R is a relation from \[\{11,\,\,12,\,\,13\}\,\,to\,\,\{8\,\,,10\,\,,12\}\] defined by \[y=\text{ }x-3\]. The relation \[{{R}^{-1}}\] is
A)
\[\{(11,8,(13,10)\}\]
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B)
\[\{(8,\,\,11\,\,(10\,\,,13)\}\]
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C)
\[\{(8,\,\,11\,\,(9,12),(10\,\,,13)\}\]
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D)
None of these
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The value of f(0), so that function \[f(x)\,\,\,=\frac{\sqrt{1+x}-{{(1+x)}^{1/3}}}{x}\]becomes continuous, is equal to
A)
\[1/6\]
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B)
\[1/4\]
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C)
2
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D)
\[1/3\]
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If \[f(x)=4{{x}^{8}}\], then
A)
\[f'\left( \frac{1}{2} \right)=f'\left( -\frac{1}{2} \right)\]
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B)
\[f'\left( \frac{1}{2} \right)=f\left( -\frac{1}{2} \right)\]
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C)
\[f\left( -\frac{1}{2} \right)=f\left( \frac{1}{2} \right)\]
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D)
\[f\left( \frac{1}{2} \right)=f'\left( -\frac{1}{2} \right)\]
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The number of real solutions of \[{{\tan }^{-1}}\sqrt{x(x-1)}+{{\sin }^{-1}}\sqrt{{{x}^{2}}+x+1}=\frac{\pi }{2}\]is
A)
Zero
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B)
One
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C)
Two
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D)
Infinite
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For a function\[F\,\,,F(0)=2\,\,,F(1)=3,F(x+2)=2\] \[-F(x+1)forx\ge 0\,\,\,,then\,\,F(5)\], is equal to
A)
\[-7\]
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B)
\[-3\]
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C)
\[17\]
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D)
\[13\]
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If A is invertible, then which of the following is not true?
A)
\[{{A}^{-2}}=|A{{|}^{-1}}\]
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B)
\[{{\left( {{A}^{-2}} \right)}^{-1}}={{\left( {{A}^{-1}} \right)}^{2}}\]
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C)
\[{{\left( A' \right)}^{-1}}=\left( {{A}^{-1}} \right)'\]
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D)
None of these
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The curve \[x+y{{e}^{xy}}\]has a tangent parallel to y-axis at the point
A)
\[\left( 0\,\,,1 \right)\]
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B)
\[\left( 1\,\,,0 \right)\]
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C)
\[\left( 1\,\,,1 \right)\]
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D)
None of these
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Six identical coins are arranged in a row. The total number of ways in which the number of heads is equal to the number of tails, is
A)
9
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B)
20
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C)
40
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D)
120
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If a, \[\alpha \,\,,\beta \,\,,\gamma \,\,,\delta \] are the roots of\[{{x}^{4}}+{{x}^{2}}+1=0,\], then the equation whose roots are\[{{\alpha }^{2}}\,\,,{{\beta }^{2}}\,\,,{{\gamma }^{2}}\,\,,{{\delta }^{2}}\] is
A)
\[{{({{x}^{2}}-x+1)}^{2}}=0\]
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B)
\[{{({{x}^{2}}+x+1)}^{2}}=0\]
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C)
\[{{x}^{4}}-{{x}^{2}}+1=0\]
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D)
\[{{x}^{2}}-x-1=0\]
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Equation of the plane parallel to the plane is \[3x-2y+4z=11\] is
A)
\[2x+3y-4z\text{ }=3~~\]
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B)
\[6x-4y+8z\text{ =}7\]
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C)
\[4x+4y-3z\text{ =}1\]
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D)
\[3x\text{ +}2y\text{ +}4z\text{ =}9\]
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The point (3, 2) is reflected in the y-axis and then moved a distance of 5 units towards the negative side of y-axis. The coordinates of the point thus obtained are
A)
(-3,-3)
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B)
(3,3)
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C)
(-3,3)
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D)
(3,-3)
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Let \[f:R\to R\] be any function. Define \[g:R\to R\,\,by\]\[g(x)=|f(x)|\forall \,x.\]. Then g is
A)
Onto if f is onto
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B)
One-one if f is one-one
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C)
Continuous if f is continuous
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D)
Differentiable if f is differentiable
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Let \[f(x)=cos\,\,\times \,\,sin2x\], then
A)
\[\min \,\,f\,\,(x)=-\frac{1}{3\sqrt{3}}\,\,for\,\,x\,\,\in [-\pi ,\pi )\]
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B)
\[\min \,\,f(x)>-\frac{9}{7}\,\,or\,\,-\frac{7}{9}\,\,for\,\,x\,\,\in [-\pi ,\pi ]\]
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C)
\[\min \,\,f\,\,(x)>-\frac{1}{9}\,\,for\,\,x\,\,\in [-\pi ,\pi ]\]
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D)
\[\min \,\,f\,\,(x)>-\frac{2}{9}\,\,for\,\,x\,\,\in [-\pi ,\pi ]\]
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\[If\int\limits_{0}^{1}{\left( 1+{{\sin }^{4}}x \right)}\left( a{{x}^{2}}+bx+c \right)dx=\int\limits_{0}^{2}{\left( 1+{{\sin }^{4}}x \right)}\] \[\left( a{{x}^{2}}+bx+c \right)dx\],then the quadratic equation
A)
At least one root in (1, 2)
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B)
No root in (1, 2)
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C)
Two equal roots in (1, 2)
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D)
Both roots imaginary
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The area bounded by \[y=x{{e}^{|x|}}\]and the lines \[|x|=1,y=0\] is
A)
4 sq. units
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B)
6 sq. units
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C)
1 sq. unit
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D)
2 sq. units
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General solution of \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{e}^{-2x}}\]
A)
\[y=\frac{1}{4}{{e}^{-2x}}+c\]
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B)
\[y={{e}^{-2x}}+cx+d\]
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C)
\[y=\frac{1}{4}{{e}^{-2x}}+cx+d\]
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D)
\[y={{e}^{-2x}}+c{{x}^{2}}+d\]
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There is a three-volume dictionary among 43 books arranged on a shelf in random order. Three books are drawn at random from the shelf. The probability that all the three volumes of the dictionary will be drawn, is
A)
\[3/12341\]
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B)
\[2/12341\]
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C)
\[1/12341\]
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D)
None of these
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If \[\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}\], and \[\overrightarrow{c}=\hat{i}+\hat{j}-2\hat{k}\]then a vector in the plane of and whose projection on a is of \[\overrightarrow{b}\]and \[\overrightarrow{c}\]whose projection on \[\overrightarrow{a}\] is of magnitude is\[\sqrt{\frac{2}{3}}\]is
A)
\[2\hat{i}\,+\,3\hat{j}-3\hat{k}\]
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B)
\[2\hat{i}\,+\,3\hat{j}+3\hat{k}\]
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C)
\[-2\hat{i}\,-\hat{j}+5\hat{k}\]
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D)
\[2\hat{i}\,-\hat{j}+5\hat{k}\]
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Consider the following statements. Assertion : The function f(x) in the figure is differentiable at x = a.
Reason (R): The function f(x) is continuous at \[x=a\]. Then, which of the following is correct?
A)
Both A and R are true and R is the correct explanation of A.
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B)
Both A and R are true but R is not the correct explanation of A.
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C)
A is true and R is false.
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D)
A is false but R is true.
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If \[\left( \alpha \,\,,\beta \right)\] is a point on the circle whose centre is on the x-axis and which touches the line \[x+y\text{ =}0\] at (2, -2), then the greatest value of \[\alpha \] is
A)
\[4-\sqrt{2}\]
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B)
6
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C)
\[4+\sqrt{2}\]
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D)
None of these
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