List of top Mathematics Questions on 3D Geometry asked in UP Board XII

Find the vector equation of a straight line passing through the point (5, 2, -4) and parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).
Find the shortest distance between the lines whose vector equations are:
\(\vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}\) and \(\vec{r} = (s+1)\hat{i} + (2s-1)\hat{j} - (2s+1)\hat{k}\).
Find the shortest distance between the lines \(\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})\) and \(\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k})\).
Find the shortest distance between the lines \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \) and \( \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \).
If the ordered pairs (2x - 3, 5) and (x, y - 1) are equal, then find the numbers x and y.
Find the direction cosine of Z-axis.
Find the shortest distance between the lines whose vector equations are \(\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}-3\hat{j}+2\hat{k})\) and \(\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu(2\hat{i}+3\hat{j}+\hat{k})\).
Find the direction-cosines of the y-axis.
The Cartesian equation of a line is \(\frac{x+3}{2} = \frac{y-5}{4} = \frac{z+6}{2}\). Find its vector equation.
Find the shortest distance between two lines \( l_1 \) and \( l_2 \) whose vector equations are: \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] and \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}). \]

Given:

\[ x \sqrt{1 + y} + y \sqrt{1 + x} + x = 0 \]

for \( -1 < x < 1 \), prove that:

\[ \frac{dy}{dx} = -\frac{1}{(1+x)^2}. \]

Find the area of the region enclosed by the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

Find the shortest distance between the lines \[ \mathbf{r}_1 = (i + 2j + k) + \lambda(i - j + k) \quad \text{and} \quad \mathbf{r}_2 = (2i - j - k) + \mu(2i + j + 2k). \] 

Find the value of \( p \) such that the lines \[ \frac{1 - x}{3} = \frac{7y - 14}{2p} = \frac{z - 3}{2} \] and \[ \frac{7 - 7x}{3p} = \frac{y - 5}{1} = \frac{6 - z}{5} \] are mutually perpendicular.
Show that the points \( (2, -1, 1) \), \( (1, -3, -5) \) and \( (3, -4, -4) \) are the vertices of a right-angled triangle.
Find the area of the circle \( x^2 + y^2 = a^2 \) surrounded by it.
(b)(ii) If the coordinates of midpoints of the sides of a triangle are: \[ (1, 5, -1), \quad (0, 4, -2), \quad (2, 3, 4), \] find the coordinates of its vertices.
(a) Find the shortest distance between the lines: \[ \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}, \quad \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}. \]
(b)(i) Find the angle between the pair of lines: \[ \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4}, \quad \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}. \]
Find the area of the circle \( x^2 + y^2 = a^2 \).
Find the shortest distance between the lines \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}) \]
Find the shortest distance between the lines \[ \vec{r} = (i + 2j + k) + \lambda(i - j + k) \] and \[ \vec{r} = (2i - j - k) + \mu(2i + j + 2k) \]
If the coordinates of mid-points of the sides of a triangle are \( (1, 5, -1), (0, 4, -2) \) and \( (2, 3, 4) \), then find the coordinates of its vertices.
Find the equation of the plane passing through the points \((-2i + 6j - 6k), (-3i + 10j - 9k), (-5i - 6j - 6k)\).
Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.