-
question_answer1)
The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{n}{1+{{n}^{2}}}+\frac{n}{4+{{n}^{2}}}+\frac{n}{9+{{n}^{2}}}+....+\frac{1}{2n} \right]\]is equal to [Bihar CEE 1994]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
1 done
clear
D)
None of these done
clear
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question_answer2)
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{1}^{3}}+{{n}^{3}}}+\frac{4}{{{2}^{3}}+{{n}^{3}}}+....+\frac{1}{2n}\]is equal to [RPET 1997]
A)
\[\frac{1}{3}{{\log }_{e}}3\] done
clear
B)
\[\frac{1}{3}{{\log }_{e}}2\] done
clear
C)
\[\frac{1}{3}{{\log }_{e}}\frac{1}{3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer3)
\[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)-f(a)}{g(x)\,-g(a)},\] [EAMCET 1994]
A)
\[\frac{9}{100}\] done
clear
B)
\[-1/2\] done
clear
C)
\[\frac{1}{99}\] done
clear
D)
\[\frac{1}{101}\] done
clear
View Solution play_arrow
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question_answer4)
\[\underset{n\to \infty }{\mathop{\lim }}\,{{\left[ \frac{n!}{{{n}^{n}}} \right]}^{1/n}}\]equals [Kurukshetra CEE 1998]
A)
e done
clear
B)
\[1/e\] done
clear
C)
\[\pi /4\] done
clear
D)
\[4/\pi \] done
clear
View Solution play_arrow
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question_answer5)
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\sum\limits_{r=1}^{2n}{\frac{r}{\sqrt{{{n}^{2}}+{{r}^{2}}}}}\] equals [IIT 1997 Re-exam]
A)
\[1+\sqrt{5}\] done
clear
B)
\[-1+\sqrt{5}\] done
clear
C)
\[-1+\sqrt{2}\] done
clear
D)
\[1+\sqrt{2}\] done
clear
View Solution play_arrow
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question_answer6)
\[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+.....\frac{1}{2n} \right]=\] [Karnataka CET 1999]
A)
0 done
clear
B)
\[{{\log }_{e}}4\] done
clear
C)
\[{{\log }_{e}}3\] done
clear
D)
\[{{\log }_{e}}2\] done
clear
View Solution play_arrow
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question_answer7)
\[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\frac{k}{{{n}^{2}}+{{k}^{2}}}}\]is equals to [Roorkee 1999]
A)
\[\frac{1}{2}\log 2\] done
clear
B)
\[x=\frac{3\pi }{4}\] done
clear
C)
\[\pi /4\] done
clear
D)
\[\pi /2\] done
clear
View Solution play_arrow
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question_answer8)
\[\underset{n\to \infty }{\mathop{\lim }}\,\,\left[ \frac{1}{n}+\frac{1}{\sqrt{{{n}^{2}}+n}}+\frac{1}{\sqrt{{{n}^{2}}+2n}}+.....+\frac{1}{\sqrt{{{n}^{2}}+(n-1)n}} \right]\] is equal to [RPET 2000]
A)
\[2+2\sqrt{2}\] done
clear
B)
\[2\sqrt{2}-2\] done
clear
C)
\[2\sqrt{2}\] done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer9)
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{1}^{p}}+{{2}^{p}}+{{3}^{p}}+.....+{{n}^{p}}}{{{n}^{p+1}}}=\] [AIEEE 2002]
A)
\[\frac{1}{p+1}\] done
clear
B)
\[\frac{1}{1-p}\] done
clear
C)
\[\frac{1}{p}-\frac{1}{p-1}\] done
clear
D)
\[\underset{x\to 0-}{\mathop{\lim }}\,f(x)=0\] done
clear
View Solution play_arrow
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question_answer10)
\[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\frac{1}{n}{{e}^{\frac{r}{n}}}}\]is [AIEEE 2004]
A)
\[e+1\] done
clear
B)
\[e-1\] done
clear
C)
\[1-e\] done
clear
D)
\[e\] done
clear
View Solution play_arrow
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question_answer11)
\[\int_{0}^{\infty }{\frac{\log \,(1+{{x}^{2}})}{1+{{x}^{2}}}}\,dx=\]
A)
\[\pi \log \frac{1}{2}\] done
clear
B)
\[\pi \log 2\] done
clear
C)
\[2\pi \log \frac{1}{2}\] done
clear
D)
\[2\pi \log 2\] done
clear
View Solution play_arrow
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question_answer12)
\[\int_{0}^{\pi /2}{{{\sin }^{2}}x{{\cos }^{3}}x}\,dx=\] [RPET 1984, 2003]
A)
0 done
clear
B)
\[\frac{2}{15}\] done
clear
C)
\[\frac{4}{15}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer13)
\[\int_{-\pi /2}^{\pi /2}{{{\sin }^{2}}x\,dx=}\]
A)
\[\pi \] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{\pi }{2}-\frac{1}{2}\] done
clear
D)
\[\pi -1\] done
clear
View Solution play_arrow
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question_answer14)
The correct evaluation of \[\int_{0}^{\pi }{\left| \,{{\sin }^{4}}x\, \right|\,dx}\] is [MP PET 1993]
A)
\[\frac{8\pi }{3}\] done
clear
B)
\[\frac{2\pi }{3}\] done
clear
C)
\[\frac{4\pi }{3}\] done
clear
D)
\[\frac{3\pi }{8}\] done
clear
View Solution play_arrow
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question_answer15)
\[\int_{0}^{\pi /2}{{{\cos }^{2}}x\,dx=}\]
A)
\[1-\frac{\pi }{4}\] done
clear
B)
\[1+\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[\frac{\pi }{2}\] done
clear
View Solution play_arrow
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question_answer16)
The points of intersection of \[{{F}_{1}}(x)=\int_{2}^{x}{(2t-5)\,dt}\] and \[{{F}_{2}}(x)=\int_{0}^{x}{2t\,dt,}\] are [IIT Screening]
A)
\[\left( \frac{6}{5},\,\frac{36}{25} \right)\] done
clear
B)
\[\left( \frac{2}{3},\,\frac{4}{9} \right)\] done
clear
C)
\[\left( \frac{1}{3},\,\frac{1}{9} \right)\] done
clear
D)
\[\left( \frac{1}{5},\,\frac{1}{25} \right)\] done
clear
View Solution play_arrow
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question_answer17)
\[\int_{0}^{\infty }{\frac{{{x}^{2}}\,dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}}=\]
A)
\[\frac{\pi }{2(a-b)}\] done
clear
B)
\[\frac{\pi }{2(b-a)}\] done
clear
C)
\[\frac{\pi }{(a+b)}\] done
clear
D)
\[\frac{\pi }{2(a+b)}\] done
clear
View Solution play_arrow
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question_answer18)
\[\int_{0}^{\infty }{\frac{{{x}^{3}}\,dx}{{{({{x}^{2}}+4)}^{2}}}=}\]
A)
0 done
clear
B)
\[\infty \] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer19)
\[\int_{0}^{\pi /2}{{{\sin }^{2m}}x\,dx=}\]
A)
\[\frac{2\,\,m\,\,!}{{{({{2}^{m}}.\,m\,\,!)}^{2}}}.\frac{\pi }{2}\] done
clear
B)
\[\frac{(2m)\,\,!}{{{({{2}^{m}}.\,m\,\,!)}^{2}}}.\frac{\pi }{2}\] done
clear
C)
\[\frac{2m\,\,!}{{{2}^{m}}.\,{{(m\,\,!)}^{2}}}.\frac{\pi }{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer20)
\[\int_{0}^{\pi /2}{{{\sin }^{5}}x\,dx=}\]
A)
\[\frac{8}{15}\] done
clear
B)
\[\frac{4}{15}\] done
clear
C)
\[\frac{8\sqrt{\pi }}{15}\] done
clear
D)
\[\frac{8\pi }{15}\] done
clear
View Solution play_arrow
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question_answer21)
\[\int_{0}^{\infty }{\frac{x\,dx}{(1+x)(1+{{x}^{2}})}}=\]
A)
\[\frac{\pi }{4}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{6}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer22)
If \[\varphi (x)=\int_{1/x}^{\sqrt{x}}{\sin ({{t}^{2}})\,dt,}\] then \[{\varphi }'(1)=\]
A)
\[\sin 1\] done
clear
B)
\[2\sin 1\] done
clear
C)
\[\frac{3}{2}\sin 1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer23)
The value of integral \[\int_{0}^{1}{\frac{{{x}^{b}}-1}{\log x}}\,dx\] is
A)
\[\log b\] done
clear
B)
\[2\log (b+1)\] done
clear
C)
\[3\log b\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer24)
The value of the integral \[\int_{-1}^{1}{\frac{d}{dx}\left( {{\tan }^{-1}}\frac{1}{x} \right)}\,dx\] is
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[-\frac{\pi }{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer25)
The least value of the function \[F(x)=\] \[\int_{5\pi /4}^{x}{(3\sin u+4\cos u)\,du}\] on the interval \[\left[ \frac{5\pi }{4},\,\,\frac{4\pi }{3} \right]\] is
A)
\[\sqrt{3}+\frac{3}{2}\] done
clear
B)
\[-2\sqrt{3}+\frac{3}{2}+\frac{1}{\sqrt{2}}\] done
clear
C)
\[\frac{3}{2}+\frac{1}{\sqrt{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer26)
\[\int_{0}^{\infty }{{{e}^{-2x}}(\sin 2x+\cos 2x)\,dx=}\]
A)
1 done
clear
B)
0 done
clear
C)
\[\frac{1}{2}\] done
clear
D)
\[\infty \] done
clear
View Solution play_arrow
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question_answer27)
\[\int_{0}^{b-c}{\,\,{f}''(x+a)\,dx=}\] [SCRA 1990]
A)
\[{f}'(a)-{f}'(b)\] done
clear
B)
\[{f}'(b-c+a)-{f}'(a)\] done
clear
C)
\[{f}'(b+c-a)+{f}'(a)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer28)
The greatest value of the function \[F(x)=\int_{1}^{x}{\,\,|t|\,dt}\] on the interval \[\left[ -\frac{1}{2},\,\,\frac{1}{2} \right]\] is given by [IIT Screening]
A)
\[\frac{3}{8}\] done
clear
B)
\[-\frac{1}{2}\] done
clear
C)
\[-\frac{3}{8}\] done
clear
D)
\[\frac{2}{5}\] done
clear
View Solution play_arrow
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question_answer29)
\[\int_{-\pi /2}^{\pi /2}{{{\sin }^{2}}x{{\cos }^{2}}x(\sin x+\cos x)\,dx=}\] [EAMCET 1992]
A)
\[\frac{2}{15}\] done
clear
B)
\[\frac{4}{15}\] done
clear
C)
\[\frac{6}{15}\] done
clear
D)
\[\frac{8}{15}\] done
clear
View Solution play_arrow
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question_answer30)
\[\int_{0}^{\infty }{\frac{dx}{{{\left( x+\sqrt{{{x}^{2}}+1} \right)}^{3}}}}=\] [EAMCET 1992]
A)
\[\frac{3}{8}\] done
clear
B)
\[\frac{1}{8}\] done
clear
C)
\[-\frac{3}{8}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer31)
The derivative of \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\frac{1}{\log t}\,dt}\], \[(x>0)\] is
A)
\[\frac{1}{3\log x}-\frac{1}{2\log x}\] done
clear
B)
\[\frac{1}{3\log x}\] done
clear
C)
\[\frac{3{{x}^{2}}}{3\log x}\] done
clear
D)
\[{{(\log x)}^{-1}}.x(x-1)\] done
clear
View Solution play_arrow
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question_answer32)
If \[f(x)=\int_{{{x}^{2}}}^{{{x}^{2}}+1}{{{e}^{-{{t}^{2}}}}}dt,\] then \[f(x)\] increases in [IIT Screening 2003]
A)
\[(2,\,\,2)\] done
clear
B)
No value of \[x\] done
clear
C)
\[(0,\,\,\infty )\] done
clear
D)
\[(-\infty ,\,\,0)\] done
clear
View Solution play_arrow
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question_answer33)
If \[f(x)=\int_{{{x}^{2}}}^{{{x}^{4}}}{\sin \sqrt{t}\,dt,}\] then \[{f}'(x)\] equals
A)
\[\sin {{x}^{2}}-\sin x\] done
clear
B)
\[4{{x}^{3}}\sin {{x}^{2}}-2x\sin x\] done
clear
C)
\[{{x}^{4}}\sin {{x}^{2}}-x\sin x\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer34)
If \[F(x)=\frac{1}{{{x}^{2}}}\int_{4}^{x}{(4{{t}^{2}}-2{F}'(t))\,dt,}\] then \[{F}'(4)\] equals
A)
32 done
clear
B)
\[\frac{32}{3}\] done
clear
C)
\[\frac{32}{9}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer35)
The value of the integral \[\sum\limits_{k=1}^{n}{\int_{0}^{1}{f(k-1+x)\,dx}}\] is
A)
\[\int_{0}^{1}{f(x)\,dx}\] done
clear
B)
\[\int_{0}^{2}{f(x)\,dx}\] done
clear
C)
\[\int_{0}^{n}{f(x)\,dx}\] done
clear
D)
\[n\int_{0}^{1}{f(x)\,dx}\] done
clear
View Solution play_arrow
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question_answer36)
The value of \[\int_{a}^{a+(\pi /2)}{({{\sin }^{4}}x+{{\cos }^{4}}x)\,dx}\] is
A)
Independent of \[a\] done
clear
B)
\[a\,{{\left( \frac{\pi }{2} \right)}^{2}}\] done
clear
C)
\[\frac{3\pi }{8}\] done
clear
D)
\[\frac{3\pi {{a}^{2}}}{8}\] done
clear
View Solution play_arrow
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question_answer37)
\[\int_{0}^{\pi }{{{\sin }^{5}}\left( \frac{x}{2} \right)\,dx}\] equals [Kurukshetra CEE 1996]
A)
\[\frac{16}{15}\] done
clear
B)
\[\frac{32}{15}\] done
clear
C)
\[\frac{8}{15}\] done
clear
D)
\[\frac{5}{6}\] done
clear
View Solution play_arrow
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question_answer38)
If \[\int_{{}}^{{}}{f(x)\,dx}=x{{e}^{-\log |x|}}+f(x),\] then \[f(x)\] is [MP PET 1997]
A)
1 done
clear
B)
0 done
clear
C)
\[c{{e}^{x}}\] done
clear
D)
\[\log x\] done
clear
View Solution play_arrow
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question_answer39)
\[\int_{0}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx}\] equals [RPET 1999]
A)
\[\frac{5\pi }{512}\] done
clear
B)
\[\frac{3\pi }{512}\] done
clear
C)
\[\frac{\pi }{512}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer40)
The value of \[\int_{\,0}^{\,\pi /2}{{{\left( \sqrt{\sin \theta }\cos \theta \right)}^{3}}d\theta }\] is [AMU 1999]
A)
2/9 done
clear
B)
2/15 done
clear
C)
8/45 done
clear
D)
5/2 done
clear
View Solution play_arrow
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question_answer41)
\[\int_{\,0}^{\,\infty }{\,\log \left( x+\frac{1}{x} \right)\frac{dx}{1+{{x}^{2}}}}\] is equal to [RPET 2000, 02]
A)
\[\pi \log 2\] done
clear
B)
\[-\pi \log 2\] done
clear
C)
\[(\pi /2)\log 2\] done
clear
D)
\[-(\pi /2)\log 2\] done
clear
View Solution play_arrow
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question_answer42)
\[\int_{\,0}^{\,\infty }{\frac{x\ln x\,dx}{{{(1+{{x}^{2}})}^{2}}}}\] is equal to [AMU 2000]
A)
0 done
clear
B)
1 done
clear
C)
\[\infty \] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer43)
If \[f(t)=\int_{\,-t}^{\,t}{\frac{dx}{1+{{x}^{2}}},}\] then \[{f}'(1)\] is [Roorkee 2000]
A)
Zero done
clear
B)
2/3 done
clear
C)
\[-\,1\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer44)
If \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\log t\,dt,\,\,(x>0),}\] then \[{F}'(x)=\] [MP PET 2001]
A)
\[(9{{x}^{2}}-4x)\log x\] done
clear
B)
\[(4x-9{{x}^{2}})\log x\] done
clear
C)
\[(9{{x}^{2}}+4x)\log x\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer45)
\[\int_{\,-\pi /2}^{\,\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx=}\] [EAMCET 2002]
A)
\[\frac{3\pi }{64}\] done
clear
B)
\[\frac{3\pi }{572}\] done
clear
C)
\[\frac{3\pi }{256}\] done
clear
D)
\[\frac{3\pi }{128}\] done
clear
View Solution play_arrow
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question_answer46)
\[\int_{\,0}^{\,1}{\frac{d}{dx}\left[ {{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right) \right]\,dx}\] is equal to [Kerala (Engg.) 2002]
A)
0 done
clear
B)
\[\pi \] done
clear
C)
\[\pi /2\] done
clear
D)
\[\pi /4\] done
clear
View Solution play_arrow
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question_answer47)
Let \[f(x)=\int_{\,1}^{\,x}{\sqrt{2-{{t}^{2}}}dt}\]. Then real roots of the equation \[{{x}^{2}}-{f}'(x)=0\] are [IIT Screening 2002]
A)
\[\pm 1\] done
clear
B)
\[\pm \frac{1}{\sqrt{2}}\] done
clear
C)
\[\pm \,\,\frac{1}{2}\] done
clear
D)
0 and 1 done
clear
View Solution play_arrow
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question_answer48)
\[\int_{\,0}^{\,\infty }{\frac{xdx}{(1+x)(1+{{x}^{2}})}=}\] [Karnataka CET 2003]
A)
0 done
clear
B)
\[\pi /2\] done
clear
C)
\[\pi /4\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer49)
\[\int_{0}^{a}{{{x}^{4}}\sqrt{{{a}^{2}}-{{x}^{2}}}}\,dx=\]
A)
\[\frac{\pi }{32}\] done
clear
B)
\[\frac{\pi }{32}{{a}^{6}}\] done
clear
C)
\[\frac{\pi }{16}{{a}^{6}}\] done
clear
D)
\[\frac{\pi }{8}{{a}^{6}}\] done
clear
View Solution play_arrow
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question_answer50)
\[\int_{0}^{a}{x{{(2ax-{{x}^{2}})}^{\frac{3}{2}}}\,dx=}\]
A)
\[{{a}^{5}}\left[ \frac{3\pi }{16}-1 \right]\] done
clear
B)
\[{{a}^{5}}\left[ \frac{3\pi }{16}+1 \right]\] done
clear
C)
\[{{a}^{5}}\left[ \frac{3\pi }{16}-\frac{1}{5} \right]\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer51)
\[\int_{0}^{a}{{{x}^{2}}{{({{a}^{2}}-{{x}^{2}})}^{3/2}}dx=}\]
A)
\[\frac{\pi {{a}^{6}}}{32}\] done
clear
B)
\[\frac{2{{a}^{5}}}{15}\] done
clear
C)
\[\frac{{{a}^{6}}}{32}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer52)
Let\[\frac{d}{dx}F(x)=\left( \frac{{{e}^{\sin x}}}{x} \right)\,;\,x>0\]. If \[\int_{\,1}^{\,4}{\frac{3}{x}{{e}^{\sin {{x}^{3}}}}dx=F(k)-F(1)}\], then one of the possible value of k, is [AIEEE 2003]
A)
15 done
clear
B)
16 done
clear
C)
63 done
clear
D)
64 done
clear
View Solution play_arrow
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question_answer53)
If \[f(x)=\int_{0}^{x}{t\sin t\,dt\,,}\] then \[{f}'(x)=\] [MNR 1982; Karnataka CET 1999]
A)
\[\cos x+x\sin x\] done
clear
B)
\[x\sin x\] done
clear
C)
\[x\cos x\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer54)
\[\underset{n\to \infty }{\mathop{\text{lim}\,}}\,\left[ \frac{1}{{{n}^{2}}}{{\sec }^{2}}\frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}{{\sec }^{2}}\frac{4}{{{n}^{2}}}+.....+\frac{1}{n}{{\sec }^{2}}1 \right]\] equals [AIEEE 2005]
A)
\[\tan 1\] done
clear
B)
\[\frac{1}{2}\tan 1\] done
clear
C)
\[\frac{1}{2}\sec 1\] done
clear
D)
\[\frac{1}{2}\text{cosec}1\] done
clear
View Solution play_arrow