Range\[\mathbf{f}\left( \mathbf{x} \right)=\frac{{{\sec }^{2}}x-\tan x}{{{\sec }^{2}}x+\tan x},\]where \[-\frac{\pi }{2}<z<\frac{\pi }{2},\]is
A)
\[\left[ \frac{1}{3},3 \right]\]
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B)
R
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C)
\[R-\left( \frac{1}{3},3 \right)\]
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D)
None of these
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The function \[\mathbf{f}\left( \mathbf{x} \right)=\frac{1}{x}\]on its domain is:
A)
Increasing function
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B)
Decreasing function
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C)
Identity function
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D)
Constant function
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\[\int{\frac{\log (x+1)-\log x}{x(x+1)}}.dx\]is equal to:
A)
\[-log\text{ }\left[ log\left( \frac{x+1}{x} \right) \right]+c\]
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B)
\[-log\text{ }log\left( \frac{x+1}{x} \right)+c\]
done
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C)
\[-{{\left\{ log\left( x+1 \right) \right\}}^{2}}-{{\left\{ log\left( x \right) \right\}}^{2}}+c\]
done
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D)
\[-\frac{1}{2}.{{\left[ \log \frac{(x+1)}{x} \right]}^{2}}+c\]
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The value of the integral \[\int\limits_{0}^{\pi }{\mathbf{x}.\mathbf{logsinx}.\mathbf{dx}}\] is:
A)
\[-\frac{{{\pi }^{2}}}{2}.log2\]
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B)
\[\frac{{{\pi }^{2}}}{2}.log2\]
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C)
\[\pi .log2\]
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D)
\[{{\pi }^{2}}.log2\]
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The order and degree of the differential equation \[{{\left( 1+3.\frac{dy}{dx} \right)}^{\frac{1}{3}}}=4.\frac{{{d}^{3}}y}{d{{x}^{3}}}\]are:
A)
\[\left( 1,\frac{3}{2} \right)\]
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B)
\[\left( 3,3 \right)\]
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C)
\[\left( 3,4 \right)\]
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D)
\[\left( 2,2 \right)\]
done
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If \[\left| \begin{align} & 6i\,\,\,\,-3i\,\,\,\,1 \\ & 4\,\,\,\,\,\,3i\,\,\,\,\,\,-1 \\ & 20\,\,\,\,3\,\,\,\,\,\,\,\,\,\,i \\ \end{align} \right|=\mathbf{x}+\mathbf{iy}\], then
A)
\[x=3,y=1\]
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B)
\[x=0,y=3\]
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C)
\[~x=1,\text{ }y=3\]
done
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D)
\[x=0,\text{ }y=0\]
done
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If \[\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\mathbf{0}\]and\[\left| \overrightarrow{\mathbf{a}} \right|=\mathbf{3},\left| \overrightarrow{\mathbf{b}} \right|=\mathbf{5},\left| \overrightarrow{\mathbf{c}} \right|=\mathbf{7}\], then angle between a and b is:
A)
\[\frac{\pi }{6}\]
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B)
\[\frac{\pi }{3}\]
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C)
\[\frac{\pi }{4}\]
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D)
\[\frac{\pi }{2}\]
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Distance between two parallel planes \[\mathbf{2x}+\mathbf{y}+\mathbf{2z}=\mathbf{8}\] and \[\mathbf{4x}+\mathbf{2y}+\mathbf{4z}+\mathbf{5}=\mathbf{0}\] is:
A)
\[\frac{3}{2}\]
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B)
\[\frac{7}{2}\]
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C)
\[\frac{5}{2}\]
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D)
\[\frac{7}{3}\]
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Two ball are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that at least one ball is red?
A)
\[\frac{5}{12}\]
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B)
\[\frac{7}{12}\]
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C)
\[\frac{1}{3}\]
done
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D)
\[\frac{1}{4}\]
done
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The domain of the function \[\mathbf{f}\left( \mathbf{x} \right)=\mathbf{lo}{{\mathbf{g}}_{10}}\left( \sqrt{\mathbf{x}-\mathbf{4}}\text{ }+\sqrt{\mathbf{6}-\mathbf{x}} \right)\]is
A)
\[\left[ 4,6 \right]\]
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B)
\[\left( 2,6 \right)\]
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C)
\[\left[ 4,6 \right)\]
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D)
\[\left( 4,6 \right]\]
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\[\int\limits_{0}^{2\pi }{\frac{dx}{1+{{e}^{\sin x}}}}\]is equal to:
A)
\[\pi \]
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B)
\[\frac{\pi }{2}\]
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C)
\[2\pi \]
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D)
\[\frac{\pi }{6}\]
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Let A be an invertible matrix, then which of the following is not true?
A)
\[{{(A')}^{-1}}={{({{A}^{-1}})}^{\grave{\ }1}}\]
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B)
\[{{\mathbf{(A)}}^{\mathbf{-1}}}\mathbf{=}{{\left| A \right|}^{-1}}\]
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C)
\[{{({{A}^{2}})}^{-1}}={{({{A}^{-1}})}^{2}}\]
done
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D)
None of these
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The degree of the differential equation\[\frac{{{d}^{3}}y}{d{{x}^{3}}}+x{{\left( \frac{dy}{dx} \right)}^{4}}=4\log \left( \frac{{{d}^{4}}y}{d{{x}^{4}}} \right)\] be:
A)
1
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B)
4
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C)
3
done
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D)
None of these
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Let \[f(x=\] \[\left\{ \begin{align} & 1,\,\,x\le -1 \\ & \left| x \right|,\,\,\,-1<x<1 \\ & 0,\,\,\,\,x\ge 1 \\ \end{align} \right.\] then
A)
f(x) is continuous at \[x=-1\]
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B)
f is differentiable at \[x=-1\]
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C)
f is differentiable \[\forall x\]
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D)
f is continuous everywhere
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\[\int\limits_{0}^{100}{\left\{ \mathbf{x}-\left[ \mathbf{x} \right] \right\}.\mathbf{dx}}\] is equal to:
A)
150
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B)
100
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C)
50
done
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D)
250
done
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If \[\mathbf{P}\left( \mathbf{a} \right)=\mathbf{0}.\mathbf{65},\mathbf{P}\left( \mathbf{b} \right)=\mathbf{0}.\mathbf{15}\],then \[\mathbf{P}\left( \mathbf{a} \right)+\mathbf{P}\left( \mathbf{b} \right)\]is equal to:
A)
1.1
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B)
1.2
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C)
1.5
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D)
1.4
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The probability that a marksman will hit a target is given as \[\frac{1}{5}\]Then, the probability of at least one hit in 10 shots is
A)
\[1-{{\left( \frac{4}{5} \right)}^{10}}\]
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B)
\[\frac{1}{{{5}^{10}}}\]
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C)
\[1-\frac{1}{{{5}^{10}}}\]
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D)
\[{{\left( \frac{4}{5} \right)}^{10}}\]
done
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The function \[\mathbf{f}\left( \mathbf{x} \right)=\frac{x}{1+\left| x \right|}\]is
A)
strictly decreasing function
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B)
strictly increasing function
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C)
neither increasing nor decreasing
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D)
None of these
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\[\int{\frac{2}{{{\left( {{e}^{x}}+{{e}^{-x}} \right)}^{2}}}.dx}=\]
A)
\[\frac{-1}{{{e}^{x}}+{{e}^{-x}}}+c\]
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B)
\[\frac{-{{e}^{x}}}{{{e}^{x}}+{{e}^{-x}}}+c\]
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C)
\[\frac{1}{{{e}^{x}}+{{e}^{-x}}}+c\]
done
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D)
\[\frac{{{e}^{x}}}{{{e}^{x}}-{{e}^{-x}}}\]
done
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Lf \[y={{e}^{\tan x}},then(co{{s}^{2}}x).\frac{{{d}^{2}}y}{d{{x}^{2}}}=\]
A)
\[(1-sin2x).{{y}_{1}}\]
done
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B)
\[\left( 1+sin2x \right).{{y}_{1}}\]
done
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C)
\[\left( 1-cosx \right).y{{,}_{1}}\]
done
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D)
\[\left( 1-cos2x \right).{{y}_{1}}\]
done
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