-
question_answer1)
If \[I\] is the greatest of the definite integrals \[{{I}_{1}}=\int_{0}^{1}{{{e}^{-x}}{{\cos }^{2}}x\,dx},\] \[{{I}_{2}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}}{{\cos }^{2}}x\,dx\] \[{{I}_{3}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}dx},\] \[{{I}_{4}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}/2}}dx},\] Then
A)
\[I={{I}_{1}}\] done
clear
B)
\[I={{I}_{2}}\] done
clear
C)
\[I={{I}_{3}}\] done
clear
D)
\[I={{I}_{4}}\] done
clear
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question_answer2)
Let \[f(x)\] be a function satisfying \[{f}'(x)=f(x)\] with \[f(0)=1\] and \[g(x)\] be the function satisfying \[f(x)+g(x)={{x}^{2}}.\] The value of integral \[\int_{0}^{1}{f(x)\,g(x)\,dx}\] is equal to [AIEEE 2003; DCE 2005]
A)
\[\frac{1}{4}(e-7)\] done
clear
B)
\[\frac{1}{4}(e-2)\] done
clear
C)
\[\frac{1}{2}(e-3)\] done
clear
D)
None of these done
clear
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question_answer3)
If \[{{I}_{m}}=\int_{1}^{x}{{{(\log x)}^{m}}dx}\] satisfies the relation \[{{I}_{m}}=k-l{{I}_{m-1}},\] then
A)
\[k=e\] done
clear
B)
\[l=m\] done
clear
C)
\[k=\frac{1}{e}\] done
clear
D)
None of these done
clear
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question_answer4)
Let \[f\] be a positive function. Let \[{{I}_{1}}=\int_{1-k}^{k}{x\,f\left\{ x(1-x) \right\}}\,dx\], \[{{I}_{2}}=\int_{1-k}^{k}{\,f\left\{ x(1-x) \right\}}\,dx\] when \[2k-1>0.\] Then \[{{I}_{1}}/{{I}_{2}}\] is [IIT 1997 Cancelled]
A)
2 done
clear
B)
\[k\] done
clear
C)
\[1/2\] done
clear
D)
1 done
clear
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question_answer5)
If \[\int_{0}^{x}{f(t)\,dt}=x+\int_{x}^{1}{t\,f(t)\,dt,}\] then the value of \[f(1)\] is [IIT 1998; AMU 2005]
A)
1/2 done
clear
B)
0 done
clear
C)
1 done
clear
D)
-1/2 done
clear
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question_answer6)
\[\int_{0}^{1}{\frac{{{x}^{7}}}{\sqrt{1-{{x}^{4}}}}dx}\] is equal to [AMU 2000]
A)
1 done
clear
B)
\[\frac{1}{3}\] done
clear
C)
\[\frac{2}{3}\] done
clear
D)
\[\frac{\pi }{3}\] done
clear
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question_answer7)
If \[n\] is any integer, then \[\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\] [IIT 1985; RPET 1995; UPSEAT 2001]
A)
\[x\] done
clear
B)
1 done
clear
C)
0 done
clear
D)
None of these done
clear
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question_answer8)
The value of the definite integral \[\int_{0}^{1}{\frac{x\,dx}{{{x}^{3}}+16}}\] lies in the interval \[[a,\,\,b].\] The smallest such interval is
A)
\[\left[ 0,\,\,\frac{1}{17} \right]\] done
clear
B)
[0, 1] done
clear
C)
\[\left[ 0,\,\,\frac{1}{27} \right]\] done
clear
D)
None of these done
clear
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question_answer9)
Let a, b, c be non-zero real numbers such that \[\int_{0}^{1}{(1+{{\cos }^{8}}x)(a{{x}^{2}}+bx+c)\,dx}=\int_{0}^{2}{(1+{{\cos }^{8}}x)(a{{x}^{2}}+bx+c)\,dx}\]Then the quadratic equation \[a{{x}^{2}}+bx+c=0\] has [IIT 1981; CEE 1993]
A)
No root in (0, 2) done
clear
B)
At least one root in (0, 2) done
clear
C)
A double root in (0, 2) done
clear
D)
None of these done
clear
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question_answer10)
If \[f(x)=\int_{-1}^{x}{|t|\,dt,}\] \[x\ge -1,\] then [MNR 1994]
A)
\[f\] and \[{f}'\] are continous for \[x+1>0\] done
clear
B)
\[f\] is continous but \[{f}'\] is not continous for \[x+1>0\] done
clear
C)
\[f\] and \[{f}'\] are not continous at \[x=0\] done
clear
D)
\[f\] is continous at \[x=0\] but \[{f}'\] is not so done
clear
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question_answer11)
Let \[g(x)=\int_{0}^{x}{f(t)\,dt}\] where \[\frac{1}{2}\le f(t)\le 1,\,t\in [0,\,1]\] and \[0\le f(t)\le \frac{1}{2}\] for \[t\in (1,\,2]\], then [IIT Screening 2000]
A)
\[-\frac{3}{2}\le g(2)<\frac{1}{2}\] done
clear
B)
\[0\le g(2)<2\] done
clear
C)
\[\frac{3}{2}<g(2)\le \frac{5}{2}\] done
clear
D)
\[2<g(2)<4\] done
clear
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question_answer12)
The value of \[\int_{\,-\,\pi }^{\,\pi }{\frac{{{\cos }^{2}}x}{1+{{a}^{x}}}dx,\,a>0,}\] is [IIT Screening 2001; AIEEE 2005]
A)
\[\pi \] done
clear
B)
\[a\pi \] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[2\pi \] done
clear
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question_answer13)
If \[f(x)=\frac{{{e}^{x}}}{1+{{e}^{x}}},\,\,\,\ {{I}_{1}}=\int_{f(-a)}^{f(a)}{xg\{x(1-x)\}dx}\], and \[{{I}_{2}}=\int_{f(-a)}^{f(a)}{g\{x(1-x))\}dx}\], then the value of \[\frac{{{I}_{2}}}{{{I}_{1}}}\] is [AIEEE 2004]
A)
1 done
clear
B)
-3 done
clear
C)
-1 done
clear
D)
2 done
clear
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question_answer14)
Let \[f:R\to R\] and \[g:R\to R\] be continuous functions, then the value of the integral \[\int_{-\pi /2}^{\pi /2}{[f(x)+f(-x)]\,\,[g(x)-g(-x)]\,dx=}\] [IIT 1990; DCE 2000; MP PET 2001]
A)
\[\pi \] done
clear
B)
1 done
clear
C)
\[-1\] done
clear
D)
0 done
clear
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question_answer15)
The numbers P, Q and \[R\] for which the function \[f(x)=P{{e}^{2x}}+Q{{e}^{x}}+Rx\] satisfies the conditions \[f(0)=-1,\] \[{f}'(\log 2)=31\] and \[\int_{0}^{\log 4}{[f(x)-Rx]\,dx=\frac{39}{2}}\] are given by
A)
\[P=2,\] \[Q=-3,\] \[R=4\] done
clear
B)
\[P=-5,\] \[Q=2,\] \[R=3\] done
clear
C)
\[P=5,\] \[Q=-2,\] \[R=3\] done
clear
D)
\[P=5,\] \[Q=-6,\] \[R=3\] done
clear
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question_answer16)
\[\left[ \sum\limits_{n=1}^{10}{\int_{-2n-1}^{2n}{{{\sin }^{27}}x\,dx}} \right]+\left[ \sum\limits_{n=1}^{10}{\int_{2n}^{2n+1}{{{\sin }^{27}}x\,dx}} \right]\] equals [MP PET 2002]
A)
\[{{27}^{2}}\] done
clear
B)
\[-54\] done
clear
C)
36 done
clear
D)
0 done
clear
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question_answer17)
Let \[\int_{0}^{1}{f(x)\,dx=1,}\] \[\int_{0}^{1}{x\,f(x)\,dx=a}\] and \[\int_{0}^{1}{{{x}^{2}}f(x)\,dx={{a}^{2}},}\] then the value of \[\int_{0}^{1}{{{(x-a)}^{2}}f(x)\,dx=}\] [IIT 1990]
A)
0 done
clear
B)
\[{{a}^{2}}\] done
clear
C)
\[{{a}^{2}}-1\] done
clear
D)
\[{{a}^{2}}-2a+2\] done
clear
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question_answer18)
Given that \[\int_{0}^{\infty }{\frac{{{x}^{2}}\,dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})({{x}^{2}}+{{c}^{2}})}=\frac{\pi }{2(a+b)(b+c)(c+a)}},\] then the value of \[\int_{0}^{\infty }{\frac{{{x}^{2}}dx}{({{x}^{2}}+4)({{x}^{2}}+9)}}\] is [Karnataka CET 1993]
A)
\[\frac{\pi }{60}\] done
clear
B)
\[\frac{\pi }{20}\] done
clear
C)
\[\frac{\pi }{40}\] done
clear
D)
\[\frac{\pi }{80}\] done
clear
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question_answer19)
If \[l(m,\,n)=\int_{0}^{1}{{{t}^{m}}{{(1+t)}^{n}}dt,}\] then the expression for \[l(m,\,n)\] in terms of \[l(m+1,\,\,n-1)\] is [IIT Screening 2003]
A)
\[\frac{{{2}^{n}}}{m+1}-\frac{n}{m+1}l(m+1,\,n-1)\] done
clear
B)
\[\frac{n}{m+1}l(m+1,\,n-1)\] done
clear
C)
\[\frac{{{2}^{n}}}{m+1}+\frac{n}{m+1}l(m+1,\,n-1)\] done
clear
D)
\[\frac{m}{n+1}l(m+1,\,n-1)\] done
clear
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question_answer20)
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+{{2}^{4}}+{{3}^{4}}+....+{{n}^{4}}}{{{n}^{5}}}\]\[-\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+{{2}^{3}}+{{3}^{3}}+....+{{n}^{3}}}{{{n}^{5}}}=\] [AIEEE 2003]
A)
\[\frac{1}{30}\] done
clear
B)
Zero done
clear
C)
\[\frac{1}{4}\] done
clear
D)
\[\frac{1}{5}\] done
clear
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question_answer21)
If \[\int_{0}^{{{t}^{2}}}{xf(x)dx=}\frac{2}{5}{{t}^{5}},\,\,t>0,\]then\[f\left( \frac{4}{25} \right)=\] [IIT Screening 2004]
A)
\[\frac{2}{5}\] done
clear
B)
\[\frac{5}{2}\] done
clear
C)
\[-\frac{2}{5}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer22)
For which of the following values of m, the area of the region bounded by the curve \[y=x-{{x}^{2}}\] and the line \[y=mx\] equals\[\frac{9}{2}\] [IIT 1999]
A)
\[-4\] done
clear
B)
\[-2\] done
clear
C)
2 done
clear
D)
4 done
clear
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question_answer23)
Area enclosed between the curve \[{{y}^{2}}(2a-x)={{x}^{3}}\] and line \[x=2a\] above x-axis is [MP PET 2001]
A)
\[\pi \,{{a}^{2}}\] done
clear
B)
\[\frac{3\pi \,{{a}^{2}}}{2}\] done
clear
C)
\[2\pi \,{{a}^{2}}\] done
clear
D)
\[3\pi \,{{a}^{2}}\] done
clear
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question_answer24)
What is the area bounded by the curves \[{{x}^{2}}+{{y}^{2}}=9\] and \[{{y}^{2}}=8x\] is [DCE 1999]
A)
0 done
clear
B)
\[\frac{2\sqrt{2}}{3}+\frac{9\pi }{2}-9{{\sin }^{-1}}\left( \frac{1}{3} \right)\] done
clear
C)
\[16\,\pi \] done
clear
D)
None of these done
clear
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question_answer25)
The area bounded by the curves \[y=\,|x|-1\] and \[y=-|x|+1\] is [IIT Screening 2002]
A)
1 done
clear
B)
2 done
clear
C)
\[2\sqrt{2}\] done
clear
D)
4 done
clear
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question_answer26)
The volume of spherical cap of height h cut off from a sphere of radius a is equal to [UPSEAT 2004]
A)
\[\frac{\pi }{3}{{h}^{2}}(3a-h)\] done
clear
B)
\[\pi (a-h)(2{{a}^{2}}-{{h}^{2}}-ah)\] done
clear
C)
\[\frac{4\pi }{3}{{h}^{3}}\] done
clear
D)
None of these done
clear
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question_answer27)
If for a real number \[y,\,\,[y]\] is the greatest integer less than or equal to \[y,\] then the value of the integral \[\int\limits_{\pi /2}^{3\pi /2}{[2\sin x]\,dx}\] is [IIT 1999]
A)
\[-\pi \] done
clear
B)
0 done
clear
C)
\[-\frac{\pi }{2}\] done
clear
D)
\[\frac{\pi }{2}\] done
clear
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question_answer28)
If \[f(x)=A\sin \left( \frac{\pi x}{2} \right)+B,\] \[{f}'\left( \frac{1}{2} \right)=\sqrt{2}\] and \[\int_{0}^{1}{f(x)\,dx=\frac{2A}{\pi },}\] then the constants \[A\] and \[B\] are respectively [IIT 1995]
A)
\[\frac{\pi }{2}\] and \[\frac{\pi }{2}\] done
clear
B)
\[\frac{2}{\pi }\] and \[\frac{3}{\pi }\] done
clear
C)
\[\frac{4}{\pi }\] and 0 done
clear
D)
0 and \[-\frac{4}{\pi }\] done
clear
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question_answer29)
If \[{{I}_{n}}=\int_{0}^{\infty }{{{e}^{-x}}{{x}^{n-1}}dx,}\] then \[\int_{0}^{\infty }{{{e}^{-\lambda x}}{{x}^{n-1}}dx=}\]
A)
\[\lambda {{I}_{n}}\] done
clear
B)
\[\frac{1}{\lambda }{{I}_{n}}\] done
clear
C)
\[\frac{{{I}_{n}}}{{{\lambda }^{n}}}\] done
clear
D)
\[{{\lambda }^{n}}{{I}_{n}}\] done
clear
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question_answer30)
\[{{I}_{n}}=\int_{\,0}^{\,\pi /4}{{{\tan }^{n}}x\,dx}\], then \[\underset{n-\infty }{\mathop{\lim }}\,n\,[{{I}_{n}}+{{I}_{n-2}}]\] equals [AIEEE 2002]
A)
1/2 done
clear
B)
1 done
clear
C)
\[\infty \] done
clear
D)
0 done
clear
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question_answer31)
The area bounded by the curves \[y=\ln x\], \[y=\ln |x|\], \[y=\,|\ln x|\] and \[y=\,|\ln |x||\] is [AIEEE 2002]
A)
4 sq. unit done
clear
B)
6 sq. unit done
clear
C)
10 sq. unit done
clear
D)
None of these done
clear
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question_answer32)
\[\int\limits_{0}^{\pi }{\,\frac{\sin \left( n+\frac{1}{2} \right)\text{ }x}{\sin x}}\,dx\], \[(n\in N)\] equals [Kurukshetra CEE 1998]
A)
\[n\pi \] done
clear
B)
\[(2n+1)\frac{\pi }{2}\] done
clear
C)
\[\pi \] done
clear
D)
0 done
clear
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question_answer33)
If \[\int_{0}^{1}{{{e}^{{{x}^{2}}}}(x-\alpha )\,dx=0,}\] then [MNR 1994; Pb. CET 2001; UPSEAT 2000]
A)
\[1<\alpha <2\] done
clear
B)
\[\alpha <0\] done
clear
C)
\[0<\alpha <1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer34)
\[\int_{\,\pi }^{\,10\pi }{\,|\sin x|dx}\] is [AIEEE 2002]
A)
20 done
clear
B)
8 done
clear
C)
10 done
clear
D)
18 done
clear
View Solution play_arrow
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question_answer35)
\[\int_{-\,\pi }^{\,\pi }{\frac{2x(1+\sin x)}{1+{{\cos }^{2}}x}dx}\] is [AIEEE 2002]
A)
\[{{\pi }^{2}}/4\] done
clear
B)
\[{{\pi }^{2}}\] done
clear
C)
0 done
clear
D)
\[\pi /2\] done
clear
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question_answer36)
On the interval \[\left[ \frac{5\pi }{3},\,\,\frac{7\pi }{4} \right],\] the greatest value of the function \[f(x)=\int_{5\pi /3}^{x}{(6\cos t-2\sin t)\,dt=}\]
A)
\[3\sqrt{3}+2\sqrt{2}+1\] done
clear
B)
\[3\sqrt{3}-2\sqrt{2}-1\] done
clear
C)
Does not exist done
clear
D)
None of these done
clear
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question_answer37)
If \[{{I}_{1}}=\int_{0}^{1}{{{2}^{{{x}^{2}}}}dx,\ }{{I}_{2}}=\int_{0}^{1}{{{2}^{{{x}^{3}}}}dx},\ {{I}_{3}}=\int_{1}^{2}{{{2}^{{{x}^{2}}}}dx}\],\[{{I}_{4}}=\int_{1}^{2}{{{2}^{{{x}^{3}}}}dx}\], then [AIEEE 2005]
A)
\[{{I}_{3}}={{I}_{4}}\] done
clear
B)
\[{{I}_{3}}>{{I}_{4}}\] done
clear
C)
\[{{I}_{2}}>{{I}_{1}}\] done
clear
D)
\[{{I}_{1}}>{{I}_{2}}\] done
clear
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question_answer38)
If \[2f(x)-3f\left( \frac{1}{x} \right)=x\], then \[\int_{1}^{2}{f(x)}\ dx\] is equal to [J & K 2005]
A)
\[\frac{3}{5}\ln 2\] done
clear
B)
\[\frac{-3}{5}(1+\ln 2)\] done
clear
C)
\[\frac{-3}{5}\ln 2\] done
clear
D)
None of these done
clear
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question_answer39)
If \[\int_{a}^{b}{{{x}^{3}}dx}=0\] and \[\int_{a}^{b}{{{x}^{2}}}dx=\frac{2}{3}\], then the value of a and b will be respectively [AMU 2005]
A)
1, 1 done
clear
B)
\[-1,-1\] done
clear
C)
\[1,-1\] done
clear
D)
\[-1,1\] done
clear
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question_answer40)
The sine and cosine curves intersects infinitely many times giving bounded regions of equal areas. The area of one of such region is [DCE 2005]
A)
\[\sqrt{2}\] done
clear
B)
\[2\sqrt{2}\] done
clear
C)
\[3\sqrt{2}\] done
clear
D)
\[4\sqrt{2}\] done
clear
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