The vector equation of the line joining the points $(-2, -3, -4)$ and $(1, -2, 4)$ is :

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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Consider the equation of the line $\vec{r} = -\hat{i} + 2\hat{k} + \mu(4\hat{i} - \hat{j} + 2\hat{k})$.

Match List-I with List-II

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Match List-I with List-II

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The points of trisection of the segment joining the points (1, 0, 2) and (1, 3, 2) are:

A. $(1, \frac{3}{2}, \frac{4}{3})$

B. (1, 1, 2)

C. (1, 2, 2)

D. $(1, \frac{3}{2}, 2)$

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Which of the following statements is/are TRUE?

(A) The angle between the lines whose direction ratios are proportional to (4, -3, 5) and (3, 4, 5) is 60°.

(B) If A(1,2,3) and B(2, 0, 5) are two points on a line, then its direction ratios are proportional to 1, -2, 2.

(C) The direction cosines of z-axis are 0, 0, 1.

(D) The vector equation of a line which passes through the point with position vector $3\hat{i}-\hat{j}+\hat{k}$ and is in the direction of $2\hat{i}+3\hat{j}+\hat{k}$ is $\vec{r} = (2\hat{i}+3\hat{j}+\hat{k}) + \lambda(3\hat{i}-\hat{j}+\hat{k})$, where $\lambda$ is a parameter.

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A line makes angle $\theta$ with $x$-axes as well as $z$-axis. If the angle $\beta$, which it makes with $y$-axis is such that $\sin^2\beta = 3\sin^2\theta$, then $\cos^2\theta$ is :