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question_answer1)
The function \[f:R\to R\] is defined by \[f\left( x \right)={{\cos }^{2}}x+{{\sin }^{4}}x\] for \[x\in R\]. Then the range of \[f(x)\]is
A)
\[\left( \frac{3}{4},1 \right]\] done
clear
B)
\[\left[ \frac{3}{4},1 \right)\] done
clear
C)
\[\left[ \frac{3}{4},1 \right]\] done
clear
D)
\[\left( \frac{3}{4},1 \right)\] done
clear
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question_answer2)
Let \[f:\left[ -\frac{\pi }{3},\frac{2\pi }{3} \right]\to [0,4]\] be a function defined as \[f(x)=\sqrt{3}\sin x-\cos x+2\].Then \[{{f}^{-1}}(x)\]is given by
A)
\[{{\sin }^{-1}}\left( \frac{x-2}{2} \right)-\frac{\pi }{6}\] done
clear
B)
\[{{\sin }^{-1}}\left( \frac{x-2}{2} \right)+\frac{\pi }{6}\] done
clear
C)
\[\frac{2\pi }{3}+{{\cos }^{-1}}\left( \frac{x-2}{2} \right)\] done
clear
D)
none of these done
clear
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question_answer3)
The domain of the function \[f(x)={{\left[ {{\log }_{10}}\left( \frac{5x-{{x}^{2}}}{4} \right) \right]}^{1/2}}\]is
A)
\[-\infty <x<\infty \] done
clear
B)
\[1\le x\le 4\] done
clear
C)
\[4\le x\le 16\] done
clear
D)
\[-1\le x\le 3\] done
clear
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question_answer4)
The range of \[f(x)=si{{n}^{-1}}(\sqrt{{{x}^{2}}+x+1})\] is
A)
\[\left( 0,\frac{\pi }{2} \right]\] done
clear
B)
\[\left( 0,\frac{\pi }{3} \right]\] done
clear
C)
\[\left[ \frac{\pi }{3},\frac{\pi }{2} \right]\] done
clear
D)
\[\left[ \frac{\pi }{6},\frac{\pi }{3} \right]\] done
clear
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question_answer5)
The range of the function \[f(x)=\left| x-1 \right|+\left| x-2 \right|,-1\le x\le 3\] is
A)
\[\left[ 1,\text{ }3 \right]\] done
clear
B)
\[[1\text{ },5]\] done
clear
C)
\[\left[ 3,\text{ }5 \right]\] done
clear
D)
none of these done
clear
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question_answer6)
If \[f(x+y)=f(x)+f(y)-xy-1\,\forall \,x,\,\,y\in R\] and\[f(1)=1,\] then the number of solutions of \[f\left( n \right)=n,\]\[n\in N\], is
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
more then 2 done
clear
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question_answer7)
The function \[f:(-\infty ,-1)\to \left( 0,{{e}^{5}} \right]\] defined by \[f(x)={{e}^{{{x}^{3-3x+2}}}}\] is
A)
many-one and onto done
clear
B)
many-one and into done
clear
C)
one-one and onto done
clear
D)
one-one and into done
clear
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question_answer8)
If \[f(x)\] is an invertible function and \[g\left( x \right)=2f\left( x \right)+5,\] then the value of \[{{g}^{-1}}(x)\] is
A)
\[2{{f}^{-1}}(x)-5\] done
clear
B)
\[\frac{1}{2{{f}^{-1}}(x)+5}\] done
clear
C)
\[\frac{1}{2}{{f}^{-1}}(x)=5\] done
clear
D)
\[{{f}^{-1}}\left( \frac{x-5}{2} \right)\] done
clear
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question_answer9)
If \[g(x)={{x}^{2}}+x-2\] and \[\frac{1}{2}gof(x)=2{{x}^{2}}-5x+2,\]then which is not a possible f(x)?
A)
\[2x-3\] done
clear
B)
\[-2x+2\] done
clear
C)
\[x-3\] done
clear
D)
None of these done
clear
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question_answer10)
A function F(x) satisfies the functional equation \[{{x}^{2}}F(x)+F(1-x)=2x-{{x}^{4}}\]for all real x. F(x) must be
A)
\[{{x}^{2}}\] done
clear
B)
\[1-{{x}^{2}}\] done
clear
C)
\[1+{{x}^{2}}\] done
clear
D)
\[{{x}^{2}}+x+1\] done
clear
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question_answer11)
Let\[f :X\to y\,f(x)=sinx+cosx+2\sqrt{2}\]be invertible. Then which \[X\to Y\]is not possible?
A)
\[\left[ \frac{\pi }{4},\frac{5\pi }{4} \right]\to [\sqrt{2},3\sqrt{2}]\] done
clear
B)
\[\left[ -\frac{3\pi }{4},\frac{\pi }{4} \right]\to [\sqrt{2},3\sqrt{2}]\] done
clear
C)
\[\left[ -\frac{3\pi }{4},\frac{3\pi }{4} \right]\to [\sqrt{2},3\sqrt{2}]\] done
clear
D)
none of these done
clear
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question_answer12)
The domain of the functions \[f(x)=\frac{1}{\sqrt{\left\{ \operatorname{sinx} \right\}+\left\{ \sin (\pi +x) \right\}}}\] where \[\left\{ \cdot \right\}\] denotes the fractional part, is
A)
\[[0,\pi ]\] done
clear
B)
\[(2n+1)\pi /2,n\in Z\] done
clear
C)
\[(0,\pi )\] done
clear
D)
none of these done
clear
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question_answer13)
Let \[g(x)=f(x)-1.\] If \[f(x)+f(1-x)=2\forall x\in R\], then \[g(x)\] is symmetrical about
A)
the origin done
clear
B)
the line\[x=\frac{1}{2}\] done
clear
C)
the point (1, 0) done
clear
D)
the point \[\left( \frac{1}{2},0 \right)\] done
clear
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question_answer14)
The range of the function f defined by \[f(x)=\left[ \frac{1}{\sin \left\{ x \right\}} \right]\] (where [.] and \[\left\{ . \right\}\], respectively, denote the greatest integer and the fractional part functions) is
A)
I, the set of integers done
clear
B)
N, the set of natural numbers done
clear
C)
W, the set of whole numbers done
clear
D)
\[\left\{ 1,2,3,4.... \right\}\] done
clear
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question_answer15)
If \[f(x)=\left\{ \begin{matrix} {{x}^{2}},\,\,\,for\,x\ge 0 \\ x,\,\,\,\,for\,x<0 \\ \end{matrix} \right.,\] then fof (x) is given by
A)
\[{{x}^{2}}for\,x\ge 0,\,\]\[x\text{ }for\text{ }x<0\] done
clear
B)
\[{{x}^{4}}for\,x\ge 0,\,\]\[{{x}^{2}}for\text{ }x<0\] done
clear
C)
\[{{x}^{4}}for\,x\ge 0,\,\]\[-{{x}^{2}}for\text{ }x<0\] done
clear
D)
\[{{x}^{4}}for\,x\ge 0,\,\]\[x\text{ }for\text{ }x<0\] done
clear
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question_answer16)
The range of \[f(x)=co{{s}^{-1}}\left( \frac{1+{{x}^{2}}}{2x} \right)+\sqrt{2-{{x}^{2}}}\]is
A)
\[\left\{ 0,\,\,1+\frac{\pi }{2} \right\}\] done
clear
B)
\[\left\{ 0,\,\,1+\pi \right\}\] done
clear
C)
\[\left\{ 1,\,\,1+\frac{\pi }{2} \right\}\] done
clear
D)
\[\left\{ 1,\,\,1+\pi \right\}\] done
clear
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question_answer17)
The range of following function is \[f(x)=\sqrt{(1-\cos x)\sqrt{(1-\cos x)\sqrt{(1-\cos x)\sqrt{...\infty }}}}\]
A)
\[\left[ 0,\text{ }1 \right]\] done
clear
B)
\[\left[ 0,\text{ }1/2 \right]\] done
clear
C)
\[\left[ 0,\text{ }2 \right]\] done
clear
D)
none of these done
clear
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question_answer18)
Which pair of functions is identical?
A)
\[{{\sin }^{-1}}(sinx)\,\,and\,\,sin\,(si{{n}^{-1}}x)\] done
clear
B)
\[{{\log }_{e}}{{e}^{x}},{{e}^{{{\log }_{e}}x}}\] done
clear
C)
\[{{\log }_{e}}{{x}^{2}},2lo{{g}_{e}}x\] done
clear
D)
None of these done
clear
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question_answer19)
A function f form the set of natural numbers to integers defined by \[f(n)=\left\{ \begin{matrix} \frac{n-1}{2},\,\,\,\text{when}\,\,n\,\,\text{odd} \\ -\frac{n}{2}\,,\,\,\,\text{when}\,\,n\,\,\text{is}\,\,\text{odd} \\ \end{matrix}\,\,\,\text{is} \right.\]
A)
one-one but not onto. done
clear
B)
onto but not one-one done
clear
C)
one-one and onto both. done
clear
D)
neither one-one nor onto. done
clear
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question_answer20)
Let \[R=\left\{ \left( 1,3 \right),\left( 4,2 \right),\left( 2,4 \right),\left( 2,3 \right),\left( 3,1 \right) \right\}\] be a relation on the set \[A=\left\{ 1,2,3,4 \right\}.\]The relation R is
A)
a function done
clear
B)
reflexive done
clear
C)
not symmetric done
clear
D)
transitive done
clear
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question_answer21)
If f is periodic, g is polynomial function, f (g(x)) is periodic, \[g\left( 2 \right)=3,\text{ and }g\left( 4 \right)=7,\] then g(6) is _______.
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question_answer22)
The function of satisfies the functional equation 3\[3f(x)+2f\left( \frac{x+59}{x-1} \right)=10x+30\]for all real \[x\ne 1.\]The value of f(7) is ______.
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question_answer23)
The total number of solutions of \[{{[x]}^{2}}=x+2\{x\}\], where [.] and {.} denote the greatest integer and the fractional part functions, respectively, is equal to ____.
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question_answer24)
If \[f\left( x+\frac{1}{2} \right)+f\left( x-\frac{1}{2} \right)=f(x)\] for all \[x\in R\], then the period of f(x) is _____.
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question_answer25)
If \[{{e}^{f(x)}}=\frac{10+x}{10-x},\,\,\] \[x\in (-10,10)\] and\[f(x)=kf\left( \frac{200x}{100+{{x}^{2}}} \right)\], then k =_____.
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