Consider 5 independent Bernoulli's trials each with probability of success p. If the probability of at least one failure is greater than or equal to \[\frac{31}{32},\]then p lies in the interval
A)
\[\left. \left( \frac{3}{4},\,\,\frac{11}{12} \right. \right]\]
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B)
\[\left[ 0,\,\,\frac{1}{2} \right]\]
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C)
\[\left. \left( \frac{11}{12},\,\,1 \right. \right]\]
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D)
\[\left. \left( \frac{1}{2},\,\,\frac{3}{4} \right. \right]\]
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E)
None of these
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Consider the following statements
P: Suman is brilliant Q: Suman is rich R: Suman is honest.
The negative of the statement. "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as
A)
\[\sim (Q\leftrightarrow (P\wedge \sim R))\]
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B)
\[\sim Q\leftrightarrow \,\sim P\wedge R\]
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C)
\[\sim (P\wedge \sim R)\leftrightarrow Q\]
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D)
\[\sim P\wedge (Q\leftrightarrow \,\sim R)\]
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E)
None of these
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Statement I: The point A (1, 0, 7) is the mirror image of the point B (1, 6, 3) in the line \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\]. Statement II: The line \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\] bisects the line segment joining A (1, 0, 7) and B (1, 6, 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 the trivial solution is the only solution of the system of equations
\[x-ky+z=0\] \[kx+3y-kz=0\] \[3x+y-z=0\]
Then, the set of all values of k is
A)
\[\{2,-3\}\]
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B)
\[R-\{2,-3\}\]
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C)
\[R-\{2\}\]
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D)
\[R-\{-3\}\]
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E)
None of these
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\[f(x)=\left\{ \begin{matrix} \frac{|x+2|}{{{\tan }^{-1}}(x+2)}, & x\ne -2 \\ 2, & x=-2 \\ \end{matrix} \right.,\] then f(x) is
A)
continuous at \[x=-2\]
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B)
not continuous at \[x=-2\]
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C)
differentiable at \[x=-2\]
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D)
continuous but not differentiable at \[x=-2\]
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E)
None of these
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Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots (4, 3). Rahul made a mistake in writing down coefficient of x to get roots (3, 2). The correct roots of equation are
A)
\[-4/-3~\]
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B)
\[6,\,\,1\]
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C)
\[4,\,\,3\]
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D)
\[-6,\,\,-1\]
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E)
None of these
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Let a, b, c be three non-zero vectors which are pairwise non-collinear. If \[a+3b\] is collinear with c and \[b+2c\] is collinear with a, then \[a+3b+6c\] is
A)
\[a+c\]
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B)
\[a\]
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C)
\[c\]
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D)
\[0\]
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E)
None of these
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If \[A\,\,(2,\,\,-3)\] and \[B\,\,(-2,\,\,1)\] are two vertices of a triangle and third vertex moves on the line \[2x+3y=9,\] then the locus of the centroid of the triangle is
A)
\[2x-3y=1\]
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B)
\[x-y=1\]
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C)
\[2x+3y=1\]
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D)
\[2x+3y=3\]
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E)
None of these
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Statement I: For each natural number n \[{{(n+1)}^{7}}-{{n}^{7}}-1\] is divisible by 7. Statement II: For each natural number n \[{{n}^{7}}-n\] is divisible by 7.
A)
Statement I is false, Statement II is true.
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B)
Statement I is true, Statement II is true; Statement II is correct explanation for Statement I.
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C)
Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
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D)
Statement I is true, Statement II is false.
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E)
None of these
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The area bounded by the curves \[y=\cos \,x\] and \[y=\sin \,x\] between the ordinates \[x=0\] and \[x=\frac{3\pi }{2}\] is
A)
\[(4\sqrt{2}-2)\,\,sq\,\,unit\]
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B)
\[(4\sqrt{2}+2)\,\,sq\,\,unit\]
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C)
\[(4\sqrt{2}-1)\,\,sq\,\,unit\]
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D)
\[(4\sqrt{2}+1)\,\,sq\,\,unit\]
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E)
None of these
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Solution of the differential equation \[x\,\,dy=y(\sin x-y)\,\,dx,\]\[0<x<\frac{\pi }{2},\]is
A)
\[\sec x=(\tan x+c)\,\,y\]
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B)
\[y\,\sec \,x=\tan x+c\]
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C)
\[y\,\,\tan \,\,x=\sec x+c\]
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D)
\[\tan \,x=(\sec x+c)\,\,y\]
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E)
None of these
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If the vectors \[\overrightarrow{a}=\hat{i}-\hat{j}+2\hat{k},\] \[\overrightarrow{b}=2\hat{i}+4\hat{j}+\hat{k}\] and \[\overrightarrow{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}\] are mutually orthogonal, then \[(\lambda ,\mu )\] is equal to
A)
\[(-3,\,2)\]
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B)
\[(2,-3)\]
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C)
\[(-2,3)\]
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D)
\[(3,-2)\]
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E)
None of these
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A line AB in three-dimensional space makes angles \[45{}^\circ \]and \[120{}^\circ \] with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle \[\theta \] with the positive z-axis, then \[\theta \] equals
A)
\[30{}^\circ \]
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B)
\[45{}^\circ \]
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C)
\[60{}^\circ \]
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D)
\[75{}^\circ \]
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E)
None of these
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For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is
A)
\[\frac{5}{2}\]
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B)
\[\frac{11}{2}\]
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C)
\[6\]
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D)
\[\frac{13}{2}\]
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E)
None of these
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For real x, let \[f(x)={{x}^{3}}+5x+1,\] then
A)
f is one-one but not onto-R
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B)
f is onto R but not one-one
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C)
f is one-one and onto R
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D)
f is neither one-one nor onto R
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E)
None of these
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The projections of a vector on the three coordinate axes are 6, -3, 2 respectively. The direction cosines of the vector are
A)
\[6,\,\,-3,\,\,2\]
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B)
\[\frac{6}{5},\]\[-\frac{3}{5},\]\[\frac{2}{5}\]
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C)
\[\frac{6}{7},\]\[-\frac{3}{7},\]\[\frac{2}{7}\]
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D)
\[-\frac{6}{7},\]\[-\frac{3}{7},\]\[\frac{2}{7}\]
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E)
None of these
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The shortest distance between the line \[y-x=1\]and the curve \[x={{y}^{2}}\] is
A)
\[\frac{3\sqrt{2}}{8}\]
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B)
\[\frac{2\sqrt{3}}{8}\]
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C)
\[\frac{3\sqrt{2}}{5}\]
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D)
\[\frac{\sqrt{3}}{4}\]
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E)
None of these
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It is given that the events A and B are such that\[P\,\,(A)=\frac{1}{4},\] \[P\,\,\left( \frac{A}{B} \right)=\frac{1}{2}\] and \[P\,\,\left( \frac{B}{A} \right)=\frac{2}{3}.\] then, \[P(B)\] is
A)
\[\frac{1}{6}\]
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B)
\[\frac{1}{3}\]
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C)
\[\frac{2}{3}\]
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D)
\[\frac{1}{2}\]
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E)
None of these
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How many real solutions does the equation \[{{x}^{7}}+14{{x}^{5}}+16{{x}^{3}}+30x-560=0\] have?
A)
7
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B)
1
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C)
3
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D)
5
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E)
None of these
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The value of \[\cot \left( \text{cose}{{\text{c}}^{-1}}\frac{5}{3}+{{\tan }^{-1}}\frac{2}{3} \right)\] is
A)
\[\frac{6}{17}\]
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B)
\[\frac{3}{17}\]
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C)
\[\frac{4}{17}\]
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D)
\[\frac{5}{17}\]
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E)
None of these
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If \[I=\int\limits_{0}^{1}{\frac{\sin x}{\sqrt{x}}}\,\,dx\]and \[J=\int\limits_{0}^{1}{\frac{\cos x}{\sqrt{x}}}\,\,dx,\] then which one of the following is true?
A)
\[I>\frac{2}{3}\,\,and\,\,J>2\]
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B)
\[I<\frac{2}{3}\,\,and\,\,J<2\]
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C)
\[I<\frac{2}{3}\,\,and\,\,J>2\]
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D)
\[I>\frac{2}{3}\,\,and\,\,J<2\]
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E)
None of these
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If \[{{\sin }^{-1}}\frac{x}{5}+\cos e{{c}^{-1}}\left( \frac{5}{4} \right)=\frac{\pi }{2},\] then a value of x is
A)
1
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B)
3
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C)
4
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D)
5
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E)
None of these
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If p and q are positive real numbers such that \[{{p}^{2}}+{{q}^{2}}=1,\] then the maximum value of (p + q) is
A)
\[2\]
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B)
\[\frac{1}{2}\]
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C)
\[\frac{1}{\sqrt{2}}\]
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D)
\[\sqrt{2}\]
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E)
None of these
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The function \[f(x)=ta{{n}^{-1}}(\sin \,x+\cos \,x)\] is an increasing function in
A)
\[(\pi /4,\pi /2)\]
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B)
\[(-\pi /2,\pi /4)\]
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C)
\[(0,\,\,\pi /2)\]
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D)
\[(-\pi /2,\pi /2)\]
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E)
None of these
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Let \[A=\left[ \begin{matrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \\ \end{matrix} \right]\]. If \[|{{A}^{2}}|\,=25,\] then \[|\alpha |\] equals
A)
\[{{5}^{2}}\]
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B)
\[1\]
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C)
\[\frac{1}{5}\]
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D)
\[5\]
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E)
None of these
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\[\int{\frac{dx}{\cos \,x+\sqrt{3}\sin \,x}}\] equals
A)
\[\frac{1}{2}\,\,\log \,\,\tan \,\,\left( \frac{x}{2}+\frac{\pi }{12} \right)+c\]
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B)
\[\frac{1}{2}\,\,\log \,\,\tan \,\,\left( \frac{x}{2}-\frac{\pi }{12} \right)+c\]
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C)
\[\log \,\,\tan \,\,\left( \frac{x}{2}+\frac{\pi }{12} \right)+c\]
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D)
\[\log \,\,\tan \,\,\left( \frac{x}{2}-\frac{\pi }{12} \right)+c\]
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E)
None of these
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The number of values of x in the interval \[\left[ 0,3\pi \right]\]satisfying the equation \[2{{\sin }^{2}}x+5\sin x-3=0\] is
A)
6
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B)
1
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C)
2
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D)
4
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E)
None of these
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The value of the integral \[\int_{3}^{6}{\frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}}\,\,dx}\] is
A)
\[\frac{3}{2}\]
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B)
\[2\]
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C)
\[1\]
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D)
\[\frac{1}{2}\]
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E)
None of these
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The function \[f(x)=\frac{x}{2}+\frac{2}{x}\] has a local minimum at
A)
\[x=-2\]
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B)
\[x=0\]
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C)
\[x=1\]
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D)
\[x=2\]
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E)
None of these
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If \[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}},\] then \[\frac{dy}{dx}\] is
A)
\[\frac{x+y}{xy}\]
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B)
\[xy\]
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C)
\[\frac{x}{y}\]
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D)
\[\frac{y}{x}\]
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E)
None of these
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The image of the point (-1, 3, 4) in the plane \[x-2y=0\] is
A)
\[(15,\,\,11,\,\,4)\]
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B)
\[\left( -\frac{17}{3},\,\,-\frac{193}{3},\,\,1 \right)\]
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C)
\[(8,\,\,4,\,\,4)\]
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D)
\[\left( -\frac{17}{3},\,\,-\frac{19}{3},\,\,4 \right)\]
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E)
None of these
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If \[A=\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right],\] then which one of the following holds for all \[n\ge 1,\] by the principle of mathematical induction?
A)
\[{{A}^{n}}={{2}^{n-1}}A+(n-1)|\]
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B)
\[{{A}^{n}}=nA+(n-1)|\]
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C)
\[{{A}^{n}}={{2}^{n-1}}A-(n-1)|\]
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D)
\[{{A}^{n}}=nA-(n-1)|\]
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E)
None of these
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A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
A)
Interval Function \[(-\infty ,\,\,-4]\] \[{{x}^{3}}+6{{x}^{2}}+6\]
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B)
Interval Function \[\left( \left. -\infty ,\,\,\frac{1}{3} \right] \right.\] \[3{{x}^{2}}-2x+1\]
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C)
Interval Function \[[2,\,\,\infty )\] \[2{{x}^{3}}-2{{x}^{2}}-12x+6\]
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D)
Interval Function \[(-\infty ,\,\,\infty )\] \[{{x}^{3}}-3{{x}^{2}}+3x+3\]
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E)
None of these
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A real valued function f(x) satisfies the functional equation \[f\,\,\left( x-y \right)=f\,(x)\,\,f(y)-f\,\,(a-x)\,\,f\,\,(a+y)\]where a is a given constant and f (0) = 1. Then \[f(2a-x)\] is equal to
A)
\[f(-x)\]
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B)
\[f(a)+f(a-x)\]
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C)
\[f(x)\]
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D)
\[-f(x)\]
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E)
None of these
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The range of the function \[f(x)={}^{7-x}{{P}_{x-3}}\] is
A)
{1, 2, 3}
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B)
{1, 2, 3, 4, 5, 6}
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C)
{1, 2, 3, 4}
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D)
{1, 2, 3, 4, 5}
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E)
None of these
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A random variable X has the probability distribution
X 1 2 3 4 5 6 7 8 P(X) 0.15 0.23 0.12 0.10 0.20 0.08 0.05 0.05
For the events E = {X is a prime number} and \[F=\{X<4\},\] the probability \[P(E\bigcup F)\] is
A)
0.87
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B)
0.77
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C)
0.35
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D)
0.50
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E)
None of these
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If \[\int{\frac{\sin x}{\sin (x-\alpha )}}\,\,dx=Ax+B\,\,\log \,\,\sin (x-\alpha )+c,\] then value of (A,B) is
A)
\[(\sin \alpha ,\,\,\cos \alpha )\]
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B)
\[(cos\alpha ,\,\,\sin \alpha )\]
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C)
\[(-\sin \alpha ,\,\,\cos \alpha )\]
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D)
\[(-cos\alpha ,\,\,\sin \alpha )\]
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E)
None of these
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On the set N of all natural numbers define the relation R by aRb if and only if the G.C.D. of a and b is 2, then R is
A)
reflexive, but not symmetric
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B)
symmetric only
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C)
reflexive and transitive
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D)
reflexive, symmetric and transitive
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E)
None of these
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Let A, B, C be three mutually independent events. Consider the two statements \[{{S}_{1}}\] and \[{{S}_{2}}\].
\[{{S}_{1}}:A\] and \[B\bigcup C\] are independent \[{{S}_{2}}:A\] and \[B\bigcap C\] are independent
Then,
A)
Both \[{{S}_{1}}\] and \[{{S}_{2}}\] are true
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B)
Only \[{{S}_{1}}\] is true
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C)
Only \[{{S}_{2}}\] is true
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D)
Neither \[{{S}_{1}}\] nor \[{{S}_{2}}\] is true
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E)
None of these
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For a moderately skewed distribution, quartile deviation and the standard deviation are related by
A)
\[S.D=\frac{2}{3}Q.D\]
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B)
\[S.D=\frac{3}{2}Q.D\]
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C)
\[S.D=\frac{3}{4}Q.D\]
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D)
\[S.D=\frac{4}{3}Q.D\]
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E)
None of these
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