If \[A={{\sin }^{2}}+{{\cos }^{4}}x\], then for all real x
A)
\[\frac{13}{16}\le A\le 1\]
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B)
\[1\le A\le 2\]
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C)
\[\frac{3}{4}\le A\le \frac{13}{16}\]
done
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D)
\[\frac{3}{4}\le A\le 1\]
done
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E)
None of these
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The coefficient of \[{{x}^{7}}\] in the expansion of \[{{(1-x-{{x}^{2}}+{{x}^{3}})}^{6}}\] is
A)
\[\,132\]
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B)
\[\,144\]
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C)
132
done
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D)
144
done
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E)
None of these
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\[\underset{x\to 2}{\mathop{\lim }}\,\,\,\left( \frac{\sqrt{1-\{\cos 2\,(x-2)\}}}{x-2} \right)\]
A)
equals \[\sqrt{2}\]
done
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B)
equals \[-\,\sqrt{2}\]
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C)
equals \[\frac{1}{\sqrt{2}}\]
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D)
does not exist
done
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E)
None of these
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If C and D are two events such that \[C\subset D\] and \[P(D)\ne 0,\] then the correct statement among the following is
A)
\[P(C|D)\ge P(C)\]
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B)
\[P(C|D)<P(C)\]
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C)
\[P(C|D)=\frac{P(D)}{P(C)}\]
done
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D)
\[P(C|D)=P(C)\]
done
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E)
None of these
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If \[\omega \,\,(\ne 1)\] is a cube root of unity and \[{{(1+\omega )}^{7}}=A+B\omega ,\] then (A, B) equals to
A)
(1, 1)
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B)
(1, 0)
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C)
\[\left( -\,1,1 \right)\]
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D)
(0, 1)
done
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E)
None of these
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Let A, B, C be pairwise independent events with \[P\,\,\left( C \right)>0\] and\[P(A\cap B\cap C)=0\]. Then,\[P({{A}^{c}}\cap {{B}^{c}}|C)\] is equal to
A)
\[P({{A}^{c}})-P(B)\]
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B)
\[P(A)-P({{B}^{c}})\]
done
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C)
\[P({{A}^{c}})+P({{B}^{c}})\]
done
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D)
\[P({{A}^{c}})-P({{B}^{c}})\]
done
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E)
None of these
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The equation of the hyperbola whose foci are \[\left( -\,2,0 \right)\] and (2, 0) and eccentricity is 2 is given by
A)
\[-3{{x}^{2}}+{{y}^{2}}=3\]
done
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B)
\[{{x}^{2}}-3{{y}^{2}}=3\]
done
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C)
\[3{{x}^{2}}-{{y}^{2}}=3\]
done
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D)
\[-{{x}^{2}}+3{{y}^{2}}=3\]
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E)
None of these
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The lines \[x+y=\left| a \right|\] and \[ax-y=1\] intersect each other in the first quadrant. Then, the set of all possible values of a is the interval
A)
\[\left( -\,1,\text{1} \right)\]
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B)
\[(0,\infty )\]
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C)
\[(1,\infty )\]
done
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D)
\[(-1,\infty )\]
done
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E)
None of these
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The distance of the point \[\left( 1,-\,5,9 \right)\] from the plane \[x-y+z=5\] measured along a straight line\[~x=y=z\] is
A)
\[3\sqrt{5}\]
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B)
\[10\sqrt{3}\]
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C)
\[5\sqrt{3}\]
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D)
\[3\sqrt{10}\]
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E)
None of these
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The number of complex numbers z such that \[\left| z-1 \right|=\left| z+1 \right|=\left| z-i \right|\] equals
A)
0
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B)
1
done
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C)
2
done
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D)
\[\infty \]
done
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E)
None of these
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A person is to count 4500 currency notes. Let \[{{a}_{n}}\]denotes the number of notes he counts in the nth minute. If \[{{a}_{1}}={{a}_{2}}=.....={{a}_{10}}=150\] and \[{{a}_{10}},\] \[{{a}_{11}}.....\] are in AP with common difference \[-\,2,\]then the time taken by him to count all notes, is
A)
24 min
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B)
34 min
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C)
125 min
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D)
135 min
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E)
None of these
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The equation of the tangent to the curve \[y=x+\frac{4}{{{x}^{2}}},\] that is parallel to the x-axis, is
A)
y = 0
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B)
y = l
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C)
y = 2
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D)
y = 3
done
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E)
None of these
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Let \[\cos \,(\alpha +\beta )=\frac{4}{5}\] and let \[sin\,(\alpha -\beta )=\frac{5}{13},\] where \[0\le \alpha ,\beta \le \frac{\pi }{4}\].Then tan \[2\alpha \] is equal to
A)
\[\frac{25}{16}\]
done
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B)
\[\frac{56}{33}\]
done
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C)
\[\frac{19}{12}\]
done
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D)
\[\frac{20}{7}\]
done
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E)
None of these
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For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is
A)
There is a regular polygon with \[\frac{r}{R}=\frac{1}{2}\]
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B)
There is a regular polygon with \[\frac{r}{R}=\frac{1}{\sqrt{2}}\]
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C)
There is a regular polygon with \[\frac{r}{R}=\frac{2}{3}\]
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D)
There is a regular polygon with \[\frac{r}{R}=\frac{\sqrt{3}}{2}\]
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E)
None of these
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View Answer play_arrow
If A, B and C are three sets such that \[A\cap B=A\cap C\] and \[A\cup B=A\cup C,\] then
A)
A = C
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B)
B = C
done
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C)
\[A\cap B=\phi \]
done
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D)
A = B
done
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E)
None of these
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If \[\left| z-\frac{4}{z} \right|=2,\] then the maximum value of |z| is equal to
A)
\[\sqrt{3}+1\]
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B)
\[\sqrt{5}+1\]
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C)
2
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D)
\[2+\sqrt{2}\]
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E)
None of these
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View Answer play_arrow
The remainder left out when \[{{8}^{2n}}-{{(62)}^{2n+1}}\]is divided by 9 is
A)
0
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B)
2
done
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C)
7
done
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D)
8
done
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E)
None of these
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A focus of an ellipse is at the origin. The directrix is the line \[x=4\] and the eccentricity is \[\frac{1}{2}\]. Then, the length of the semi-major axis is
A)
\[\frac{8}{3}\]
done
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B)
\[\frac{2}{3}\]
done
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C)
\[\frac{4}{3}\]
done
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D)
\[\frac{5}{3}\]
done
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E)
None of these
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View Answer play_arrow
The conjugate of a complex number is \[\frac{1}{i+1}\]Then, that complex number is
A)
\[-\frac{1}{i-1}\]
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B)
\[\frac{1}{i+1}\]
done
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C)
\[-\frac{1}{i+1}\]
done
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D)
\[\frac{1}{i-1}\]
done
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E)
None of these
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View Answer play_arrow
How many numbers can be formed from the digits 1, 2, 3, 4 when the repetition is not allowed?
A)
\[{}^{4}{{P}_{4}}\]
done
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B)
\[{}^{4}{{P}_{3}}\]
done
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C)
\[{}^{4}{{P}_{1}}+{}^{4}{{P}_{2}}+{}^{4}{{P}_{3}}\]
done
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D)
\[{}^{4}{{P}_{1}}+{}^{4}{{P}_{2}}+{}^{4}{{P}_{3}}+{}^{4}{{P}_{4}}\]
done
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E)
None of these
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View Answer play_arrow
Match the column in the table given below:
S. No. Sets Roster Form 1. {\[x:{{x}^{2}}-3=0\] and x is a rational number} a. {4} 2. {\[x:x\] is an even prime number} b. \[\left\{ -\,5,5 \right\}\] 3. {\[x:3<x<5,\] x is a natural number} c. {*} 4. {\[x:{{x}^{2}}=25,\] and x is an odd integer} d. {2}
A)
1 - c, 2 - d, 3 - a, 4 - b
done
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B)
1 - d, 2 - c, 3 - a, 4 - b
done
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C)
1 - c, 2 - a, 3 - d, 4 - b
done
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D)
1 - b, 2 - d, 3 - a, 4 - c
done
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E)
None of these
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View Answer play_arrow
Angle between the lines \[2x-y-15=0\] and \[~3x+y+4=0\] is
A)
\[60{}^\circ \]
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B)
\[180{}^\circ \]
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C)
\[90{}^\circ \]
done
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D)
\[45{}^\circ \]
done
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E)
None of these
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View Answer play_arrow
If the line \[x+2by+7=0\] is a diameter of the circle \[{{x}^{2}}+{{y}^{2}}-6x+2y=0,\] then b =
A)
\[-\,1\]
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B)
3
done
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C)
5
done
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D)
\[-\,5\]
done
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E)
None of these
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View Answer play_arrow
If a, b, c are in G.P, then
A)
\[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are in G.P
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B)
\[{{a}^{2}}(b+c),{{c}^{2}}(a+b),{{b}^{2}}(a+c)\] are in G.P
done
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C)
\[\frac{a}{a+b},\frac{b}{c+a},\frac{c}{a+b}\] are in G.P
done
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D)
All of these
done
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E)
None of these
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View Answer play_arrow
The distance between the lines \[3x-2y=1\] and \[6x+9=4y\] is
A)
\[\frac{1}{\sqrt{52}}\]
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B)
\[\frac{11}{\sqrt{52}}\]
done
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C)
\[\frac{4}{\sqrt{13}}\]
done
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D)
\[\frac{6}{\sqrt{13}}\]
done
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E)
None of these
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View Answer play_arrow
If the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] represents two lines \[y={{m}_{1}}x\] and \[y={{m}_{2}}x,\] then
A)
\[{{m}_{1}}+{{m}_{2}}=\,\frac{2h}{b}\,\,and\,\,{{m}_{1}}{{m}_{2}}=-\frac{a}{b}\]
done
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B)
\[{{m}_{1}}+{{m}_{2}}=\frac{2h}{b}\,\,and\,\,{{m}_{1}}{{m}_{2}}=\frac{a}{b}\]
done
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C)
\[{{m}_{1}}+{{m}_{2}}=\frac{2h}{b}\,\,and\,\,{{m}_{1}}{{m}_{2}}=-ab\]
done
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D)
\[{{m}_{1}}+{{m}_{2}}=\frac{-2h}{b}\,\,and\,\,{{m}_{1}}{{m}_{2}}=\frac{a}{b}\]
done
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E)
None of these
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View Answer play_arrow
How many words can be formed using the letter A thrice, the letter B twice and the letter C thrice?
A)
500
done
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B)
560
done
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C)
580
done
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D)
520
done
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E)
None of these
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View Answer play_arrow
The triangle formed by the points (0, 7, 10), \[\left( -\,1,6,6 \right),\] \[\left( -\,4,\text{9},6 \right)\] is
A)
Equilateral
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B)
Isosceles
done
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C)
Right angled
done
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D)
Right angled isosceles
done
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E)
None of these
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View Answer play_arrow
Argument and modulus of \[\frac{1+i}{1-i}\] are respectively
A)
\[\frac{-\pi }{2}\,and\,\,1\]
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B)
\[\frac{\pi }{2}\,\,and\,\,\sqrt{2}\]
done
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C)
\[\frac{\pi }{2}\,\,and\,\,1\]
done
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D)
\[0\,\,and\,\,\sqrt{2}\]
done
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E)
None of these
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View Answer play_arrow
X and Y are two sets such that \[X\cup Y\]has 18 elements, X has 8 elements and Y has 15 elements; how many elements does \[X\cap Y\]have?
A)
23
done
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B)
5
done
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C)
15
done
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D)
18
done
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E)
None of these
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View Answer play_arrow
If \[\frac{3x+4}{{{x}^{2}}-3x+2}=\frac{A}{x-2}-\frac{B}{x-1},\] then(A, B)=
A)
(7, 10)
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B)
(10, 7)
done
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C)
\[\left( 10,-\,7 \right)\]
done
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D)
\[\left( -\,10,7 \right)\]
done
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E)
None of these
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View Answer play_arrow
If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+.......+{{C}_{n}}{{x}^{n}},\] then the value of \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+.......+(n+1)\,{{C}_{n}}\], will be
A)
\[(n+2)\,{{2}^{n-1}}\]
done
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B)
\[(n+1)\,{{2}^{n}}\]
done
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C)
\[(n+1)\,{{2}^{n-1}}\]
done
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D)
\[(n+2)\,{{2}^{n}}\]
done
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E)
None of these
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The mid-point of the line joining the points \[\left( -\,10,8 \right)\] and \[\left( -\,6,12 \right)\] divides the line joining the points \[\left( 4,-\,2 \right)\] and \[\left( -\,2,4 \right)\] in the ratio of:
A)
1 : 2 internally
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B)
1 : 2 externally
done
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C)
2 : 1 internally
done
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D)
2 : 1 externally
done
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E)
None of these
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View Answer play_arrow
If f(a) = 2, f?(a) = 1, \[g\left( a \right)=-\,1;\] g'(a) = 2, then \[\underset{x\to a}{\mathop{\lim }}\,\,\,\frac{g(x)\,f(a)-g(a)\,f(x)}{x-a}\] is equal to
A)
3
done
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B)
5
done
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C)
0
done
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D)
\[-\,3\]
done
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E)
None of these
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View Answer play_arrow
Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at
A)
Town B
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B)
45 km from town A
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C)
Town A
done
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D)
45 km from town B
done
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E)
None of these
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View Answer play_arrow
If \[{}^{n-1}{{C}_{r}}={{({{k}^{2}}-3)}^{n}}{{C}_{r+1}},\] then k \[\in \]
A)
\[(-\,\infty ,-\,2]\]
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B)
\[[2,\infty ]\]
done
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C)
\[[-\,\sqrt{3},\sqrt{3}]\]
done
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D)
\[(\sqrt{3},2]\]
done
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E)
None of these
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View Answer play_arrow
One angle of an isosceles \[\Delta \] is \[120{}^\circ \]and radius of its in circle\[=\sqrt{3}\]. Then the area of the triangle in sq. units is
A)
\[7+12\sqrt{3}\]
done
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B)
\[12-7\sqrt{3}\]
done
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C)
\[12+7\sqrt{3}\]
done
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D)
\[4\pi \]
done
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E)
None of these
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View Answer play_arrow
Direction: Read the given passage carefully and answer the questions that follow. A, B, C, D, E, F are members of a family. They are engineer, stenographer, doctor, draughtsman, lawyer and judge (not in order). A, the engineer is married to the lady stenographer. The judge is married to the lawyer. F, the draughtsman is the son of B and brother of E. C, the lawyer is the daughter-in-law of D. E is the unmarried doctor. D is the grandmother of F. There are two married couples in the family.
What is the profession of B?
A)
Judge
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B)
Lawyer
done
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C)
Draughtsman
done
clear
D)
Cannot be determined
done
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E)
None of these
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View Answer play_arrow
Direction: Read the given passage carefully and answer the questions that follow. A, B, C, D, E, F are members of a family. They are engineer, stenographer, doctor, draughtsman, lawyer and judge (not in order). A, the engineer is married to the lady stenographer. The judge is married to the lawyer. F, the draughtsman is the son of B and brother of E. C, the lawyer is the daughter-in-law of D. E is the unmarried doctor. D is the grandmother of F. There are two married couples in the family.
Which of the following is/are a couple/ couples?
A)
AD only
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B)
BC only
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C)
Both AD and BC
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D)
Both AC and BD
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E)
None of these
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View Answer play_arrow
Direction: Read the given passage carefully and answer the questions that follow. A, B, C, D, E, F are members of a family. They are engineer, stenographer, doctor, draughtsman, lawyer and judge (not in order). A, the engineer is married to the lady stenographer. The judge is married to the lawyer. F, the draughtsman is the son of B and brother of E. C, the lawyer is the daughter-in-law of D. E is the unmarried doctor. D is the grandmother of F. There are two married couples in the family.
What is the profession of D?
A)
Judge
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B)
Stenographer
done
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C)
Doctor
done
clear
D)
Cannot be determined
done
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E)
None of these
done
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View Answer play_arrow