Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i + \widehat j - \widehat k.$$ A vector in the plane of $$\overrightarrow a $$ and $$\overrightarrow b $$ whose projection on $$\overrightarrow c $$ is $${1 \over {\sqrt 3 }},$$ is

Let $$\theta \in \left( {0,{\pi \over 4}} \right)$$ and $${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }},\,\,\,\,{t_2} = \,\,{\left( {\tan \theta } \right)^{\cot \theta }}$$, $${t_3}\, = \,\,{\left( {\cot \theta } \right)^{\tan \theta }}$$ and $${t_4}\, = \,\,{\left( {\cot \theta } \right)^{\cot \theta }},$$then

Let $${\overrightarrow A }$$ be vector parallel to line of intersection of planes $${P_1}$$ and $${P_2}.$$ Planes $${P_1}$$ is parallel to the vectors $$2\widehat j + 3\widehat k$$ and $$4\widehat j - 3\widehat k$$ and that $${P_2}$$ is parallel to $$\widehat j - \widehat k$$ and $$3\widehat i + 3\widehat j,$$ then the angle between vector $${\overrightarrow A }$$ and a given vector $$2\widehat i + \widehat j - 2\widehat k$$ is

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