$$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ Column-$$I$$
(A)$$\,\,\,\,\,$$ The number of solutions of the equations
$$\,\,\,\,\,$$$$x\,{e^{\sin x}} - \cos x = 0$$ in the interval $$\left( {0,{\pi \over 2}} \right)$$
(B)$$\,\,\,\,\,$$ Value(s) of $$k$$ for which the planes $$kx+4y+z=0,$$ $$4x+ky+2z=0$$
$$\,\,\,\,\,$$ and $$2x+2y+z=0$$ intersect in a straight line
(C)$$\,\,\,\,\,$$ Value(s) of $$k$$ for which $$\,\left| {x - 1} \right| + \left| {x - 2} \right| + \left| {x + 1} \right| + \left| {x + 2} \right| = 4k$$
$$\,\,\,\,\,$$has integer solutions(s)
(D)$$\,\,\,\,\,$$ If $$y'=y+1$$ and $$y(0)=1, $$ then value(s) of $$y$$($$ln$$ $$2$$)
$$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $$\,\,\,\,\,$$Column-$$II$$
(p)$$\,\,\,\,\,$$ $$1$$
(q)$$\,\,\,\,\,$$ $$2$$
(r)$$\,\,\,\,\,$$ $$3$$
(s)$$\,\,\,\,\,$$ $$4$$
(t)$$\,\,\,\,\,$$ $$5$$
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$
(A)$$\,\,\,\,$$ Root(s) of the equations $$2{\sin ^2}\theta + {\sin ^2}2\theta = 2$$
(B)$$\,\,\,\,$$ Points of discontinuity of the unction $$f\left( x \right) = \left[ {{{6x} \over \pi }} \right]\cos \left[ {{{3x} \over \pi }} \right],$$ $$f$$ where $$\left[ y \right]$$ denotes the largest integer less than or equal to $$y$$
(C)$$\,\,\,\,$$ Volume of the parallelopiped with its edges represented by the vectors $$\,\widehat i + \widehat j,\widehat i + 2\widehat j$$ and $$\widehat i + \widehat j + \pi \widehat k$$
(D)$$\,\,\,\,$$ Angle between vector $${\overrightarrow a }$$ and $${\overrightarrow b }$$ where $${\overrightarrow a },$$ $${\overrightarrow b }$$ and $${\overrightarrow c }$$ are unit vectors satisfying $$\overrightarrow a + \overrightarrow b + \sqrt 3 \,\,\overrightarrow c = \overrightarrow 0 $$
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$II$$
(p)$$\,\,\,\,$$ $${\pi \over 6}$$
(q)$$\,\,\,\,$$ $${\pi \over 4}$$
(r)$$\,\,\,\,$$ $${\pi \over 3}$$
(s)$$\,\,\,\,$$ $${\pi \over 2}$$
(t)$$\,\,\,\,$$ $$\pi $$
Match the conditions/expressions in Column $$I$$ with statements in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS.$$
$$\,\,\,$$ Column $$I$$
(A)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$
(B)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(C)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(D)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$
$$\,\,\,$$ Column $$II$$
(p)$$\,\,\,$$ the equations represents planes meeting only at asingle point
(q)$$\,\,\,$$ the equations represents the line $$x=y=z.$$
(r)$$\,\,\,$$ the equations represent identical planes.
(s) $$\,\,\,$$ the equations represents the whole of the three dimensional space.
(p)$$\,\,\,$$ $$2$$
(q)$$\,\,\,$$ $${4 \over 3}$$
(r)$$\,\,\,$$ $$\left| {\int\limits_0^1 {\sqrt {1 - xdx} } } \right| + \left| {\int\limits_{ - 1}^0 {\sqrt {1 + xdx} } } \right|$$
(s)$$\,\,\,$$ $$1$$