Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation: $$$\matrix{ {3x - y - z = 0} \cr { - 3x + z = 0} \cr { - 3x + 2y + z = 0} \cr } $$$

Then the number of such points for which $$x^2 + {y^2} + {z^2} \le 100$$ is

For $$0 < \theta < {\pi \over 2},$$ the solution (s) of $$$\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{\left( {m - 1} \right)\pi } \over 4}} \right)\,\cos ec\,\left( {\theta + {{m\pi } \over 4}} \right) = 4\sqrt 2 } $$$ is (are)

Match the statements/expressions in Column I with the values given in Column II:

	Column I		Column II
(A)	The number of solutions of the equation $$x{e^{\sin x}} - \cos x = 0$$ in the interval $$\left( {0,{\pi \over 2}} \right)$$	(P)	1
(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	(Q)	2
(C)	Value(s) of $$k$$ for which $$|x - 1| + |x - 2| + |x + 1| + |x + 2| = 4k$$ has integer solution(s)	(R)	3
(D)	If $$y' = y + 1$$ and $$y(0) = 1$$ then value(s) of $$y(\ln 2)$$	(S)	4
		(T)	5

Match the statements/expressions in Column I with the values given in Column II:

	Column I		Column II
(A)	Root(s) of the expression $$2{\sin ^2}\theta + {\sin ^2}2\theta = 2$$	(P)	$${\pi \over 6}$$
(B)	Points of discontinuity of the function $$f(x) = \left[ {{{6x} \over \pi }} \right]\cos \left[ {{{3x} \over \pi }} \right]$$, where $$[y]$$ denotes the largest integer less than or equal to y	(Q)	$${\pi \over 4}$$
(C)	Volume of the parallelopiped with its edges represented by the vectors $$\widehat i + \widehat j + \widehat i + 2\widehat j$$ and $$\widehat i + \widehat j + \pi \widehat k$$	(R)	$${\pi \over 3}$$
(D)	Angle between vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ where $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors satisfying $$\overrightarrow a + \overrightarrow b + \sqrt 3 \overrightarrow c = \overrightarrow 0 $$	(S)	$${\pi \over 2}$$
		(T)	$$\pi $$