The line \[\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\]
A)
lies in \[3x+2y+6z-12=0\]
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B)
is parallel to \[2x+y-2z=11\]
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C)
is perpendicular to \[4x+7y+6z=0\]
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D)
passes through \[(-2,-3,-4)\]
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The function \[f(\theta )=\frac{d}{d\theta }\int\limits_{0}^{\theta }{\frac{dx}{1-\cos \theta \cos x}satisfies}\]
A)
\[\frac{df}{d\theta }+2f(\theta )cot\theta =0\]
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B)
\[\frac{df}{d\theta }-2f(\theta )cot\theta =0\]
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C)
\[\frac{df}{d\theta }+2f(\theta )=0\]
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D)
\[\frac{df}{d\theta }-2f(\theta )=0\]
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The integral \[\int\limits_{-1/2}^{1/2}{\left( [x]+In\left( \frac{1+x}{1-x} \right) \right)}\,\,dx=\]
A)
\[-\frac{1}{2}\]
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B)
\[0\]
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C)
\[2\,\,in\left( \frac{1}{2} \right)\]
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D)
\[1\]
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All the points on the curve \[{{y}^{2}}=4a\left( x+a\sin \frac{x}{a} \right)\] at which the tangents are parallel to the axis of x, lie on a
A)
Circle
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B)
Parabola
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C)
Line
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D)
None of these
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Find the value of 'a' for which the volume of parallelepiped formed by the vectors \[\hat{i}+a\hat{j}+\hat{k},\hat{j}+a\hat{k}\,\,\,\,and\,\,\,a\hat{i}+\hat{k}\]is minimum.
A)
\[0\]
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B)
\[\frac{1}{\sqrt{3}}\]
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C)
\[\sqrt{3}\]
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D)
\[2\sqrt{3}\]
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If \[y={{\sin }^{3}}x,\,\,\,then\frac{{{d}^{4}}y}{d{{x}^{4}}}\,\,\,at\,\,\,x=\frac{\pi }{2}\] is equal to
A)
-15
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B)
12
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C)
-6
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D)
21
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If f, g are functions from R to R such that \[f(x)=[x]and\,\,g(x)=\frac{2+3x}{4}\] then
A)
f is one- one and g is onto
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B)
g is one-one but is not onto
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C)
f is neither one-one nor onto
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D)
None of these
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If \[\left( \omega \ne 1 \right)\] is a cube root of unity, then
A)
2
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B)
1
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C)
\[i\]
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D)
\[\omega \]
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Find the maximum value of \[f(x)=-xlo{{g}_{e}}x\,\,in(0,1)\]_________.
A)
\[1/e\]
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B)
\[e\]
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C)
\[1/{{e}^{2}}\]
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D)
\[{{e}^{2}}\]
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\[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,\frac{\int\limits_{1}^{x}{|t-1|dt}}{\sin (x-1)}=\]
A)
0
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B)
1
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C)
-1
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D)
None of these
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If \[{{\sin }^{-1}}\frac{x}{3}+{{\sin }^{-1}}\frac{y}{4}=\frac{\pi }{6}\] then the value of \[\frac{{{x}^{2}}}{9}+\frac{xy}{4\sqrt{3}}+\frac{{{y}^{2}}}{16}\] is
A)
\[\frac{3}{4}\]
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B)
\[\frac{1}{2}\]
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C)
\[\frac{1}{4}\]
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D)
None of these
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If
is non-invertible, then a =
A)
2
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B)
1
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C)
0
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D)
-1
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If each term of a G.P. is positive and each term is the sum of its two succeeding terms, then the common ratio of the G.P. is
A)
\[\left( \frac{\sqrt{5}-1}{2} \right)\]
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B)
\[\left( \frac{\sqrt{5}+1}{2} \right)\]
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C)
\[-\left( \frac{\sqrt{5}+1}{2} \right)\]
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D)
\[\left( \frac{1-\sqrt{5}}{2} \right)\]
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If the coefficient of \[{{x}^{3}}\] in the expansion of \[{{\left( 1+ax \right)}^{4}}\] is 32, then a equals
A)
2
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B)
3
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C)
4
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D)
6
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\[{{5}^{6}}-1\] is divisible by
A)
13
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B)
19
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C)
31
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D)
37
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In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is
A)
\[4+2\sqrt{3}\]
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B)
\[6+4\sqrt{3}\]
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C)
\[12+\frac{7\sqrt{3}}{4}\]
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D)
\[3+\frac{7\sqrt{3}}{4}\]
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The value of \[\underset{x\to 0}{\mathop{lim}}\,\frac{\sqrt{1/2(1-cos2x)}}{x}\] is equal to
A)
0
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B)
-1
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C)
1
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D)
None of these
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The locus of all points on the curve \[{{y}^{2}}=4a\left( x+a\,\,\sin \left( \frac{x}{a} \right) \right)\]at which the tangent is
A)
Straight line
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B)
Ellipse
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C)
Parabola
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D)
Circle
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Consider the following statements. Assertion : In any triangle, \[a\cos A+b\cos B+c\cos C\le s.\] Reason (R):In any triangle, \[\sin \,\,\left( \frac{A}{2} \right)\sin \left( \frac{B}{2} \right)\sin \left( \frac{C}{2} \right)\le \frac{1}{8}\] 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 the non-zero vectors \[\overrightarrow{a}\] and \[\overrightarrow{b}\] are perpendicular to each other, then the solution of the equation \[\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\] is
A)
\[\overrightarrow{r}=x\overrightarrow{a}+\frac{1}{\overrightarrow{a}.\overrightarrow{a}}\left( \overrightarrow{a}\times \overrightarrow{b} \right)\]
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B)
\[\overrightarrow{r}=x\overrightarrow{b}+\frac{1}{\overrightarrow{b}.\overrightarrow{b}}\left( \overrightarrow{a}\times \overrightarrow{b} \right)\]
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C)
\[\overrightarrow{r}=x\left( \overrightarrow{a}\times \overrightarrow{b} \right)\]
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D)
None of these
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