3-D Geometry — Class 12 Mathematics PYQs

If $(1,-1,2)$ and $(2,1,-1)$ are the end points of a diameter of a sphere
$x^2+y^2+z^2+2ux+2vy+2wz-1=0$,
then what is $u+v+w$ equal to?

If $<l,m,n>$ are the direction cosines of a normal to the plane
$2x-3y+6z+4=0$,
then what is the value of
$49(7l^2+m^2-n^2)$?

If the direction cosines $<l,m,n>$ of a line are connected by relation
$l+2m+n=0,2l-2m+3n=0$, then what is the value of $l^2+m^2-n^2$?

Direction : Consider the following for the two (02) items that follow :

Let $S$ be the line of intersection of two planes $x+y+z=1$ and $2x+3y-4z=8$.

Which of the following are the direction ratios of $S$?

Direction : Consider the following for the two (02) items that follow :

Let $S$ be the line of intersection of two planes $x+y+z=1$ and $2x+3y-4z=8$.

If $⟨l,m,n⟩$ are direction cosines of $S$, then what is the value of $43(l^2-m^2-n^2)$?

Consider the following for the next two (02) items that follow :

Let $A(1,-1,2)$ and $B(2,1,-1)$ be the end points of the diameter of the sphere $x^2+y^2+z^2+2ux+2vy+2wz-1=0$.

What is $u+v+w$ equal to?

Consider the following for the next two (02) items that follow :

Let $A(1,-1,2)$ and $B(2,1,-1)$ be the end points of the diameter of the sphere $x^2+y^2+z^2+2ux+2vy+2wz-1=0$.

If $P(x,y,z)$ is any point on the sphere, then what is $P{A}^2+P{B}^2$ equal to?

Consider the following for the next two (02) items that follow :

Consider two lines whose direction ratios are $(2,-1,2)$ and $(k,3,5)$. They are inclined at an angle $\frac{\pi }{4}$.

What is the value of $k$?

Consider the following for the next two (02) items that follow :

Consider two lines whose direction ratios are $(2,-1,2)$ and $(k,3,5)$. They are inclined at an angle $\frac{\pi }{4}$.

What are the direction ratios of a line which is perpendicular to both the lines?

Consider the following for the next two (02) items that follow :

The position vectors of two points $A$ and $B$ are $\hat{i}-\hat{j}$ and $\hat{j}+\hat{k}$ respectively.

Consider the following points :

1. $(-1,-3,1)$
2. $(-1,3,2)$
3. $(-2,5,3)$

Which of the above points lie on the line joining $A$ and $B$?