Consider a line $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$, then which of the following statements are correct? (A) it passes through point (9, -4, 12) (B) it passes through point (1, 4, -1) (C) its direction cosine's are $\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}$ (D) its Cartesian equation is $\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$ Choose the **correct** answer from the options given below:

If the lines $\dfrac{x+1}{3}=\dfrac{y+3}{5}=\dfrac{z+\lambda}{7}$ and $\dfrac{x-2}{1}=\dfrac{y-4}{3}=\dfrac{z-6}{5}$ are intersecting , then the value of '$\lambda$' is

Acute angle between the lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ and $\frac{x-1}{4} = \frac{y+1}{-3} = \frac{z+10}{5}$ is:

Direction ratios of the line perpendicular to the lines $\frac{x-3}{2}=\frac{y+7}{-3}=\frac{z-2}{1}$ and $\frac{x+2}{1}=\frac{y+3}{2}=\frac{z-5}{-2}$ are

If $\theta$ is the angle between the lines $L_1$ and $L_2$, whose direction cosines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$ respectively, then

Match List-I with List-II

Choose the correct answer from the options given below:

The equation of plane that contains line $\frac{x - 1}{-1} = \frac{y + 1}{2} = \frac{z - 1}{3}$ and also pass through point $(0, 1, 0)$ is :

Find the root of perpendicular from origin to the line $\frac{x - 1}{3} = \frac{y - 2}{-2} = \frac{z + 1}{3}$.