Domain of the function \[f(x)=\sqrt{lo{{g}_{0.5}}(3x-8)-lo{{g}_{0.5}}({{x}^{2}}+4)}\]is
A)
\[\left( \frac{8}{3},\infty \right)\]
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B)
\[\left( -\infty ,\frac{8}{3} \right)\]
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C)
\[\left( -\infty ,\infty \right)\]
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D)
\[\left( 0,\infty \right)\]
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The value of c in Lagrange's theorem for the function \[\mathbf{f}\left( \mathbf{x} \right)=\mathbf{logsinx}\]in the interval \[\left[ \frac{\pi }{6},\frac{5\pi }{6} \right]\]is:
A)
\[\frac{\pi }{3}\]
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B)
\[\frac{\pi }{2}\]
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C)
\[\frac{\pi }{4}\]
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D)
None of these
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\[\int{{{e}^{{{\tan }^{-1x}}}}}\left( 1+\frac{x}{1+{{x}^{2}}} \right)\]Is equal to:
A)
\[\frac{1}{2}x{{e}^{{{\tan }^{-1}}x}}+C\]
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B)
\[x.{{e}^{{{\tan }^{-1}}x}}+C\]
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C)
\[\frac{1}{2}.{{e}^{{{\tan }^{-1}}x}}+C\]
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D)
\[{{e}^{{{\tan }^{-1}}x}}+C\]
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\[\int\limits_{0}^{\pi }{\mathbf{log}\left( 1+\mathbf{cosx} \right).\mathbf{dx}}\] is
A)
\[\pi l\mathbf{og}2\]
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B)
\[-\pi l\mathbf{og}2\]
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C)
\[\frac{\pi }{2}l\mathbf{og}2\]
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D)
\[{{\pi }^{2}}l\mathbf{og}2\]
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If AB=A and BA=B, then B2 is equal to
A)
B
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B)
A
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C)
0
done
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D)
1
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The value of
is:
A)
\[abc\]
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B)
\[a+b+c\]
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C)
4abc
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D)
0
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If the sum of two unit vectors is a unit vector, then the angle between them is equal to:
A)
\[\frac{\pi }{3}\]
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B)
\[\frac{\pi }{6}\]
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C)
\[\frac{\pi }{2}\]
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D)
\[\frac{2\pi }{3}\]
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The direction cosines of a line equally inclined to the axis be
A)
\[\frac{1}{3},\frac{1}{3},\frac{1}{3}\]
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B)
\[\frac{-1}{3},\frac{-1}{3},\frac{-1}{3}\]
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C)
\[\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2}\]
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D)
\[\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}\]
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Seven white balls and three black balls are rounding placed in row. The probability that no two black balls are placed adjacent is
A)
\[\frac{7}{15}\]
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B)
\[\frac{1}{3}\]
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C)
\[\frac{2}{15}\]
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D)
\[\frac{4}{15}\]
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\[ta{{n}^{-1}}x+co{{t}^{-1}}\left( x+1 \right)=\]
A)
[a] \[ta{{n}^{-1}}({{x}^{2}}+x+1)\]
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B)
(b)\[co{{t}^{-1}}({{x}^{2}}-x+1)\]
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C)
\[co{{t}^{-1}}({{x}^{2}}+x+1)\]
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D)
\[ta{{n}^{-1}}({{x}^{2}}-x+1)\]
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If x is real, then the maximum value of \[\mathbf{3}-\mathbf{6x}-\mathbf{8}{{\mathbf{x}}^{\mathbf{2}}}\] is:
A)
\[\frac{17}{8}\]
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B)
\[\frac{15}{8}\]
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C)
\[\frac{29}{8}\]
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D)
\[\frac{33}{8}\]
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In tossing 10 coins the probability of getting exactly 5 heads is
A)
\[\frac{193}{256}\]
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B)
\[\frac{9}{128}\]
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C)
\[\frac{1}{2}\]
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D)
\[\frac{63}{256}\]
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The slope of the tangent to the curve \[\mathbf{x}={{\mathbf{t}}^{\mathbf{2}}}-\mathbf{3t}+\mathbf{5}\]and \[\mathbf{y}=\mathbf{2}{{\mathbf{t}}^{\mathbf{2}}}-\mathbf{2t}+\mathbf{6}\text{ }\mathbf{at}\,\mathbf{t}=\mathbf{2}\]is:
A)
6
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B)
5
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C)
4
done
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D)
1
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Let \[\mathbf{f}\left( \mathbf{x}+\mathbf{y} \right)=\mathbf{f}\left( \mathbf{x} \right).\mathbf{f}\left( \mathbf{y} \right),\forall \mathbf{x},\mathbf{y}\]; where \[f(0)\ne 0.\,if\,f(5)=2\]and \[\mathbf{f}'\left( \mathbf{0} \right)=\mathbf{3}\], then f'(5) is equal to
A)
2
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B)
4
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C)
6
done
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D)
8
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If \[\mathbf{f}\left( \mathbf{x} \right)=\left\{ \begin{align} & x,\,\mathbf{when}\text{ }\mathbf{x}\text{ }\mathbf{is}\text{ }\mathbf{rational} \\ & 1-x,\mathbf{when}\text{ }\mathbf{x}\text{ }\mathbf{is}\text{ }\mathbf{irrational} \\ \end{align} \right.\] then
A)
f(x) is continuous \[\forall x\]
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B)
f(x) is differentiable \[\forall x\]
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C)
f(x) is continuous at \[x=\frac{1}{2}\]
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D)
f(x) is discontinuous at \[x=\frac{1}{2}\]
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, then \[\underset{x\to 0}{\mathop{lim}}\,\left[ \frac{f'x}{x} \right]\]is:
A)
0
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B)
2
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C)
\[-2\]
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D)
6
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The binomial distribution whose mean is 10 and S.D. is \[2\sqrt{2}\] is:
A)
\[{{\left( \frac{1}{5}+\frac{4}{5} \right)}^{\frac{1}{50}}}\]
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B)
\[{{\left( \frac{1}{5}+\frac{4}{5} \right)}^{50}}\]
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C)
\[{{\left( \frac{4}{5}-\frac{1}{5} \right)}^{50}}\]
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D)
\[{{\left( \frac{4}{5}+\frac{5}{1} \right)}^{50}}\]
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Let \[\overrightarrow{\mathbf{a}}=\mathbf{2}\widehat{\mathbf{i}}-\mathbf{3}\widehat{\mathbf{j}}-\mathbf{6}\widehat{\mathbf{k}}\]and \[\overrightarrow{\mathbf{b}}=-\mathbf{2}\widehat{\mathbf{i}}-\mathbf{2}\widehat{\mathbf{j}}-\widehat{\mathbf{k}}\], then the value of the ratio of the projection of a on b and projection of b on a is equal to
A)
\[\frac{3}{7}\]
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B)
\[\frac{7}{3}\]
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C)
3
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D)
7
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The equation of the plane which cuts equal intercepts of unit length on the coordinate axes is
A)
\[x+y-z+1=0\]
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B)
(b)\[x+y+z=1\]
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C)
\[~x+y+z+1=0\]
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D)
\[x-y-z=1\]
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The inverse of the function \[\mathbf{f}\left( \mathbf{x} \right)=\frac{{{a}^{x}}-{{a}^{-x}}}{{{a}^{x}}+{{a}^{-x}}}\]is
A)
\[\frac{1}{2}{{\log }_{a}}\left( \frac{1+x}{1-x} \right)\]
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B)
\[\frac{1}{2}{{\log }_{a}}\left( \frac{1-x}{1+x} \right)\]
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C)
\[\frac{1}{2}{{\log }_{e}}\left( \frac{1+x}{1-x} \right)\]
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D)
\[\frac{1}{2}{{\log }_{e}}\left( \frac{1-x}{1+x} \right)\]
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