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:

Let $L_1$ and $L_2$ lines such that $L_1:\vec{r}=\hat{i}+\hat{j}+\lambda(2\hat{i}-\hat{j}+\hat{k})$, $L_2:\vec{r}=2\hat{i}+\hat{j}-\hat{k}+\mu(4\hat{i}-2\hat{j}+2\hat{k})$ and $\theta$ be the acute angle between $L_1$ and $L_2$, then

Match the LIST-I with LIST-II

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The shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 8\hat{k})$ is equal to

For a given line L: $\dfrac{x}{2}=\dfrac{y+1}{3}=\dfrac{-z-3}{-5}$, which of the following is/are True ?

(A) The direction ratio of line L is (2, 3, -5)

(B) The perpendicular distance of the point (1, 1, 3) on the line L is $\sqrt{3}$

(C) The foot of perpendicular from the point (1, 1, 3) on line L is (2, 2, 1)

(D) The image of point (1, 1, 3) with respect to line L is (3, 3, 1)

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The equation of line passing through the points (2, -1, 3) and (4, 2, 1) is

(A) $\frac{x-2}{2} = \frac{y+1}{3} = \frac{z-3}{-2}$

(B) $\frac{x-4}{3} = \frac{y-2}{-2} = \frac{z-1}{2}$

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

(D) $\frac{x-4}{-2} = \frac{y-2}{-3} = \frac{z-1}{2}$

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Equation of a line passing through (1, 2, -3) which is parallel to the line $\frac{x+2}{3} = \frac{y-3}{-4} = \frac{z+5}{5}$ is

If the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{\lambda}$ and $\frac{x}{1} = \frac{y}{2\lambda} = \frac{z}{3}$ are perpendicular to each other, then $\lambda$ is equal to