and \[{{f}_{n}}(x)=x\,x\,x...x\](n digits), then \[{{f}^{2}}_{n}(3)+{{f}_{n}}(2)=\]
A)
\[2{{f}_{2n}}(1)\] done
clear
B)
\[{{f}^{2}}_{n}(1)\] done
clear
C)
\[{{f}_{2n}}(1)\] done
clear
D)
\[-{{f}_{2n}}(4)\] done
clear
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question_answer17)
If \[p(n)={{n}^{3}}+{{(n+1)}^{3}}+{{(n+2)}^{3}}\]where \[n\in N\]then p(n) is divisible by
A)
9 done
clear
B)
7 done
clear
C)
13 done
clear
D)
15 done
clear
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question_answer18)
The sum to n terms of 1+3+5..is
A)
\[n\] done
clear
B)
\[2n-1\] done
clear
C)
\[{{n}^{2}}\] done
clear
D)
\[{{n}^{3}}\] done
clear
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question_answer19)
The product of \[\cos \theta \cos 2\theta \cos 3\theta ...\cos ({{2}^{n-1}}\theta )\]is
A)
\[\frac{\sin ({{2}^{n}}\theta )}{{{2}^{n}}\sin \theta }\] done
clear
B)
\[\frac{cos({{2}^{n}}\theta )}{{{2}^{n}}\cos \theta }\] done
clear
C)
\[\frac{cosn\theta }{{{2}^{n}}\cos \theta }\] done
clear
D)
\[\frac{sin({{2}^{n}}\theta )}{\sin \theta }\] done
clear
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question_answer20)
If 'n' is a positive integer, then \[{{n}^{3}}+2n\]is divisible by
A)
3 done
clear
B)
2 done
clear
C)
6 done
clear
D)
15 done
clear
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question_answer21)
The sum \[\frac{{{1}^{2}}}{2}+\frac{{{1}^{2}}+{{2}^{2}}}{3}+\frac{{{1}^{2}}+{{2}^{2}}+{{3}^{3}}}{4}+...\] to 10 terms is _________.
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question_answer22)
Sum to 7 terms of \[{{1.3}^{2}}+{{3.5}^{2}}+{{5.7}^{2}}+...\] is ________.
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question_answer23)
It is given that \[\frac{1}{{{1}^{4}}}+\frac{1}{{{2}^{4}}}+\frac{1}{{{3}^{4}}}+...\]to \[\infty =\frac{{{\pi }^{4}}}{90}\]. Then \[\frac{1}{{{1}^{4}}}+\frac{1}{{{3}^{4}}}+\frac{1}{{{5}^{4}}}+...\]\[\infty \] is equal to __________.
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question_answer24)
The least value of n other than 1 for which the inequality \[{{2}^{n}}>{{n}^{2}}\]holds good is _________.
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question_answer25)
Sum of \[{{1}^{2}}-{{2}^{2}}+{{3}^{2}}-{{4}^{2}}+{{5}^{2}}-+...+{{21}^{2}}\] is ________.
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