The reflection of the point $(\alpha, \beta, \gamma)$ in the xz plane is

Two points are moving along the lines :

$L_1: \vec{r} = (2\hat{i}+3\hat{j}+\hat{k}) + \lambda(\hat{i}+2\hat{j}+\hat{k})$

$L_2: \vec{r} = (\hat{i}+\hat{j}) + \mu(2\hat{i}+\hat{j}-\hat{k})$

where $\lambda$ and $\mu$ are real parameters , then which of the following statements are correct ?

A. The direction ratios of the line perpendicular to both lines are $<-1,1,-1>$.

B. The direction cosines of the second line $L_2$ are $<\dfrac{2}{\sqrt6},\dfrac{-1}{\sqrt6},\dfrac{-1}{\sqrt6}>$.

C. Angle between two lines is $\pi/3$.

D. The lines are skew lines.

Choose the correct answer from the options given below:

A line in space is uniquely determined if

(A) It passes through a given point and has a given direction.

(B) It passes through two given points.

(C) It passes through a given point.

(D) It has a given direction.

Choose the correct answer from the options given below:

The acute angle between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ and $\frac{x-1}{1} = \frac{y+1}{-3} = \frac{z-2}{2}$ is

Match List I with List II

Choose the correct answer from the options given below:

If $\alpha, \beta$ and $\gamma$ are angle of inclinations of a line with x, y and z axes respectively, then the value of $2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$ is

The angle between the lines $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{j} - 2\hat{k} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :