3-D Geometry — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
$P(-2,-3,5)$, $Q(4,-1,5)$, $R(6,-4,8)$ and $S(2,-6,10)$ are four points.

Consider the following statements :
I. PQ is parallel to RS.
II. PR is perpendicular to QS.
Which of the statements given above is/are correct ?

Directions : For the next two (02) items that follow :
$P(-2,-3,5)$, $Q(4,-1,5)$, $R(6,-4,8)$ and $S(2,-6,10)$ are four points.

What is $(P{Q}^2+2Q{S}^2-2P{R}^2)$ equal to ?

Directions : For the next two (02) items that follow :
A line L passing through the point $(-1,2,-3)$ is perpendicular to the plane P given by $2x+3y+z+5=0$.

What is the equation of the line L ?

Directions : For the next two (02) items that follow :
A line L passing through the point $(-1,2,-3)$ is perpendicular to the plane P given by $2x+3y+z+5=0$.

What are the direction ratios of a line M parallel to the plane P ?

Directions : For the next two (02) items that follow :
The equation of the sphere S is $x^2+y^2+z^2-4x-6y-12z+k=0$.

What is the radius of the sphere passing through origin and concentric with the sphere S ?

Directions : For the next two (02) items that follow :
The equation of the sphere S is $x^2+y^2+z^2-4x-6y-12z+k=0$.

If the radius of the sphere S is 8 units, what is the value of k ?

If a line in $3$ dimensions makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of the coordinate axes, then what is $\cos (\alpha +\beta )\cos (\alpha -\beta )$ equal to?

$A(1,2,-1)$, $B(2,5,-2)$ and $C(4,4,-3)$ are three vertices of a rectangle. What is the area of the rectangle?

$ABC$ is a triangle right-angled at $B$. If $A(k,1,-1)$, $B(2k,0,2)$ and $C(2+2k,k,1)$ are the vertices of the triangle, then what is the value of $k$?

If a line $\frac{x+1}{p}=\frac{y-1}{q}=\frac{z-2}{r}$, where $p=2q=3r$, makes an angle $\theta$ with the positive direction of $y$-axis, then what is $\cos 2\theta$ equal to?