Match the statements/expressions given in Column-$$I$$ with the values given in Column-$$II.$$

$$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ 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)$$\,\,\,\,\,$$ $$5$$

Match the statements / expressions given in Column-$$I$$ with the values given in Column-$$II.$$

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ 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)$$\,\,\,\,$$ $$\pi $$

Consider the following linear equations $$ax+by+cz=0;$$ $$\,\,\,$$ $$bx+cy+az=0;$$ $$\,\,\,$$ $$cx+ay+bz=0$$

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.

Match the folowing :
(A)$$\,\,\,$$Two rays $$x + y = \left| a \right|$$ and $$ax - y=1$$ intersects each other in the
$$\,\,\,\,\,\,\,\,\,\,$$first quadrant in interval $$a \in \left( {{a_0},\,\,\infty } \right),$$ the value of $${{a_0}}$$ is
(B)$$\,\,\,$$ Point $$\left( {\alpha ,\beta ,\gamma } \right)$$ lies on the plane $$x+y+z=2.$$
$$\,\,\,\,\,\,\,\,\,\,\,$$Let $$\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k,\widehat k \times \left( {\widehat k \times \overrightarrow a } \right) = 0,$$ then $$\gamma = $$
(C)$$\,\,\,$$$$\left| {\int\limits_0^1 {\left( {1 - {y^2}} \right)dy} } \right| + \left| {\int\limits_1^0 {\left( {{y^2} - 1} \right)dy} } \right|$$
(D)$$\,\,\,$$If $$\sin A\,\,\sin B\,\,\sin C + \cos A\,\,\cos B = 1,$$ then the value of $$\sin C = $$

(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)$$\,\,\,$$ $$1$$