3D Geometry · NDA previous year questions

What is the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0)?

1. A.$\frac{\pi }{6}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{2}$

If $l,m,n$ are the direction cosines of the line $x-1=2(y+3)=1-z$, then what is $l^4+m^4+n^4$ equal to?

1. A.1
2. B.$\frac{11}{27}$
3. C.$\frac{13}{27}$
4. D.4

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

1. A.(0, 1, 2)
2. B.(0, 0, 3)
3. C.(1, 1, 1)
4. D.(-1, 1, 3)

What is the length of the diameter of the sphere whose centre is at $(1,-2,3)$ and which touches the plane $6x-3y+2z-4=0$ ?

1. A.1 unit
2. B.2 units
3. C.3 units
4. D.4 units

What is the perpendicular distance from the point $(2,3,4)$ to the line $\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}$ ?

1. A.6 units
2. B.5 units
3. C.3 units
4. D.2 units

If a line has direction ratios $⟨a+b,b+c,c+a⟩$, then what is the sum of the squares of its direction cosines?

1. A.$(a+b+c{)}^2$
2. B.$2(a+b+c)$
3. C.3
4. D.1

Into how many compartments do the coordinate planes divide the space?

1. A.2
2. B.4
3. C.8
4. D.16

What is the equation of the plane which cuts an intercept 5 units on the $z$-axis and is parallel to $xy$-plane?

1. A.$x+y=5$
2. B.$z=5$
3. C.$z=0$
4. D.$x+y+z=5$

If the line $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}$ lies on the plane $2x-4y+z=7$, then what is the value of $k$?

1. A.2
2. B.3
3. C.5
4. D.7

There are 10 points in a plane. No three of these points are in a straight line. What is the total number of straight lines which can be formed by joining the points?

1. A.90
2. B.45
3. C.40
4. D.30

What is the radius of the sphere $x^2+y^2+z^2-6x+8y-10z+1=0$?

1. A.5
2. B.2
3. C.7
4. D.3

The equation of the plane passing through the intersection of the planes $2x+y+2z=9$, $4x-5y-4z=1$ and the point $(3,2,1)$ is

1. A.$10x-2y+2z=28$
2. B.$10x+2y+2z=28$
3. C.$10x+2y-2z=28$
4. D.$10x-2y-2z=24$

The distance between the parallel planes $4x-2y+4z+9=0$ and $8x-4y+8z+21=0$ is

1. A.$\frac{1}{4}$
2. B.$\frac{1}{2}$
3. C.$\frac{3}{2}$
4. D.$\frac{7}{4}$

What are the direction cosines of z-axis?

1. A.$⟨1,1,1⟩$
2. B.$⟨1,0,0⟩$
3. C.$⟨0,1,0⟩$
4. D.$⟨0,0,1⟩$

What is the distance of the point $(2,3,4)$ from the plane $3x-6y+2z+11=0$?

1. A.1 unit
2. B.2 units
3. C.3 units
4. D.4 units

Consider the following statements: 1. The angle between the planes $2x-y+z=1$ and $x+y+2z=3$ is $\frac{\pi }{3}$. 2. The distance between the planes $6x-3y+6z+2=0$ and $2x-y+2z+4=0$ is $\frac{10}{9}$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the number of triangles that can be formed by choosing the vertices from a set of 12 points in a plane, seven of which lie on the same straight line ?

1. A.185
2. B.175
3. C.115
4. D.105

What is $C(n,r)+2C(n,r-1)+C(n,r-2)$ equal to ?

1. A.$C(n+1,r)$
2. B.$C(n-1,r+1)$
3. C.$C(n,r+1)$
4. D.$C(n+2,r)$

What is the equation of the plane passing through the points $(-2,6,-6)$, $(-3,10,-9)$ and $(-5,0,-6)$?

1. A.$2x-y-2z=2$
2. B.$2x+y+3z=3$
3. C.$x+y+z=6$
4. D.$x-y-z=3$

A sphere of constant radius $r$ through the origin intersects the coordinate axes in A, B and C. What is the locus of the centroid of the triangle ABC?

1. A.$x^2+y^2+z^2=r^2$
2. B.$x^2+y^2+z^2=4r^2$
3. C.$9(x^2+y^2+z^2)=4r^2$
4. D.$3(x^2+y^2+z^2)=2r^2$

What is the equation to the sphere whose centre is at $(-2,3,4)$ and radius is 6 units?

1. A.$x^2+y^2+z^2+4x-6y-8z=7$
2. B.$x^2+y^2+z^2+6x-4y-8z=7$
3. C.$x^2+y^2+z^2+4x-6y-8z=4$
4. D.$x^2+y^2+z^2+4x+6y+8z=4$

The length of the normal from origin to the plane $x+2y-2z=9$ is equal to

1. A.2 units
2. B.3 units
3. C.4 units
4. D.5 units

The point of intersection of the line joining the points $(-3,4,-8)$ and $(5,-6,4)$ with the $XY$-plane is

1. A.$(\frac{7}{3},-\frac{8}{3},0)$
2. B.$(-\frac{7}{3},-\frac{8}{3},0)$
3. C.$(-\frac{7}{3},\frac{8}{3},0)$
4. D.$(\frac{7}{3},\frac{8}{3},0)$

If the angle between the lines whose direction ratios are $⟨2,-1,2⟩$ and $⟨x,3,5⟩$ is $\frac{\pi }{4}$, then the smaller value of $x$ is

1. A.52
2. B.4
3. C.2
4. D.1

A variable plane passes through a fixed point $(a,b,c)$ and cuts the axes in $A,B$ and $C$ respectively. The locus of the centre of the sphere $OABC$, $O$ being the origin, is

1. A.$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
2. B.$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$
3. C.$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2$
4. D.$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=2$