If R and R' are symmetric relation (Not disjoint) on a set A, then the relation \[\mathbf{R}\cap \mathbf{R}'\]is
A)
reflexive
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B)
symmetric
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C)
transitive
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D)
None of these
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If A and Bare any two sets then \[\mathbf{A}\cup \left( \mathbf{A}\cap \mathbf{B} \right)\]is equal to
A)
\[{{B}^{c}}\]
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B)
\[{{A}^{c}}\]
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C)
B
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D)
A
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A boy contains 10 apples and of which 4 be rotten. Two apples are taken out together. If one of them is found to be good, the probability that other is also good is
A)
\[\frac{8}{15}\]
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B)
\[\frac{5}{18}\]
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C)
\[\frac{2}{3}\]
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D)
\[\frac{1}{3}\]
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\[\frac{2}{1}+\frac{4}{3}+\frac{6}{5}+.......\infty \]is equal to.
A)
\[e+1\]
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B)
\[e-1\]
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C)
\[{{e}^{-1}}\]
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D)
e
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There are 4 candidates for the post of a lecturer in physics and one is to be selected by votes of 5 men. The number of ways in which the votes can be given is
A)
1024
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B)
1050
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C)
1016
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D)
1025
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If w is a complex cube root of unity, then
A)
\[{{w}^{6}}=w\]
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B)
\[{{w}^{14}}={{w}^{2}}\]
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C)
\[{{w}^{4}}=1\]
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D)
None of these
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The product of n positive numbers is unity, then their sum is:-
A)
a positive integer
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B)
divisible by n
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C)
equal to \[n+\frac{1}{n}\]
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D)
never less than n
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In a triangle ABC, If \[\mathbf{cotA},\mathbf{cotB},\mathbf{cotC}\]are in A.P. than\[\mathbf{a},\mathbf{b},\mathbf{c}\]are in-
A)
G.P.
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B)
A. P.
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C)
H.P.
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D)
None of these
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If tan\[\alpha =\frac{1}{3}\]and\[\mathbf{tan}\frac{\beta }{2}=\frac{1}{2}\], then tan \[\left( \alpha +\beta \right)\]=..........
A)
2
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B)
\[-1\]
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C)
3
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D)
1
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\[\underset{h\to 0}{\mathop{Lt}}\,\frac{\mathbf{sin}\sqrt{\mathbf{x}+\mathbf{h}}-\mathbf{sin}\sqrt{\mathbf{x}}}{h}\] is equal to
A)
\[\frac{\cos \sqrt{x}}{2\sqrt{x}}\]
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B)
\[\sin \sqrt{x}\]
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C)
\[cos\sqrt{x}\]
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D)
\[\frac{1}{2}sin\sqrt{x}\]
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if \[\mathbf{f}\left( \mathbf{x} \right)=lo{{g}_{x}}(l\mathbf{og}x)\]then \[f'\left( x \right)\]at\[x=e\]is equal to
A)
e
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B)
\[\frac{1}{e}\]
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C)
\[{{e}^{2}}\]
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D)
\[\frac{1}{{{e}^{2}}}\]
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Three lines \[\mathbf{px}+\mathbf{qy}+\mathbf{r}=\mathbf{0};\mathbf{qx}+\mathbf{ry}+\mathbf{p}=\mathbf{0}\]and \[\mathbf{rx}+\mathbf{py}+\mathbf{q}=\mathbf{0}\]are concurrent; If
A)
\[p+q+r=pqr\]
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B)
\[{{p}^{2}}+{{q}^{2}}+{{r}^{2}}=pr+rq\]
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C)
\[{{p}^{3}}+{{q}^{3}}+{{r}^{3}}=3pqr\]
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D)
None of these
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If \[\alpha +\beta =\frac{\pi }{2}\]and \[\beta +\gamma =\alpha \]then tan a equal to.
A)
\[2tan\text{ }\beta tan\,\gamma \]
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B)
\[tan\text{ }\beta +2\text{ }tan\text{ }\gamma \]
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C)
\[tan\text{ }\beta -2\text{ }tan\text{ }\gamma \]
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D)
\[2\left( tan\text{ }\beta +tan\text{ }\gamma \text{ } \right)\]
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The equation of the directrix of the parabola \[{{\mathbf{y}}^{\mathbf{2}}}+\mathbf{4y}+\mathbf{4x}+\mathbf{2}=\mathbf{0}\]is
A)
\[2x=3\]
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B)
\[~x=3\]
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C)
\[3x=2\]
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D)
\[x=-3\]
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Let a, b be the roots of the equations \[(x-a)(x-b)+c,c\ne 0,\] Then the roots of the equations - \[\left( \mathbf{x}-\mathbf{a} \right)\left( \mathbf{x}-\mathbf{b} \right)+\mathbf{c}=\mathbf{0}\]are:-
A)
\[\left( a,c \right)\]
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B)
\[\left( b,c \right)\]
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C)
\[\left( a+c,b+c \right)\]
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D)
\[a,b\]
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A function \[\mathbf{f}:\mathbf{R}\to \mathbf{R},\mathbf{f}\left( \mathbf{x} \right)={{\mathbf{x}}^{\mathbf{2}}}+\mathbf{x}\], is f
A)
one-one onto
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B)
one-one into
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C)
Many-one into
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D)
Many-one onto
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If\[\alpha \And \beta \]be the roots of the quadratic equation \[{{x}^{2}}+2x+3=0\]then the value of \[\frac{{{\alpha }^{2}}}{\beta }+\frac{{{\beta }^{2}}}{\alpha }\]is
A)
\[\frac{10}{3}\]
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B)
\[\frac{3}{10}\]
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C)
\[\frac{2}{3}\]
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D)
None of these
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If \[\mathbf{y}=\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}\left( \frac{2x}{1+{{x}^{2}}} \right)\], then \[\frac{dy}{dx}\]is equal to
A)
\[\frac{2}{1+{{x}^{2}}}\], when \[-1<x<1\]
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B)
\[\frac{2}{1+{{x}^{2}}}\] when \[x<-1\,or\,x>1\]
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C)
\[\frac{-2}{1+{{x}^{2}}}\]when\[-1<x<1\]
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D)
None of these
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A and B are two fixed points. The locus of a point p such that \[\angle \mathbf{APB}\] is a right angle, is
A)
\[2{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]
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B)
\[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]
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C)
\[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]
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D)
\[2{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]
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If \[\alpha +\beta =\mathbf{9}{{\mathbf{0}}^{{}^\circ }}\], then the maximum value of sin \[\alpha \].sin \[\beta \]. is
A)
1
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B)
\[\frac{1}{2}\]
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C)
\[\frac{3}{2}\]
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D)
\[\frac{-1}{2}\]
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