-
question_answer1)
If \[f(x)=\frac{{{\cos }^{2}}x+{{\sin }^{4}}x}{{{\sin }^{2}}x+{{\cos }^{4}}x}\] for \[x\in R\], then \[f(2002)=\] [EAMCET 2002]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer2)
If \[f:R\to R\] satisfies \[f(x+y)=f(x)+f(y)\], for all \[x,\ y\in R\] and \[f(1)=7\], then \[\sum\limits_{r=1}^{n}{f(r)}\] is [AIEEE 2003]
A)
\[\frac{7n}{2}\] done
clear
B)
\[\frac{7(n+1)}{2}\] done
clear
C)
\[7n(n+1)\] done
clear
D)
\[\frac{7n(n+1)}{2}\] done
clear
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question_answer3)
Suppose \[f:[2,\ 2]\to R\] is defined by \[f(x)=\left\{ \begin{align} & -1\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\ -2\le x\le 0 \\ & x-1\ \ \ \ \ \text{for}\ 0\le x\le 2 \\ \end{align} \right.\], then \[\{x\in (-2,\ 2):x\le 0\] and \[f(|x|)=x\}=\] [EAMCET 2003]
A)
\[\{-1\}\] done
clear
B)
{0} done
clear
C)
\[\{-1/2\}\] done
clear
D)
\[\varphi \] done
clear
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question_answer4)
If \[f(x)=sgn ({{x}^{3}})\], then [DCE 2001]
A)
f is continuous but not derivable at \[x=0\] done
clear
B)
\[f'({{0}^{+}})=2\] done
clear
C)
\[f'({{0}^{-}})=1\] done
clear
D)
f is not derivable at \[x=0\] done
clear
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question_answer5)
If \[f:R\to R\] and \[g:R\to R\] are given by \[f(x)=\ |x|\] and \[g(x)=\ |x|\] for each \[x\in R\], then \[\{x\in R\ :g(f(x))\le f(g(x))\}=\] [EAMCET 2003]
A)
\[Z\cup (-\infty ,\ 0)\] done
clear
B)
\[(-\infty ,0)\] done
clear
C)
Z done
clear
D)
R done
clear
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question_answer6)
For a real number \[x,\ [x]\] denotes the integral part of x. The value of \[\left[ \frac{1}{2} \right]+\left[ \frac{1}{2}+\frac{1}{100} \right]+\left[ \frac{1}{2}+\frac{2}{100} \right]+....+\left[ \frac{1}{2}+\frac{99}{100} \right]\] is [IIT Screening 1994]
A)
49 done
clear
B)
50 done
clear
C)
48 done
clear
D)
51 done
clear
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question_answer7)
If function \[f(x)=\frac{1}{2}-\tan \left( \frac{\pi x}{2} \right)\]; \[(-1<x<1)\] and \[g(x)=\sqrt{3+4x-4{{x}^{2}}}\], then the domain of gof is [IIT 1990]
A)
\[(-1,\ 1)\] done
clear
B)
\[\left[ -\frac{1}{2},\ \frac{1}{2} \right]\] done
clear
C)
\[\left[ -1,\ \frac{1}{2} \right]\] done
clear
D)
\[\left[ -\frac{1}{2},\ -1 \right]\] done
clear
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question_answer8)
The domain of the function \[f(x)=\frac{1}{{{\log }_{10}}(1-x)}+\sqrt{x+2}\] is [DCE 2000]
A)
\[]-3,\ -2.5[\cup ]-2.5,\ -2[\] done
clear
B)
\[[-2,\ 0[\cup ]0,\ 1[\] done
clear
C)
]0,1[ done
clear
D)
None of these done
clear
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question_answer9)
The domain of definition of the function \[y(x)\] given by \[{{2}^{x}}+{{2}^{y}}=2\] is [IIT Screening 2000; DCE 2001]
A)
(0, 1] done
clear
B)
[0, 1] done
clear
C)
\[(-\infty ,\ 0]\] done
clear
D)
\[(-\infty ,\ 1)\] done
clear
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question_answer10)
Let \[f(x)=(1+{{b}^{2}}){{x}^{2}}+2bx+1\] and \[m(b)\] the minimum value of \[f(x)\]for a given b. As b varies, the range of m is [IIT Screening 2001]
A)
[0, 1] done
clear
B)
\[\left( 0,\ \frac{1}{2} \right]\] done
clear
C)
\[\left[ \frac{1}{2},\ 1 \right]\] done
clear
D)
\[(0,\ 1]\] done
clear
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question_answer11)
The range of the function \[f(x){{=}^{7-x}}{{P}_{x-3}}\] is [AIEEE 2004]
A)
{1, 2, 3, 4, 5} done
clear
B)
(1, 2, 3, 4, 5, 6) done
clear
C)
{1, 2, 3, 4} done
clear
D)
{1, 2, 3} done
clear
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question_answer12)
Let \[2{{\sin }^{2}}x+3\sin x-2>0\] and \[{{x}^{2}}-x-2<0\] (x is measured in radians). Then x lies in the interval [IIT 1994]
A)
\[\left( \frac{\pi }{6},\ \frac{5\pi }{6} \right)\] done
clear
B)
\[\left( -1,\ \frac{5\pi }{6} \right)\] done
clear
C)
\[(-1,\ 2)\] done
clear
D)
\[\left( \frac{\pi }{6},\ 2 \right)\] done
clear
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question_answer13)
Let \[f(x)={{(x+1)}^{2}}-1,\ \ (x\ge -1)\]. Then the set \[S=\{x:f(x)={{f}^{-1}}(x)\}\] is [IIT 1995]
A)
Empty done
clear
B)
{0, -1} done
clear
C)
{0, 1, -1} done
clear
D)
\[\left\{ 0,\ -1,\ \frac{-3+i\sqrt{3}}{2},\ \frac{-3-i\sqrt{3}}{2} \right\}\] done
clear
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question_answer14)
If f is an even function defined on the interval (-5, 5), then four real values of x satisfying the equation \[f(x)=f\left( \frac{x+1}{x+2} \right)\] are [IIT 1996]
A)
\[\frac{-3-\sqrt{5}}{2},\ \frac{-3+\sqrt{5}}{2},\ \frac{3-\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2}\] done
clear
B)
\[\frac{-5+\sqrt{3}}{2},\ \frac{-3+\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2},\ \frac{3-\sqrt{5}}{2}\] done
clear
C)
\[\frac{3-\sqrt{5}}{2},\ \frac{3+\sqrt{5}}{2},\ \frac{-3-\sqrt{5}}{2},\ \frac{5+\sqrt{3}}{2}\] done
clear
D)
\[-3-\sqrt{5},\ -3+\sqrt{5},\ 3-\sqrt{5},\ 3+\sqrt{5}\] done
clear
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question_answer15)
If \[f(x)={{\sin }^{2}}x+{{\sin }^{2}}\left( x+\frac{\pi }{3} \right)+\cos x\cos \left( x+\frac{\pi }{3} \right)\] and \[g\left( \frac{5}{4} \right)=1\], then \[(gof)(x)=\] [IIT 1996]
A)
-2 done
clear
B)
-1 done
clear
C)
2 done
clear
D)
1 done
clear
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question_answer16)
If \[g(f(x))=|\sin x|\] and \[f(g(x))={{(\sin \sqrt{x})}^{2}}\], then [IIT 1998]
A)
\[f(x)={{\sin }^{2}}x,\ g(x)=\sqrt{x}\] done
clear
B)
\[f(x)=\sin x,\ g(x)=|x|\] done
clear
C)
\[f(x)={{x}^{2}},\ g(x)=\sin \sqrt{x}\] done
clear
D)
f and g cannot be determined done
clear
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question_answer17)
If \[f(x)=3x+10\], \[g(x)={{x}^{2}}-1\], then \[{{(fog)}^{-1}}\] is equal to [UPSEAT 2001]
A)
\[{{\left( \frac{x-7}{3} \right)}^{1/2}}\] done
clear
B)
\[{{\left( \frac{x+7}{3} \right)}^{1/2}}\] done
clear
C)
\[{{\left( \frac{x-3}{7} \right)}^{1/2}}\] done
clear
D)
\[{{\left( \frac{x+3}{7} \right)}^{1/2}}\] done
clear
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question_answer18)
If \[f:R\to R\]and \[g:R\to R\] are defined by \[f(x)=2x+3\]and \[g(x)={{x}^{2}}+7\], then the values of x such that \[g(f(x))=8\] are [EAMCET 2000, 03]
A)
1, 2 done
clear
B)
-1, 2 done
clear
C)
-1, -2 done
clear
D)
1, -2 done
clear
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question_answer19)
\[\underset{x\to 1}{\mathop{\lim }}\,(1-x)\tan \left( \frac{\pi x}{2} \right)=\] [IIT 1978, 84; RPET 1997, 2001; UPSEAT 2003; Pb. CET 2003]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\pi \] done
clear
C)
\[\frac{2}{\pi }\] done
clear
D)
0 done
clear
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question_answer20)
True statement for \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{2+3x}-\sqrt{2-3x}}\] is [BIT Ranchi 1982]
A)
Does not exist done
clear
B)
Lies between 0 and \[\frac{1}{2}\] done
clear
C)
Lies between \[\frac{1}{2}\] and 1 done
clear
D)
Greater then 1 done
clear
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question_answer21)
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{e}^{x}}}=0\] for [IIT 1992]
A)
No value of n done
clear
B)
n is any whole number done
clear
C)
\[n=0\] only done
clear
D)
\[n=2\] only done
clear
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question_answer22)
\[\underset{n\to \infty }{\mathop{\lim }}\,\sin [\pi \sqrt{{{n}^{2}}+1}]=\]
A)
\[\infty \] done
clear
B)
0 done
clear
C)
Does not exist done
clear
D)
None of these done
clear
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question_answer23)
If [.] denotes the greatest integer less than or equal to x, then the value of \[\underset{x\to 1}{\mathop{\lim }}\,(1-x+[x-1]+[1-x])\]is
A)
0 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
None of these done
clear
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question_answer24)
The values of a and b such that \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x(1+a\cos x)-b\sin x}{{{x}^{3}}}=1\], are [Roorkee 1996]
A)
\[\frac{5}{2},\ \frac{3}{2}\] done
clear
B)
\[\frac{5}{2},\ -\frac{3}{2}\] done
clear
C)
\[-\frac{5}{2},\ -\frac{3}{2}\] done
clear
D)
None of these done
clear
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question_answer25)
If \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{x}^{a}}}{{{x}^{x}}-{{a}^{a}}}=-1\], then [EAMCET 2003]
A)
\[a=1\] done
clear
B)
\[a=0\] done
clear
C)
\[a=e\] done
clear
D)
None of these done
clear
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question_answer26)
If \[{{x}_{1}}=3\]and\[x>0\]then \[\underset{n\to \infty }{\mathop{\lim }}\,{{x}_{n}}\] is equal to
A)
-1 done
clear
B)
2 done
clear
C)
\[\sqrt{5}\] done
clear
D)
3 done
clear
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question_answer27)
The value of \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\int_{\pi /2}^{x}{t\,dt}}{\sin (2x-\pi )}\]is [MP PET 1998]
A)
\[\infty \] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[\frac{\pi }{8}\] done
clear
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question_answer28)
The \[\underset{x\to 0}{\mathop{\lim }}\,{{(\cos x)}^{\cot x}}\]is [RPET 1999]
A)
-1 done
clear
B)
0 done
clear
C)
1 done
clear
D)
None of these done
clear
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question_answer29)
The integer n for which \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{(\cos x-1)\,(\cos x-{{e}^{x}})}{{{x}^{n}}}\] is a finite non-zero number is [IIT Screening 2002]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer30)
If f is strictly increasing function, then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f({{x}^{2}})-f(x)}{f(x)-f(0)}\] is equal to [IIT Screening 2004]
A)
0 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer31)
If \[f(x)=\left\{ \begin{align} & {{x}^{2}}-3,\ 2<x<3 \\ & 2x+5,\ 3<x<4 \\ \end{align} \right.\], the equation whose roots are \[\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f(x)\]and\[\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f(x)\]is [Orissa JEE 2004]
A)
\[{{x}^{2}}-7x+3=0\] done
clear
B)
\[{{x}^{2}}-20x+66=0\] done
clear
C)
\[{{x}^{2}}-17x+66=0\] done
clear
D)
\[{{x}^{2}}-18x+60=0\] done
clear
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question_answer32)
The function\[f(x)=[x]\cos \left[ \frac{2x-1}{2} \right]\pi ,\,\]where\[[.]\] denotes the greatest integer function, is discontinuous at [IIT 1995]
A)
All x done
clear
B)
No x done
clear
C)
All integer points done
clear
D)
x which is not an integer done
clear
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question_answer33)
Let \[f(x)\]be defined for all \[x>0\]and be continuous. Let \[f(x)\]satisfy \[f\left( \frac{x}{y} \right)=f(x)-f(y)\]for all x, y and \[f(e)=1,\]then [IIT 1995]
A)
\[f(x)=\ln x\] done
clear
B)
\[f(x)\]is bounded done
clear
C)
\[f\left( \frac{1}{x} \right)\to 0\]as\[x\to 0\] done
clear
D)
\[x\,f(x)\to 1\]as \[x\to 0\] done
clear
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question_answer34)
The value of \[p\] for which the function \[f(x)=\left\{ \begin{align} & \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12{{(\log 4)}^{3}},\,\,x=0 \\ \end{align} \right.\]may be continuous at \[x=0\], is [Orissa JEE 2004]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer35)
The function \[f(x)={{[x]}^{2}}-[{{x}^{2}}]\], (where [y] is the greatest integer less than or equal to y),is discontinuous at [IIT 1999]
A)
All integers done
clear
B)
All integers except 0 and 1 done
clear
C)
All integers except 0 done
clear
D)
All integers except 1 done
clear
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question_answer36)
If \[f(x)\,=\,\left\{ \begin{matrix} x{{e}^{-\,\left( \frac{1}{|\,x\,|}\,+\,\frac{1}{x} \right)}}, & x\ne 0 \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,, & x=0 \\ \end{matrix} \right.\] , then \[f(x)\,\] is [AIEEE 2003]
A)
Continuous as well as differentiable for all x done
clear
B)
Continuous for all x but not differentiable at \[x=0\] done
clear
C)
Neither differentiable nor continuous at \[x=0\] done
clear
D)
Discontinuous every where done
clear
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question_answer37)
Let \[f(x)=\frac{1-\tan x}{4x-\pi },\ x\ne \frac{\pi }{4},\ \ x\in \left[ 0,\frac{\pi }{2} \right]\], If \[f(x)\]is continuous in \[\left[ 0,\frac{\pi }{2} \right]\], then \[f\left( \frac{\pi }{4} \right)\]is [AIEEE 2004]
A)
-1 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[-\frac{1}{2}\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer38)
Let \[g(x)=x.\,f(x),\]where \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] at \[x=0\] [IIT Screening 1994; UPSEAT 2004]
A)
g is differentiable but g' is not continuous done
clear
B)
g is differentiable while f is not done
clear
C)
Both f and g are differentiable done
clear
D)
g is differentiable and g' is continuous done
clear
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question_answer39)
The function \[f(x)=\max [(1-x),\,(1+x),\,2],\] \[x\in (-\infty ,\,\infty ),\]is [IIT 1995]
A)
Continuous at all points done
clear
B)
Differentiable at all points done
clear
C)
Differentiable at all points except at \[x=1\]and \[x=-1\] done
clear
D)
Continuous at all points except at \[x=1\]and \[x=-1\]where it is discontinuous done
clear
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question_answer40)
The function \[f(x)\,=\,|x|+|x-1|\] is [RPET 1996; Kurukshetra CEE 2002]
A)
Continuous at \[x=1,\] but not differentiable at \[x=1\] done
clear
B)
Both continuous and differentiable at \[x=1\] done
clear
C)
Not continuous at \[x=1\] done
clear
D)
Not differentiable at \[x=1\] done
clear
View Solution play_arrow