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

If the incident ray on a surface is along the unit vector $$\widehat v\,\,,$$ the reflected ray is along the unit vector $$\widehat w\,\,$$ and the normal is along unit vector $$\widehat a\,\,$$ outwards. Express $$\widehat w\,\,$$ in terms of $$\widehat a\,\,$$ and $$\widehat v\,\,.$$

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are distinct vectors such that
$$\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d \,.$$ Prove that
$$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) \ne 0\,\,i.e.\,\,\,\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c \ne \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $$

If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$$ are three non-coplanar unit vectors and $$\alpha ,\beta ,\gamma $$ are the angles between $$\overrightarrow u $$ and $$\overrightarrow v $$ and $$\overrightarrow w ,$$ $$\overrightarrow w $$ and $$\overrightarrow u $$ respectively and $$\overrightarrow x ,\overrightarrow y ,\overrightarrow z ,$$ are unit vectors along the bisectors of the angles $$\alpha ,\,\,\beta ,\,\,\gamma $$ respectively. Prove that $$\,\left[ {\overrightarrow x \times \overrightarrow y \,\,\overrightarrow y \times \overrightarrow z \,\,\overrightarrow z \times \overrightarrow x } \right] = {1 \over {16}}{\left[ {\overrightarrow u \,\,\overrightarrow v \,\,\overrightarrow w } \right]^2}\,{\sec ^2}{\alpha \over 2}{\sec ^2}{\beta \over 2}{\sec ^2}{\gamma \over 2}.$$