The fundamental period of the function \[f(x)=2cos\frac{1}{3}(x-\pi )\] is
A)
6\[\pi \]
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B)
4\[\pi \]
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C)
2\[\pi \]
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D)
\[\pi \]
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If\[0<\theta <\frac{\pi }{2}\], then minimum value of\[\frac{{{\cos }^{3}}\theta }{\sin \theta }+\frac{{{\sin }^{3}}\theta }{\cos \theta }\] is
A)
\[\sqrt{3}\]
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B)
\[1/3\]
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C)
1
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D)
None of these
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lf A=\[ta{{n}^{-1}}\left( \frac{x\sqrt{3}}{2k-x} \right)\] and B=\[ta{{n}^{-1}}\left( \frac{2x-k}{k\sqrt{3}} \right)\] tan- then the value of A - B is
A)
\[0{}^\circ \]
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B)
\[45{}^\circ \]
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C)
\[60{}^\circ \]
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D)
\[30{}^\circ \]
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The number of solutions of the system of equations \[3x-2y+z=5,6x-4y+2z=10\] and \[9x-6y+3z=15\] is
A)
0
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B)
1
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C)
2
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D)
Infinite
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The function
,is continuous at \[x=0\] if
A)
\[a=0~~\]
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B)
\[a=3/5~~\]
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C)
\[a=2\]
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D)
\[a=5/3\]
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If\[f(x)=co{{t}^{-1}}\left( \frac{{{x}^{x}}-{{x}^{-x}}}{2} \right)\],then \[f'(1)=\]
A)
\[-\log 2\]
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B)
\[\log 2\]
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C)
1
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D)
-1
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On the curve \[f(x)={{x}^{3}}\], the point at which tangent line is parallel to the chord through the points \[A(-1,-1)\]and \[B(2,8)\] is
A)
\[\left( -1,1 \right)\]
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B)
\[\left( 1,-1 \right)\]
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C)
\[\left( -1,-1 \right)\]
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D)
\[\left( 1,1 \right)\]
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Seven persons draw lottery for occupancy of 6 seats inside a first class railway compartment. The probability that 2 specified persons will obtain opposite seats, is
A)
\[1/7\]
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B)
\[2/7\]
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C)
\[5/7\]
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D)
None of these
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For a real number y, [y] denotes the greatest integer less than or equal to y, then , tan \[f(x)=\frac{\tan \left( \pi \left[ x-\pi \right] \right)}{1+{{\left[ x \right]}^{2}}}\]is
A)
Discontinuous at some x
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B)
Continuous at all x, but \[f'(x)\] does not exist for some x
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C)
\[f'(x)\]exists for all x but \[f''(x)\]does not exist
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D)
\[f'(x)\]exists for all x
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\[\int {{e}^{3\log x}}{{({{x}^{4}}+1)}^{-1}}dx=\]
A)
\[\log \left( {{x}^{4}}+1 \right)+C\]
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B)
\[\frac{1}{4}\log \left( {{x}^{4}}+1 \right)+C\]
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C)
\[-\log \left( {{x}^{4}}+1 \right)+C\]
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D)
None of these
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In a parallelopiped, the ratio of the sum of the squares on the four diagonals, to the sum of the squares on the three coterminus edges is
A)
1
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B)
2
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C)
3
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D)
4
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If \[f'(x)={{(x-a)}^{2n}}{{(x-b)}^{2m+1}}\] where \[m,n\in N\] then for \[f(x)\]
A)
x = a is a point of minimum
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B)
x = a is a point of maximum
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C)
x = a is neither a point of maximum or minimum
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D)
None of these
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Area bounded by the curve \[y=(x-1)(x-2)(x-3)\] and x-axis lying between the ordinates x = 0 and x = 3 is equal to (in sq. units)
A)
\[9/4\]
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B)
\[11/4\]
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C)
\[11/12\]
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D)
None of these
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The plane \[ax+by+cz=1\] meets the coordinate axes in A, B and C. The centroid of \[\Delta ABC\]is
A)
\[\left( 3a,3b,3c \right)\]
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B)
\[\left( \frac{a}{3},\frac{b}{3},\frac{c}{3} \right)\]
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C)
\[\left( \frac{3}{a},\frac{3}{b},\frac{3}{c} \right)\]
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D)
\[\left( \frac{1}{3a},\frac{1}{3b},\frac{1}{3c} \right)\]
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Value of \[\int\limits_{-2}^{2}{\frac{{{\sin }^{2}}x}{-2\left[ \frac{x}{\pi } \right]+\frac{1}{2}}dx}\], (where [x] is the greatest integer function) is
A)
1
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B)
0
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C)
4 - sin 4
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D)
None of these
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The solution of the differential equation \[2x\frac{dy}{dx}-y=3\] represent the family of
A)
Straight lines
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B)
Circles
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C)
Parabolas
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D)
Ellipses
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Find the coefficient of \[{{x}^{3}}\] in the expansion of \[\left( 1+x+2{{x}^{2}} \right){{\left( 2{{x}^{2}}-\frac{1}{3x} \right)}^{9}}\]
A)
\[-\frac{143}{13}\]
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B)
\[-\frac{224}{29}\]
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C)
\[-\frac{161}{27}\]
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D)
\[-\frac{224}{27}\]
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Solve : \[\sin x+\sin 2x>0,x\in \left[ -\pi ,\pi \right]\]
A)
\[\left( -\pi ,\frac{-2\pi }{3} \right)\cup \left( 0,\frac{2\pi }{3} \right)\]
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B)
\[\left( -\pi ,\frac{-\pi }{3} \right)\cup \left( 0,\frac{\pi }{3} \right)\]
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C)
\[\left( \frac{-\pi }{2},\frac{\pi }{2} \right)\]
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D)
\[\left( -\pi \frac{-2\pi }{3} \right)\cup \left[ 0,\frac{2\pi }{3} \right]\]
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A traveller starts from a certain place on a certain day and travels 1 km on the first day and on subsequent days, he travels 2 km more than the previous day. After 3 days, a second traveller sets out from the same place and on his first day he travel) 12 km and on subsequent days he travels 1 km more than the previous day. On how many days will the second traveller be ahead of the first?
A)
7
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B)
6
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C)
8
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D)
9
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Evaluate
A)
\[5\sqrt{3}(\sqrt{6}-5)\]
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B)
\[5\sqrt{3}\]
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C)
0
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D)
\[3(\sqrt{5}-\sqrt{6})\]
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