List of top Mathematics Questions on Three Dimensional Geometry asked in CUET (UG)

Consider the line \[ \vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k}) \]

Match List-I with List-II:

List-IList-II
(A) A point on the given line(I) \(\left(-\tfrac{1}{\sqrt{21}}, \tfrac{2}{\sqrt{21}}, -\tfrac{4}{\sqrt{21}}\right)\)
(B) Direction ratios of the line(II) (4, -2, -2)
(C) Direction cosines of the line(III) (1, -2, 4)
(D) Direction ratios of a line perpendicular to given line(IV) (-1, 2, -4)
The shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$ is equal to
If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to
The shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$ is equal to
If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

Consider the line \[ \vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k}) \]

Match List-I with List-II:

List-IList-II
(A) A point on the given line(I) \(\left(-\tfrac{1}{\sqrt{21}}, \tfrac{2}{\sqrt{21}}, -\tfrac{4}{\sqrt{21}}\right)\)
(B) Direction ratios of the line(II) (4, -2, -2)
(C) Direction cosines of the line(III) (1, -2, 4)
(D) Direction ratios of a line perpendicular to given line(IV) (-1, 2, -4)