List of top Mathematics Questions on Three Dimensional Geometry asked in MHT CET

If the position vectors of the points A and B are 3$\hat {i}$ + $\hat {j}$ + 2$\hat {k}$ and $\hat {i}$ -2$\hat {j}$ -4$\hat {k}$ respectively, then the equation of the plane through B and perpendicular to AB is

The angle between two lines x +1 =y + 3 =z - 4 and $\frac {x-4}{1}$ = $\frac {y+2}{2}$ = $\frac {z+1}{2}$ is

In a triangle ABC, with usual notations ∠A = 60°, then (1 + $\frac {a}{c}$ + $\frac {b}{c}$)(1 + $\frac {c}{b}$ - $\frac {a}{b}$) = ?

The area of triangle with vertices $(1,2,0)$, $(1,0,a)$ and $(0,3,1)$ is $\sqrt{6}$ sq. units, then the values of '$a$' are

The co-ordinates of the foot of the perpendicular drawn from the point $2\hat{\text{i}} - \hat{\text{j}} + 5\hat{\text{k}}$ to the line $\vec{\text{r}} = (11\hat{\text{i}} - 2\hat{\text{j}} - 8\hat{\text{k}}) + \lambda(10\hat{\text{i}} - 4\hat{\text{j}} - 11\hat{\text{k}})$ are

If the lines $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}$ and $\frac{x-2}{1} = \frac{y+m}{2} = \frac{z-2}{1}$ intersect each other, then the value of $m$ is

If $A(3,2,-1)$ and $B(1,4,3)$, then the equation of the plane which bisects the segment $AB$ perpendicularly is

The equation of the line passing through the point $(1,2,3)$ and perpendicular to the lines $\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3}$ and $\vec r = \lambda(-3\hat i + 2\hat j + 5\hat k)$ is

If the lines given by $ \vec{r} = 2\hat{i} + \lambda(\hat{i} + 2\hat{j} + m\hat{k}) $ and $ \vec{r} = \hat{i} + \mu(2\hat{i} + \hat{j} + 6\hat{k}) $ are perpendicular, then the value of $ m $ is

The length of the perpendicular to the plane \[ \vec r \cdot (i - 2j + 3k) = 14 \] from the origin is

The direction cosines of a line which lies in the $ ZOX $ plane and makes an angle of $ 30^\circ $ with the $ Z $-axis are

The p.d.f. of a continuous random variable $ X $ is given by \[ f(x) = \begin{cases} \frac{1}{2} & \text{if } 0<x<2, \\ 0 & \text{otherwise}, \end{cases} \] and if \[ a = P(X \leq \frac{1}{2}), \quad b = P(X>\frac{1}{2}), \] then the relation between $ a $ and $ b $ is

The equation of the line passing through the point $ (2, 3, -4) $ and perpendicular to the XOZ plane is

The foot of the perpendicular drawn from the origin to the plane $ x + y + 3z - 4 = 0 $ is

If $ A(-1,2,3) $, $ B(3,-2,1) $, $ C(2,1,3) $ and $ D(-1,-2,4) $ are the vertices of a tetrahedron, then its volume is

The distance of the point $ (3,4,5) $ from the point of intersection of the line \[ \frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2} \] and the plane $ x+y+z=2 $ is

If $A$ and $B$ are two angles such that $A,B\in(0,\pi)$ and they are not supplementary angles such that $\sin A-\sin B=0$, then

The equation of a plane passing through the intersection of the planes $x + 2y - 3z + 2 = 0$ and $6x + y + z + 1 = 0$ and parallel to the line $x - 1 = y + 2 = 7 - z$ is

The angle between the lines $ \frac{x - 1}{4} = \frac{y - 3}{1} = \frac{z}{8} $ and $ \frac{x - 2}{2} = \frac{y + 1}{2} = \frac{z - 4}{1} $ is

The integrating factor of the differential equation \[ (1 + x^2) \, dt = \left( \tan^{-1} x - t \right) \, dx \text{ is} \]

If the origin is the centroid of the triangle whose vertices are $ A(2, p, -3) $, $ B(q, -2, 5) $ and $ C(-5, 1, r) $, then

If the direction cosines of a line are $ \frac{1}{c}, \frac{1}{c}, \frac{1}{c} $, then

If the line $ \vec{r} = (i - 2j + 3k) + \lambda(2i + j + 2k) $ is parallel to the plane $ \vec{r} = (3i - 2j - mk) = 5 $, then the value of $ m $ is

If the Cartesian equation of the line is \[ x - 1 = 2y + 3 = 3 - z, \] then its vector equation is

The shortest distance between the lines $ 1 + x = 2y = -12z $ and $ x = y + 2 = 6z - 6 $ is