Let R be a relation defined by R = {(a, b): a b, a, b e R}. The relation R is
A)
Reflexive, symmetric and transitive
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B)
Reflexive, transitive but not symmetric
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C)
Symmetric, transitive but not reflexive
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D)
Neither transitive nor reflexive but symmetric
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If \[Se{{c}^{-1}}x=\cos e{{c}^{-1}}y\],then the value of \[{{\cos }^{-1}}\frac{1}{x}+{{\cos }^{-1}}\frac{1}{y}\]
A)
\[\frac{\pi }{4}\]
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B)
\[-\frac{\pi }{2}\]
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C)
\[\pi \]
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D)
\[\frac{\pi }{2}\]
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If the derivative of the
is everywhere continuous, then
A)
\[a\,=2\,\,,\,\,b\,=3\]
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B)
\[a\,=3\,\,,\,\,b\,=2\]
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C)
\[a=-2,b=-3\]
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D)
\[a\,=-3\,\,,\,\,b\,=-2.\]
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Let R f be a function with domain X and range Y. Let \[A,\text{ }B\text{ }\subseteq X\]and \[C,\text{ }D\text{ }\subseteq \text{ Y}\]. Which of the following is not true?
A)
\[f(A\cup B)=f(A)\cup f(B)\]
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B)
\[f(A\cap B)=f(A)\cap f(B)\]
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C)
\[{{f}^{-1}}(C\cup D)={{f}^{-1}}(C)\cup {{f}^{-1}}(D)\]
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D)
\[{{f}^{-1}}(C\,\cap D)={{f}^{-1}}(C)\cap {{f}^{-1}}(D)\]
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If a, b, c are non-zero real numbers, then
vanishes when
A)
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\]
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B)
\[\frac{1}{a}-\frac{1}{b}-\frac{1}{c}=0\]
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C)
\[\frac{1}{b}-\frac{1}{c}-\frac{1}{a}=0\]
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D)
\[\frac{1}{b}-\frac{1}{c}+\frac{1}{a}=0\]
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If \[f(x)=ta{{n}^{-1}}g\left( x \right)\](where \[g\left( x \right)\] is monotonically increasing for \[0<\times <\pi /2\], then f(x) is
A)
Increasing in \[\left( 0,\frac{\pi }{2} \right)\]
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B)
Decreasing in \[\left( 0,\frac{\pi }{2} \right)\]
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C)
Increasing in \[\left( 0,\pi \right)\]
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D)
Decreasing in \[\left( \frac{\pi }{4},\frac{\pi }{2} \right)\]
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If \[{{A}_{1}}\] is the area of the parabola \[{{y}^{2}}=4ax\] lying between vertex and the latus rectum and \[{{A}_{2}}\]is the area between the latus rectum and the double ordinate \[x=2\]a, then \[\frac{{{A}_{1}}}{{{A}_{2}}}=\]
A)
\[2\sqrt{2-1}\]
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B)
\[\frac{1}{7}\left( 2\sqrt{2+1} \right)\]
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C)
\[\frac{1}{7}\left( 2\sqrt{2-1} \right)\]
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D)
None of these
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Let\[\alpha =\left( x+4y \right)\overrightarrow{a}\text{ +}\left( 2x+y+1 \right)\overrightarrow{b}\text{ }\]and \[\text{ }\beta =\left( y-2x+2 \right)\overrightarrow{a}+\left( 2x-3y-1 \right)\overrightarrow{b},\] where \[\overrightarrow{a}\] and \[\overrightarrow{b}\] are non-zero, non-collinear vectors. If\[3\alpha =2\beta ,\], then
A)
\[x=2,y=1~\]
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B)
\[x=2,y=1~\]
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C)
\[x=-1,y=2~\]
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D)
\[x\text{ }=\text{ }2,\text{ }y\text{ }=-1\]
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The set of points where the function f(x) = x |x| is differentiable is
A)
\[\left( -\infty ,\infty \right)\]
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B)
\[R-\{0\}\]
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C)
\[\left( 0,\infty \right)\]
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D)
None of these
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The domain of the function \[f\left( x \right)\frac{{{x}^{2}}}{\sqrt{\left( a-x \right)(x-b)}},\left( b>a \right)\]
A)
\[\left[ a,b \right)\]
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B)
\[\left[ a,b \right]\]
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C)
\[(a,\text{ }b]\]
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D)
\[(a,\text{ }b)\]
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The equation of a normal passing through \[\left( -2,\text{ }3 \right)\]and parallel to the tangent at origin for circle \[{{x}^{2}}+\text{ }{{y}^{2}}\text{ }+\text{ }x\text{ -}y=0\] is
A)
\[x-y-1=0\]
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B)
\[x-y+3=0~~\]
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C)
\[x-y+5=0\]
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D)
\[x-y-5=0\]
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\[\int{\frac{\cot x}{\sqrt[3]{\sin x}}dx=}\]
A)
\[\frac{-3}{\sqrt[3]{\sin x}}+C\]
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B)
\[\frac{-2}{{{\sin }^{3}}x}+C\]
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C)
\[\frac{3}{{{\sin }^{1/3}}x}+C\]
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D)
None of these
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Find the equation of tangent to the curve\[y=({{2}^{x}}-1){{e}^{2\left( 1-x \right)}}\] at the point of its maximum.
A)
\[y=1\]
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B)
\[x=1\]
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C)
\[y=-1\]
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D)
\[x=-1\]
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The probability that Krishna will be alive 10 years hence, is 7/15 and the probability that Hari will be alive after 10 years, is 7/10. The probability that both Krishna and Hari will be alive 10 years hence, is
A)
\[\frac{21}{150}\]
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B)
\[\frac{24}{150}\]
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C)
\[\frac{49}{150}\]
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D)
\[\frac{56}{150}\]
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Equation of a line through (1, 2, -3) and parallel to the line\[\frac{X-2}{1}=\frac{Y+1}{3}=\frac{Z-1}{4}\]
A)
\[\frac{x-1}{1}=\frac{y-2}{3}=\frac{z+3}{4}\]
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B)
\[\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-1}{-3}\]
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C)
\[\frac{x-1}{1}=\frac{y-3}{2}=\frac{z-4}{-3}\]
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D)
None of these
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For all values of A, B, C, and P, Q,R, the value of
A)
\[\frac{10}{3}\]
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B)
\[\frac{5}{2}\]
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C)
\[1\]
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D)
\[0\]
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Find the integral solutions for \[x:{{(1-i)}^{x}}={{2}^{x}}\]
A)
0
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B)
1
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C)
2
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D)
3
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Find the equation of that diameter which bisects the chord \[7x+\text{ }y-20=0\] of the hyperbola \[\frac{{{x}^{2}}}{3}-\frac{{{y}^{2}}}{7}=7\]
A)
\[\frac{1}{3}x-y=0\]
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B)
\[-\frac{1}{3}x=y\]
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C)
\[y+3x=0\]
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D)
None of these
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If \[\left| \overrightarrow{a} \right|\text{=}3,\left| \overrightarrow{b} \right|=1,\left| \overrightarrow{c} \right|=4\] and \[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,\] then find the value of \[\overrightarrow{a\,.\,}\overrightarrow{b}+\overrightarrow{b}\,.\,\overrightarrow{c}+\overrightarrow{c\,}.\,\overrightarrow{a}.\]
A)
13
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B)
42
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C)
0
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D)
-13
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A pair of dice is rolled together till a sum of either 5 or 7 is obtained. Find the probability that 5 comes before 7.
A)
\[\frac{1}{3}\]
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B)
\[\frac{1}{5}\]
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C)
\[\frac{2}{3}\]
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D)
\[\frac{2}{5}\]
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