If A = [\[x:x\]is a multiple of 3] and B = [\[x:x\]is a multiple of 5], then \[A-B\]is (\[\overline{A}\] means complement of A)
A)
\[\overline{A}\bigcap B\]
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B)
\[A\bigcap \overline{B}\]
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C)
\[\overline{A}\bigcap \overline{B}\]
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D)
\[\overline{A\bigcap B}\]
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E)
None of these
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If \[a={{\log }_{24}}12,\] \[b={{\log }_{36}}24\] and \[c={{\log }_{48}}36,\]then 1 + abc is equal to
A)
2ab
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B)
2ac
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C)
2bc
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D)
0
done
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E)
None of these
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If \[x=\sqrt{7}+\sqrt{3}\] and \[xy=4,\] then \[{{x}^{4}}+{{y}^{4}}=\]
A)
200
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B)
352
done
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C)
368
done
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D)
400
done
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E)
None of these
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The partial fraction of \[\frac{{{x}^{2}}}{{{(x-1)}^{3}}(x-2)}\] are
A)
\[\frac{-1}{{{(x-1)}^{3}}}+\frac{3}{{{(x-1)}^{2}}}-\frac{4}{(x-1)}+\frac{4}{(x-2)}\]
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B)
\[\frac{-1}{{{(x-1)}^{3}}}-\frac{3}{{{(x-1)}^{2}}}+\frac{4}{(x-1)}+\frac{4}{(x-2)}\]
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C)
\[\frac{-1}{{{(x-1)}^{3}}}+\frac{-3}{{{(x-1)}^{2}}}+\frac{-4}{(x-1)}+\frac{4}{(x-2)}\]
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D)
All of these
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E)
None of these
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If \[{{x}^{2/3}}-7{{x}^{1/3}}+10=0,\] then x =
A)
\[\{125\}\]
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B)
\[\{8\}\]
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C)
\[\phi \]
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D)
\[\{125,8\}\]
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E)
None of these
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If \[{{x}^{2}}-3x+2\] be a factor of \[{{x}^{4}}-p{{x}^{2}}+q,\] then (p, q) =
A)
(3, 4)
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B)
(4, 5)
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C)
(4, 3)
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D)
(5, 4)
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E)
None of these
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If \[y=\sqrt{\frac{1+\tan x}{1-\tan x}},\] then \[\frac{dy}{dx}=\]
A)
\[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\]
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B)
\[\sqrt{\frac{1-\tan x}{1+\tan x}}{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\]
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C)
\[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}\sec \left( \frac{\pi }{4}+x \right)\]
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D)
All of these
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E)
None of these
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If \[y\sqrt{{{x}^{2}}+1}=\log \left\{ \sqrt{{{x}^{2}}+1}+x \right\},\] then \[({{x}^{2}}+1)\frac{dy}{dx}+xy+1=\]
A)
3
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B)
2
done
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C)
1
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D)
0
done
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E)
None of these
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If \[{{x}^{p}}{{y}^{q}}={{(x+y)}^{p+q}},\] then \[\frac{dy}{dx}=\]
A)
\[\frac{x}{y}\]
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B)
\[\frac{y}{x}\]
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C)
\[\frac{-y}{x}\]
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D)
\[-\frac{x}{y}\]
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E)
None of these
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The differential coefficient of \[{{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}}\] w.r.t. \[{{\sin }^{-1}}\frac{2x}{1+{{x}^{2}}}\] is
A)
1
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B)
\[-1\]
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C)
0
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D)
2
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E)
None of these
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A point moves in a straight line during the time \[t=0\]to \[t=3\]according to the law \[S=15t-2{{t}^{2}}\]. The average velocity is
A)
3
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B)
9
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C)
15
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D)
27
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E)
None of these
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The point on the curve \[{{y}^{2}}=2(x-3)\] at which the normal is parallel to the line \[y-2x+1=0\] is
A)
\[\left( 5,2 \right)\]
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B)
\[\left( -\frac{1}{2},-2 \right)\]
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C)
\[\left( 5,-2 \right)\]
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D)
\[\left( \frac{3}{2},2 \right)\]
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E)
None of these
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The function \[{{x}^{5}}-5{{x}^{4}}+5{{x}^{3}}-10\] has a maximum, when x =
A)
0
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B)
1
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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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For which interval the given function \[f(x)=-2{{x}^{3}}-9{{x}^{2}}-12x+1\] is decreasing
A)
\[(-2,\,\,\infty )\]
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B)
\[(-2,\,\,-1)\]
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C)
\[(-\infty ,\,\,-1)\]
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D)
\[(-\infty ,-2)\,\,\And \,\,(-1,\infty )\]
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E)
None of these
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\[\int{\frac{{{e}^{5\log x}}-{{e}^{4\log x}}}{{{e}^{3\log x}}-{{e}^{2\log x}}}\,\,dx=}\]
A)
\[e{{.3}^{-3x}}+c\]
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B)
\[{{e}^{3}}\log x+c\]
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C)
\[\frac{{{x}^{3}}}{3}+c\]
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D)
All of these
done
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E)
None of these
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\[\int{\frac{cos\sqrt{x}\,dx}{\sqrt{x}}}\]
A)
\[2\cos \sqrt{x}+C\]
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B)
\[2\sin \sqrt{x}+C\]
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C)
\[\sin \sqrt{x}+C\]
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D)
\[\frac{1}{2}\,\,\cos \sqrt{x}+C\]
done
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E)
None of these
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If an antiderivative of f(x) is \[{{e}^{x}}\] and that of g(x) is \[\cos x\]then\[\int{f(x)\cos x\,\,dx+\int{g(x){{e}^{x}}\,\,dx=}}\]
A)
\[f\,\,(x)\,\,g\,\,(x)+C\]
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B)
\[f\,\,(x)+g\,\,(x)+C\]
done
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C)
\[{{e}^{x}}\cos x+C\]
done
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D)
\[{{e}^{x}}\cos x+f(x)\,\,g(x)+C\]
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E)
None of these
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\[\int{\frac{dx}{{{x}^{2}}+4x+13}}\] is equal to
A)
\[\log ({{x}^{2}}+4x+13)+C\]
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B)
\[\frac{1}{3}{{\tan }^{-1}}\left( \frac{x+2}{3} \right)+C\]
done
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C)
\[\log (2x+4)+C\]
done
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D)
\[\frac{2x+4}{{{({{x}^{2}}+4x+13)}^{2}}}+C\]
done
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E)
None of these
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The value of,\[\int_{0}^{{{\sin }^{2}}x}{{{\sin }^{-1}}\sqrt{t\,\,}dt+\int_{0}^{{{\cos }^{2}}x}{{{\cos }^{-1}}\sqrt{t}\,\,dt}}\] is
A)
\[\frac{\pi }{2}\]
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B)
\[1\]
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C)
\[\frac{\pi }{4}\]
done
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D)
\[\frac{\pi }{8}\]
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E)
None of these
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If \[{{I}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\theta \,d\,\theta ,}\] then \[{{I}_{8}}+{{I}_{6}}\] equals
A)
\[\frac{1}{4}\]
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B)
\[\frac{1}{5}\]
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C)
\[\frac{1}{6}\]
done
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D)
\[\frac{1}{7}\]
done
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E)
None of these
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Area included between the two curves \[{{y}^{2}}=4ax\] and \[{{x}^{2}}=4ay,\] is
A)
\[\frac{32}{3}{{a}^{2}}\,\,sq.unit\]
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B)
\[\frac{16}{3}a\,\,sq.unit\]
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C)
\[\frac{32}{3}\,\,sq.unit\]
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D)
\[\frac{16}{3}{{a}^{2}}\,\,sq.unit\]
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E)
None of these
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If \[{{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z=\pi ,\] then \[\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=\]
A)
\[0\]
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B)
\[1\]
done
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C)
\[\frac{1}{xyz}\]
done
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D)
\[xyz\]
done
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E)
None of these
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If \[\angle \,A=90{}^\circ \] in the triangle ABC, then \[{{\tan }^{-1}}\left( \frac{c}{a+b} \right)+{{\tan }^{-1}}\left( \frac{b}{a+c} \right)\]
A)
\[0\]
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B)
\[1\]
done
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C)
\[\frac{\pi }{4}\]
done
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D)
\[\frac{\pi }{8}\]
done
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E)
None of these
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If \[u=\log \,\tan \left( \frac{\pi }{4}+\frac{x}{2} \right),\] is equal to
A)
\[\sec x\]
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B)
\[cosec\,x\]
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C)
\[\tan x\]
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D)
\[\sin x\]
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E)
None of these
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The xy - plane divides the line joining the points \[\left( -1,3,4 \right)\] and \[\left( 2,-5,6 \right)\]
A)
Internally in the ratio 2 : 3
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B)
Internally in the ratio 3 : 2
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C)
Externally in the ratio 2 : 3
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D)
Externally in the ratio 3 : 2
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E)
None of these
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If \[y={{(x+\sqrt{1+{{x}^{2}}})}^{n}},\] then \[(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}=x\frac{dy}{dx}\] is
A)
\[{{n}^{2}}y\]
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B)
\[-{{n}^{2}}y\]
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C)
\[-y\]
done
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D)
\[2{{x}^{2}}y\]
done
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E)
None of these
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The solution of the equation \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{e}^{-2x}}\]
A)
\[\frac{{{e}^{-2x}}}{4}\]
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B)
\[\frac{{{e}^{-2x}}}{4}+cx+d\]
done
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C)
\[\frac{1}{4}{{e}^{-2x}}+c{{x}^{2}}+d\]
done
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D)
\[\frac{1}{4}{{e}^{-2x}}+c+d\]
done
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E)
None of these
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A problem in Mathematics is given to three students A, B, C and their respective probability of solving the problem is\[\frac{1}{2},\]\[\frac{1}{3}\]and\[\frac{1}{4}\]. Probability that the problem is solved, is
A)
\[\frac{3}{4}\]
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B)
\[\frac{1}{2}\]
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C)
\[\frac{2}{3}\]
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D)
\[\frac{1}{3}\]
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E)
None of these
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If the vectors \[\overrightarrow{a},\] \[\overrightarrow{b}\] and \[\overrightarrow{c}\] from the sides BC, CA and AB respectively of a triangle ABC, then
A)
\[\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{b}=0\]
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B)
\[\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{b}\]
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C)
\[\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0\]
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D)
\[\overrightarrow{a}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{0}\]
done
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E)
None of these
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If\[\left| \begin{matrix} a & {{a}^{2}} & 1+{{a}^{3}} \\ b & {{b}^{2}} & 1+{{b}^{3}} \\ c & {{c}^{2}} & 1+{{c}^{3}} \\ \end{matrix} \right|=0\]and vectors\[(1,a,{{a}^{2}}),\]\[(1,b,{{b}^{2}})\] and \[(1,c,{{c}^{2}})\] are non-coplanar, then the product abc equals
A)
\[0\]
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B)
\[-1\]
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C)
\[1\]
done
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D)
\[2\]
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E)
None of these
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If \[A=\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right]\] and \[{{A}^{2}}=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right],\] then
A)
\[\alpha ={{a}^{2}}+{{b}^{2}},\] \[\beta =ab\]
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B)
\[\alpha ={{a}^{2}}+{{b}^{2}},\] \[\beta =2ab\]
done
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C)
\[\alpha ={{a}^{2}}+{{b}^{2}},\] \[\beta ={{a}^{2}}-{{b}^{2}}\]
done
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D)
\[\alpha =2ab,\] \[\beta ={{a}^{2}}+{{b}^{2}}\]
done
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E)
None of these
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The lines \[\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}\] and\[\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-5}{1}\] are coplanar, if
A)
\[k=0\,\,or-1\]
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B)
\[k=1\,\,or-1\]
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C)
\[k=0\,\,or-3\]
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D)
\[k=3\,\,or-3\]
done
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E)
None of these
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The shortest distance from the plane \[12x+4y+3z=327\] to the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+4x-2y-6z-155\]is
A)
26
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B)
39
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C)
13
done
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D)
\[11\frac{4}{13}\]
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E)
None of these
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If the mean deviations about the median of the numbers a, 2a, ........5a is 50, then | a | equals to
A)
3
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B)
4
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C)
5
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D)
2
done
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E)
None of these
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Consider the system of linear equations
\[{{x}_{1}}+2\,{{x}_{2}}+{{x}_{3}}=3\] \[2{{x}_{1}}+3\,{{x}_{2}}+{{x}_{3}}=3\] \[3{{x}_{1}}+5\,{{x}_{2}}+2{{x}_{3}}=1\]
The system has
A)
Infinite number of solutions
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B)
Exactly 3 solutions
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C)
A unique solution
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D)
No solution
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E)
None of these
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A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
A)
19 months
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B)
20 months
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C)
21 months
done
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D)
18 months
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E)
None of these
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Statement I: The number of ways distributing 10 identical balls in 4 distinct boxes such that no box is empty is \[{}^{9}{{C}_{3}}\] . Statement II: The number of ways of choosing any 3 places from 9 different places is \[{}^{9}{{C}_{3}}\] .
A)
Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
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B)
Statement I is true, Statement II is false.
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C)
Statement I is false. Statement II is true.
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D)
Statement I is true. Statement II is true; Statement II is a correct explanation for Statement I.
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E)
None of these
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If \[f:\left( -1,1 \right)\to R\] be a differentiable function with \[f\left( 0 \right)=-1\] and\[f'\left( 0 \right)=1\]. Let\[g(x)={{[f(2f(x)+2)]}^{2}}\]. Then g'(0) is equal to
A)
4
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B)
\[-4\]
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C)
0
done
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D)
\[-2\]
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E)
None of these
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Six books are labelled A, B, C, D, E and F and are placed side by side. Books B, C, E and F have green covers while others have yellow covers. Books A, B and D are new while the rest are old volumes. Books A, B and C are law reports while the rest are medical extracts. Which two books are old medical extracts and have green covers?
A)
B and C
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B)
E and F
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C)
C and E
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D)
C and F
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E)
None of these
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Out of a total of 120 musicians in a club, 5 % can play all the three instruments, guitar, violin and flute. It so happens that the number of musicians who can play any two and only two of the above instruments is 30. The number of musicians who play the guitar alone is 40. What is the total number of those who can play violin alone or flute alone?
A)
45
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B)
44
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C)
38
done
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D)
30
done
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E)
None of these
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