If \[M=\left[ \frac{1}{2}\,\,\,\frac{2}{3} \right]\] and \[{{M}^{2}}-\lambda \,\,M-{{I}_{2}}=0,\] then \[\lambda =\]
A)
\[-2\]
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B)
2
done
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C)
\[-4\]
done
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D)
4
done
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E)
None of these
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If \[A=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right],\] then \[{{A}^{2}}=\]
A)
Unit matrix
done
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B)
Null matrix
done
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C)
A
done
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D)
\[~-A\]
done
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E)
None of these
done
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If \[O\,\,(A)=2\times 3,\] \[O\,\,(B)=3\times 2,\] and \[O\,\,(C)=3\times 3,\] which one of the following is not defined
A)
\[CB+A'\]
done
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B)
\[BAC\]
done
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C)
\[C\,\,(A+B')'\]
done
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D)
\[C\,\,(A+B')\]
done
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E)
None of these
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If A and B are two matrices and \[(A+B)\,\,(A-B)={{A}^{2}}-{{B}^{2}},\] then
A)
\[AB=BA\]
done
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B)
\[{{A}^{2}}+{{B}^{2}}={{A}^{2}}-{{B}^{2}}\]
done
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C)
\[A'\,\,B'=AB\]
done
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D)
All of these
done
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E)
None of these
done
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The inverse of \[\left[ \begin{matrix} 2 & -3 \\ -4 & 2 \\ \end{matrix} \right]\] is
A)
\[\frac{-1}{8}\,\,\left[ \begin{matrix} 2 & 3 \\ 4 & 2 \\ \end{matrix} \right]\]
done
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B)
\[\frac{-1}{8}\,\,\left[ \begin{matrix} 3 & 2 \\ 2 & 4 \\ \end{matrix} \right]\]
done
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C)
\[\frac{1}{8}\,\,\left[ \begin{matrix} 2 & 3 \\ 4 & 2 \\ \end{matrix} \right]\]
done
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D)
\[\frac{1}{8}\,\,\left[ \begin{matrix} 3 & 2 \\ 2 & 4 \\ \end{matrix} \right]\]
done
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E)
None of these
done
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If \[A=\left[ \begin{matrix} -1 & 2 \\ 2 & -1 \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 3 \\ 1 \\ \end{matrix} \right],\] \[AX=B,\] then x=
A)
\[[5]\]
done
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B)
\[\frac{1}{3}\left[ \begin{matrix} 5 \\ 7 \\ \end{matrix} \right]\]
done
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C)
\[\frac{1}{3}[5\,\,\,7]\]
done
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D)
\[\left[ \begin{matrix} 5 \\ 7 \\ \end{matrix} \right]\]
done
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E)
None of these
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If \[A=\left[ \begin{matrix} 2 & 4 & 5 \\ 4 & 8 & 10 \\ -6 & -12 & -15 \\ \end{matrix} \right],\] then rank of A is equal to
A)
0
done
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B)
1
done
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C)
2
done
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D)
3
done
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E)
None of these
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If \[{{\cos }^{-1}}\,\,\left( \frac{1}{x} \right)=\theta ,\]then \[\tan \theta =\]
A)
\[\frac{1}{\sqrt{{{x}^{2}}-1}}\]
done
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B)
\[\sqrt{{{x}^{2}}-1}\]
done
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C)
\[\sqrt{1-{{x}^{2}}}\]
done
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D)
\[\sqrt{{{x}^{2}}-1}\]
done
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E)
None of these
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View Answer play_arrow
The principal value of \[{{\sin }^{-1}}\,\,\left( -\frac{\sqrt{3}}{2} \right)\] is
A)
\[\frac{-2\pi }{3}\]
done
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B)
\[\frac{-\pi }{3}\]
done
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C)
\[\frac{4\pi }{3}\]
done
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D)
\[\frac{5\pi }{3}\]
done
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E)
None of these
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\[{{\tan }^{-1}}2x+{{\tan }^{-1}}3x=\frac{\pi }{{}},\] then x=
A)
\[-1\]
done
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B)
\[\frac{1}{6}\]
done
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C)
\[-1,\frac{1}{6}\]
done
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D)
All of these
done
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E)
None of these
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If \[{{\tan }^{-1}}\frac{1}{2}+{{\tan }^{-1}}\frac{1}{3}=\]
A)
\[0\]
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B)
\[\frac{\pi }{4}\]
done
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C)
\[\frac{\pi }{2}\]
done
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D)
\[\pi \]
done
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E)
None of these
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View Answer play_arrow
If the distance between the points (a, 2) and (3, 4) be 8, then a =
A)
\[2+3\sqrt{15}\]
done
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B)
\[2-3\sqrt{15}\]
done
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C)
\[2\pm 3\sqrt{15}\]
done
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D)
\[3\pm 2\sqrt{15}\]
done
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E)
None of these
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There vertices of a parallelogram taken in order are \[\left( -1,-6 \right),\]\[\left( 2,-5 \right)\] and (7, 2). The fourth vertex is
A)
(1, 4)
done
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B)
(4, 1)
done
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C)
(1, 1)
done
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D)
(4. 4)
done
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E)
None of these
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If the position vectors of P and Q are \[(i\text{ }+3j-7k)\] and \[\left( 5i-2j+4k \right),\] then \[|\overrightarrow{PQ}|\] is
A)
\[\sqrt{158}\]
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B)
\[\sqrt{160}\]
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C)
\[\sqrt{161}\]
done
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D)
\[\sqrt{162}\]
done
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E)
None of these
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If \[\hat{a}\]and \[\hat{b}\] are unit vectors such that \[[\hat{a},\hat{b},\hat{a}\times \hat{b}]=\frac{1}{4},\] then angle between \[\hat{a}\] and \[\hat{b}\] is.
A)
\[\frac{\pi }{3}\]
done
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B)
\[\frac{\pi }{4}\]
done
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C)
\[\frac{\pi }{6}\]
done
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D)
\[\frac{\pi }{2}\]
done
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E)
None of these
done
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View Answer play_arrow
The direction cosines of the line \[4x-4=1-3y=2z-1\] are
A)
\[\frac{3}{\sqrt{56}},\,\,\frac{4}{\sqrt{56}},\,\,\frac{6}{\sqrt{56}}\]
done
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B)
\[\frac{3}{\sqrt{29}},\,\,\frac{-4}{\sqrt{29}},\,\,\frac{6}{\sqrt{29}}\]
done
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C)
\[\frac{3}{\sqrt{61}},\,\,\frac{-4}{\sqrt{61}},\,\,\frac{6}{\sqrt{61}}\]
done
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D)
\[\frac{4}{\sqrt{29}},\,\,\frac{-3}{\sqrt{29}},\,\,\frac{2}{\sqrt{29}}\]
done
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E)
None of these
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View Answer play_arrow
Perpendicular distance of the point (3, 4, 5) from the y-axis, is
A)
\[\sqrt{34}\]
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B)
\[\sqrt{41}\]
done
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C)
4
done
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D)
5
done
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E)
None of these
done
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View Answer play_arrow
The angle between the straight lines \[\frac{x+1}{2}=\frac{y-2}{5}=\frac{z+3}{4}\] and \[\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{-3}\]
A)
\[45{}^\circ \]
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B)
\[30{}^\circ \]
done
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C)
\[60{}^\circ \]
done
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D)
\[90{}^\circ \]
done
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E)
None of these
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View Answer play_arrow
If two planes intersect, then the shortest distance between the planes is.
A)
\[cos\,\,\theta \]
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B)
\[cos\,\,90{}^\circ \]
done
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C)
\[sin\,\,90{}^\circ \]
done
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D)
All of these
done
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E)
None of these
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View Answer play_arrow
The line \[\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\] is parallel to the plane.
A)
\[3x+4y+5z=7\]
done
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B)
\[2x+y-2z=0\]
done
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C)
\[x+y-z=2\]
done
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D)
\[2x+3y+4z=0\]
done
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E)
None of these
done
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View Answer play_arrow
\[\frac{d}{dx}\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}=\]
A)
\[{{\sec }^{2}}x\]
done
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B)
\[-{{\sec }^{2}}\,\,\left( \frac{\pi }{4}-x \right)\]
done
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C)
\[{{\sec }^{2}}\,\,\left( \frac{\pi }{4}+x \right)\]
done
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D)
\[{{\sec }^{2}}\,\,\left( \frac{\pi }{4}-x \right)\]
done
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E)
None of these
done
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View Answer play_arrow
\[\frac{d}{dx}\left( \frac{\sec x+\tan x}{\sec x-\tan x} \right)=\]
A)
\[\frac{2\cos x}{{{(1-\sin x)}^{2}}}\]
done
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B)
\[\frac{\cos x}{{{(1-\sin x)}^{2}}}\]
done
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C)
\[\frac{2\cos x}{1-\sin x}\]
done
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D)
All of these
done
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E)
None of these
done
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View Answer play_arrow
If \[y=\sqrt{\frac{1+{{e}^{x}}}{1-{{e}^{x}}}}\] then \[\frac{dy}{dx}=\]
A)
\[\frac{{{e}^{x}}}{(1-{{e}^{x}})\,\,\sqrt{1-{{e}^{2x}}}}\]
done
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B)
\[\frac{{{e}^{x}}}{(1-{{e}^{x}})\,\,\sqrt{1-{{e}^{x}}}}\]
done
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C)
\[2\pm 3\sqrt{15}\]
done
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D)
\[3\pm 2\sqrt{15}\]
done
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E)
None of these
done
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View Answer play_arrow
If \[f\left( x \right)=sin\text{ }log\,x,\] then the value of \[f\left( xy \right)+f\left( \frac{x}{y} \right)-f(x).\,\cos \,\,\log y\] is equal to
A)
\[1\]
done
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B)
\[0\]
done
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C)
\[-1\]
done
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D)
\[\sin \,\,\log x.\,\,\cos \,\,log\,y\]
done
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E)
None of these
done
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View Answer play_arrow
If \[f(x)=\cos \,\,(\log x),\] then \[f(x)f(y)-\frac{1}{2}[f(x\text{/}y)+f(xy)]=\]
A)
\[-1\]
done
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B)
\[\frac{1}{2}\]
done
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C)
\[-2\]
done
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D)
All of these
done
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E)
None of these
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View Answer play_arrow
If \[f(x)=\frac{x}{x-1}=\frac{1}{y},\] then f(y)=
A)
\[x\]
done
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B)
\[\frac{2}{5}\]
done
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C)
\[-\frac{2}{5}\]
done
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D)
All of these
done
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E)
None of these
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View Answer play_arrow
If \[f(x)=\left\{ \frac{\sin \,\,2x}{5x} \right.,when\,\,x\ne 0\] is continuous at x = 0, then the value of k will be
A)
1
done
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B)
\[\frac{2}{5}\]
done
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C)
\[-\frac{2}{5}\]
done
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D)
All of these
done
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E)
None of these
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If \[f(x)=\frac{x}{1+|x|}\] then f(0)=
A)
0
done
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B)
1
done
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C)
2
done
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D)
3
done
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E)
None of these
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View Answer play_arrow
\[\int{\frac{dx}{2\cos ec2x}}\]
A)
\[\frac{\cos 2x}{4}+c\]
done
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B)
\[\frac{\sin 2x}{4}+c\]
done
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C)
\[-\frac{\sin 2x}{4}+c\]
done
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D)
\[-\frac{\cos 2x}{4}+c\]
done
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E)
None of these
done
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View Answer play_arrow
The value of \[\int{\frac{1}{{{(x-5)}^{2}}}\,\,dx}\] is
A)
\[\frac{1}{x-5}+c\]
done
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B)
\[-\frac{1}{x-5}+c\]
done
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C)
\[\frac{2}{{{(x-5)}^{3}}}+c\]
done
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D)
All of these
done
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E)
None of these
done
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View Answer play_arrow
If \[\int{x\,\sin \,\,x\,dx=-x\,\cos \,x+A,}\] then A =
A)
\[\sin x\]+ constant
done
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B)
\[\cos \,x\]+ constant
done
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C)
Constant
done
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D)
All of these
done
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E)
None of these
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\[\int_{0}^{\pi /2}{{{e}^{x}}\sin x\,\,dx=}\]
A)
\[\frac{1}{2}\,\,({{e}^{\pi /2}}-1)\]
done
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B)
\[\frac{1}{2}\,\,({{e}^{\pi /2}}+1)\]
done
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C)
\[\frac{1}{2}\,\,(1-{{e}^{\pi /2}})\]
done
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D)
\[2\,\,({{e}^{\pi /2}}+1)\]
done
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E)
None of these
done
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View Answer play_arrow
The order and degree of the differential equation \[\sqrt{\frac{dy}{dx}}-4\frac{dy}{dx}-7x=0\] are
A)
\[1\,\,and\,\,\frac{1}{2}\]
done
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B)
2 and 1
done
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C)
1 and 1
done
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D)
1 and 2
done
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E)
None of these
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The differential equation of all straight lines passing through the origin is
A)
\[y=\sqrt{x\frac{dy}{dx}}\]
done
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B)
\[\frac{dy}{dx}=y+x\]
done
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C)
\[\frac{dy}{dx}=\frac{y}{x}\]
done
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D)
All of these
done
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E)
None of these
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View Answer play_arrow
Solution of differential equation \[2xy\frac{dy}{dx}={{x}^{2}}+3{{y}^{2}}\] is
A)
\[{{x}^{3}}+{{y}^{2}}=p{{x}^{2}}\]
done
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B)
\[\frac{{{x}^{2}}}{2}+\frac{{{y}^{3}}}{x}={{y}^{2}}+P\]
done
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C)
\[{{x}^{2}}+{{y}^{3}}=p{{x}^{2}}\]
done
clear
D)
\[{{x}^{2}}+{{y}^{3}}=p{{x}^{3}}\]
done
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E)
None of these
done
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View Answer play_arrow
A problem of mathematics is given to three students whose chances of solving the problem are \[\frac{1}{3},\]\[\frac{1}{4}\]and\[\frac{1}{5}\] respectively. The probability that the question will be solved is
A)
\[\frac{2}{3}\]
done
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B)
\[\frac{3}{4}\]
done
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C)
\[\frac{4}{5}\]
done
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D)
\[\frac{3}{5}\]
done
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E)
None of these
done
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The probability that A speaks truth is \[\frac{4}{5},\] while this probability for B is\[\frac{3}{4}\] . The Probability that they contradict each other when asked to speak on a fact is
A)
\[\frac{4}{5}\]
done
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B)
\[\frac{1}{5}\]
done
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C)
\[\frac{7}{20}\]
done
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D)
\[\frac{3}{20}\]
done
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E)
None of these
done
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Let A and B be two events such that \[P\left( A \right)=0.3\] and \[P\left( A\cup B \right)=0.8\]. If A and B are independent events, then P(B)=
A)
\[\frac{5}{6}\]
done
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B)
\[\frac{5}{7}\]
done
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C)
\[\frac{3}{5}\]
done
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D)
\[\frac{2}{5}\]
done
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E)
None of these
done
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Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is 11, if 5 appears one the first?
A)
\[\frac{1}{36}\]
done
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B)
\[\frac{1}{6}\]
done
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C)
\[\frac{5}{6}\]
done
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D)
All of these
done
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E)
None of these
done
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View Answer play_arrow
8 coins are tosssed simultaneously. The probability of getting at least 6 heads is
A)
\[\frac{57}{64}\]
done
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B)
\[\frac{229}{256}\]
done
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C)
\[\frac{7}{64}\]
done
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D)
\[\frac{37}{256}\]
done
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E)
None of these
done
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