3-D Geometry — NDA Mathematics PYQs

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?

What is the equation of the plane passing through the point $(1,1,1)$ and perpendicular to the line whose direction ratios are $⟨3,2,1⟩$?

Consider the following for the two (02) items that follow :
A plane $P$ is parallel to the line having direction ratios $⟨1,3,2⟩$ and contains the line of intersection of the planes $6x+4y-5z=2$ and $x-2y+3z=0$.

Which of the following are the direction ratios of the line of intersection of the given planes?

Consider the following for the two (02) items that follow :
A plane $P$ is parallel to the line having direction ratios $⟨1,3,2⟩$ and contains the line of intersection of the planes $6x+4y-5z=2$ and $x-2y+3z=0$.

What is the equation of the plane $P$?

Consider the following for the two (02) items that follow :
Suppose $S$ is the sphere with the smallest radius that passes through the points $A(1,0,0)$, $B(0,1,0)$ and $C(0,0,1)$.

What is the radius of $S$?

Consider the following for the two (02) items that follow :
Suppose $S$ is the sphere with the smallest radius that passes through the points $A(1,0,0)$, $B(0,1,0)$ and $C(0,0,1)$.

On which one of the following planes does the centre of $S$ lie?

Consider the following for the two (02) items that follow :
Let $A(1,-1,0)$, $B(-2,1,8)$ and $C(-1,2,7)$ are three consecutive vertices of a parallelogram $ABCD$.

What is the fourth vertex $D$?