If a line makes angles $90^{\circ}, 60^{\circ}$ and $\theta$ with $\mathrm{x}, \mathrm{y}$ and z axes respectively, where $\theta$ is acute, then the value of $\theta$ is

The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is

If lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-5}{1}=\frac{z-6}{-5}$ are mutually perpendicular, then $k$ is equal to

The distance between the two planes $2 x+3 y+4 z=4$ and $4 x+6 y+8 z=12$ is

The sine of the angle between the straight line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{4-z}{-5}$ are the plane $2 x-2 y+z=5$ is

The equation $x y=0$ in three-dimensional space represents

The plane containing the point $(3,2,0)$ and the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$ is

If a line makes an angle of $$\frac{\pi}{3}$$ with each $$X$$ and $$Y$$ axis, then the acute angle made by $$\mathrm{Z}$$-axis is

The length of perpendicular drawn from the point $$(3,-1,11)$$ to the line $$\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$$ is

The equation of the plane through the points $$(2,1,0),(3,2,-2)$$ and $$(3,1,7)$$ is

The point of intersection of the line $$x+1=\frac{y+3}{3}=\frac{-z+2}{2}$$ with the plane $$3 x+4 y+5 z=10$$ is

If $$(2,3,-1)$$ is the foot of the perpendicular from $$(4,2,1)$$ to a plane, then the equation of the plane is

The octant in which the point (2, $$-$$4, $$-7$$) lies is

The coordinates of foot of the perpendicular drawn from the origin to the plane $$2 x-3 y+4 z=29$$ are

The angle between the pair of lines $$\frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4}$$ and $$\frac{x+1}{1}=\frac{y-4}{4}=\frac{z-5}{2}$$ is

The distance of the point whose position vector is $$(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})$$ from the plane $$\mathbf{r} \cdot(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=4$$ is

The equation of the line joining the points $$(-3,4,11)$$ and $$(1,-2,7)$$ is

The angle between the lines whose direction cosines are $$\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)$$ and $$\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)$$

If a plane meets the coordinate axes at $$A, B$$ and $$C$$ in such a way that the centroid of $$\triangle A B C$$ is at the point $$(1,2,3)$$, then the equation of the plane is

The area of the quadrilateral $$A B C D$$ when $$A(0,4,1), B(2,3,-1), C(4,5,0)$$ and $$D(2,6,2)$$ is equal to

The mid points of the sides of triangle are $$(1,5,-1)(0,4,-2)$$ and $$(2,3,4)$$ then centroid of the triangle

The point $$(1,-3,4)$$ lies in the octant

The distance of the point $$(1,2,-4)$$ from the line $$\frac{x-3}{2}=\frac{y-3}{3}=\frac{z+5}{6}$$ is

The sine of the angle between the straight line $$\frac{x-2}{3}=\frac{3-y}{-4}=\frac{z-4}{5}$$ and the plane $$2 x-2 y+z=5$$ is

If a line makes an angle of with each of $$X$$ and $$Y$$-axis, then the acute angle made by Z-axis is

Foot of the perpendicular drawn from the point $$(1,3,4)$$ to the plane $$2 x-y+z+3=0$$ is

Acute angel between the line $$\frac{(x-5)}{2}=\frac{y+1}{-1}=\frac{z+4}{1}$$ and the plane $$3 x-4 y-z+5=0$$ is

The distance of the point $$(1,2,1)$$ from the line $$\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{2}$$ is.

$$X Y$$ plane divides the line joining the points $$A(2,3,-5)$$ and $$B(-1,-2,-3)$$ in the ratio